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# Interval (music)

Melodic and harmonic intervals.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In Western music, intervals are most commonly differences between notes of a diatonic scale. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C and D. Intervals can be arbitrarily small, and even imperceptible to the human ear.

In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.

In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). Examples include the minor third or perfect fifth. These names identify not only the difference in semitones between the upper and lower notes but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G and G–A.

## Size

Example: Perfect octave on C in equal temperament and just intonation: 2/1 = 1200 cents.

The size of an interval (also known as its width or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.

### Frequency ratios

The size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (unison), 2:1 (octave), 5:3 (major sixth), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Intervals with small-integer ratios are often called just intervals, or pure intervals.

Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone equal temperament. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, although it is very close to the size of the corresponding just intervals. For instance, an equal-tempered fifth has a frequency ratio of 2712:1, approximately equal to 1.498:1, or 2.997:2 (very close to 3:2). For a comparison between the size of intervals in different tuning systems, see § Size of intervals used in different tuning systems.

### Cents

The standard system for comparing interval sizes is with cents. The cent is a logarithmic unit of measurement. If frequency is expressed in a logarithmic scale, and along that scale the distance between a given frequency and its double (also called octave) is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament (12-TET), a tuning system in which all semitones have the same size, the size of one semitone is exactly 100 cents. Hence, in 12-TET the cent can be also defined as one hundredth of a semitone.

Mathematically, the size in cents of the interval from frequency f1 to frequency f2 is

${\displaystyle n=1200\cdot \log _{2}\left({\frac {f_{2}}{f_{1}}}\right)}$

## Main intervals

The table shows the most widely used conventional names for the intervals between the notes of a chromatic scale. A perfect unison (also known as perfect prime) is an interval formed by two identical notes. Its size is zero cents. A semitone is any interval between two adjacent notes in a chromatic scale, a whole tone is an interval spanning two semitones (for example, a major second), and a tritone is an interval spanning three tones, or six semitones (for example, an augmented fourth). Rarely, the term ditone is also used to indicate an interval spanning two whole tones (for example, a major third), or more strictly as a synonym of major third.

Intervals with different names may span the same number of semitones, and may even have the same width. For instance, the interval from D to F is a major third, while that from D to G is a diminished fourth. However, they both span 4 semitones. If the instrument is tuned so that the 12 notes of the chromatic scale are equally spaced (as in equal temperament), these intervals also have the same width. Namely, all semitones have a width of 100 cents, and all intervals spanning 4 semitones are 400 cents wide.

The names listed here cannot be determined by counting semitones alone. The rules to determine them are explained below. Other names, determined with different naming conventions, are listed in a separate section. Intervals smaller than one semitone (commas or microtones) and larger than one octave (compound intervals) are introduced below.

Number of
semitones
Minor, major,
or perfect
intervals
Short Augmented or
diminished
intervals
Short Widely used
alternative names
Short Audio
0 Perfect unison P1 Diminished second d2
1 Minor second m2 Augmented unison A1 Semitone, half tone, half step S
2 Major second M2 Diminished third d3 Tone, whole tone, whole step T
3 Minor third m3 Augmented second A2 Trisemitone
4 Major third M3 Diminished fourth d4
5 Perfect fourth P4 Augmented third A3
6 Diminished fifth d5 Tritone TT
Augmented fourth A4
7 Perfect fifth P5 Diminished sixth d6
8 Minor sixth m6 Augmented fifth A5
9 Major sixth M6 Diminished seventh d7
10 Minor seventh m7 Augmented sixth A6
11 Major seventh M7 Diminished octave d8
12 Perfect octave P8 Augmented seventh A7

## Interval number and quality

Main intervals from C

In Western music theory, an interval is named according to its number (also called diatonic number) and quality. For instance, major third (or M3) is an interval name, in which the term major (M) describes the quality of the interval, and third (3) indicates its number.

### Number

Staff, with staff positions indicated.
Fifth from C to G in the A major scale.

The number of an interval is the number of letter names or staff positions (lines and spaces) it encompasses, including the positions of both notes forming the interval. For instance, the interval C–G is a fifth (denoted P5) because the notes from C to the G above it encompass five letter names (C, D, E, F, G) and occupy five consecutive staff positions, including the positions of C and G. The table and the figure above show intervals with numbers ranging from 1 (e.g., P1) to 8 (e.g., P8). Intervals with larger numbers are called compound intervals.

There is a one-to-one correspondence between staff positions and diatonic-scale degrees (the notes of diatonic scale). This means that interval numbers can also be determined by counting diatonic scale degrees, rather than staff positions, provided that the two notes that form the interval are drawn from a diatonic scale. Namely, C–G is a fifth because in any diatonic scale that contains C and G, the sequence from C to G includes five notes. For instance, in the A-major diatonic scale, the five notes are C–D–E–F–G (see figure). This is not true for all kinds of scales. For instance, in a chromatic scale, the notes from C to G are eight (C–C–D–D–E–F–F–G). This is the reason interval numbers are also called diatonic numbers, and this convention is called diatonic numbering.

If one adds any accidentals to the notes that form an interval, by definition the notes do not change their staff positions. As a consequence, any interval has the same interval number as the corresponding natural interval, formed by the same notes without accidentals. For instance, the intervals C–G (spanning 8 semitones) and C–G (spanning 6 semitones) are fifths, like the corresponding natural interval C–G (7 semitones).

Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval C–C, a perfect unison, is called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval C–D is a second, but D is only one staff position, or diatonic-scale degree, above C. Similarly, C–E is a third, but E is only two staff positions above C, and so on. As a consequence, joining two intervals always yields an interval number one less than their sum. For instance, the intervals C–E and E–G are thirds, but joined together they form a fifth (C–G), not a sixth. Similarly, a stack of three thirds, such as C–E, E–G, and G–B, is a seventh (C–B), not a ninth.

This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see § Compound intervals below.

### Quality

Intervals formed by the notes of a C major diatonic scale.

The name of any interval is further qualified using the terms perfect (P), major (M), minor (m), augmented (A), and diminished (d). This is called its interval quality. It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in chromatic contexts. The quality of a compound interval is the quality of the simple interval on which it is based.

#### Perfect

Perfect intervals on C.  ,  ,  ,  .

Perfect intervals are so-called because they were traditionally considered perfectly consonant, although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was contrapuntal.[vague] Conversely, minor, major, augmented or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or dissonances.

Within a diatonic scale all unisons (P1) and octaves (P8) are perfect. Most fourths and fifths are also perfect (P4 and P5), with five and seven semitones respectively. One occurrence of a fourth is augmented (A4) and one fifth is diminished (d5), both spanning six semitones. For instance, in a C-major scale, the A4 is between F and B, and the d5 is between B and F (see table).

By definition, the inversion of a perfect interval is also perfect. Since the inversion does not change the pitch class of the two notes, it hardly affects their level of consonance (matching of their harmonics). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, the inversion of an augmented interval is a diminished interval.

#### Major and minor

Major and minor intervals on C.  ,  ,  ,  ,  ,  ,  ,

As shown in the table, a diatonic scale defines seven intervals for each interval number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given interval number always occur in two sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise, the larger version is called major, the smaller one minor. For instance, since a 7-semitone fifth is a perfect interval (P5), the 6-semitone fifth is called "diminished fifth" (d5). Conversely, since neither kind of third is perfect, the larger one is called "major third" (M3), the smaller one "minor third" (m3).

Within a diatonic scale, unisons and octaves are always qualified as perfect, fourths as either perfect or augmented, fifths as perfect or diminished, and all the other intervals (seconds, thirds, sixths, sevenths) as major or minor.

#### Augmented and diminished

Augmented and diminished intervals on C.  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,

Augmented intervals are wider by one semitone than perfect or major intervals, while having the same interval number (i.e., encompassing the same number of staff positions). Diminished intervals, on the other hand, are narrower by one semitone than perfect or minor intervals of the same interval number. For instance, an augmented third such as C–E spans five semitones, exceeding a major third (C–E) by one semitone, while a diminished third such as C–E spans two semitones, falling short of a minor third (C–E) by one semitone.

The augmented fourth (A4) and the diminished fifth (d5) are the only augmented and diminished intervals that appear in diatonic scales (see table).

### Example

Neither the number, nor the quality of an interval can be determined by counting semitones alone. As explained above, the number of staff positions must be taken into account as well.

For example, as shown in the table below, there are four semitones between A and B, between A and C, between A and D, and between A and E, but

• A–B is a second, as it encompasses two staff positions (A, B), and it is doubly augmented, as it exceeds a major second (such as A–B) by two semitones.
• A–C is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
• A–D is a fourth, as it encompasses four staff positions (A, B, C, D), and it is diminished, as it falls short of a perfect fourth (such as A–D) by one semitone.
• A-E is a fifth, as it encompasses five staff positions (A, B, C, D, E), and it is triply diminished, as it falls short of a perfect fifth (such as A–E) by three semitones.
Number
of semitones
Interval name Staff positions
1 2 3 4 5
4 doubly augmented second (AA2) A B
4 major third (M3) A   C
4 diminished fourth (d4) A     D
4 triply diminished fifth (ddd5) A       E

## Shorthand notation

Intervals are often abbreviated with a P for perfect, m for minor, M for major, d for diminished, A for augmented, followed by the interval number. The indications M and P are often omitted. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. The tritone, an augmented fourth or diminished fifth is often TT. The interval qualities may be also abbreviated with perf, min, maj, dim, aug. Examples:

• m2 (or min2): minor second,
• M3 (or maj3): major third,
• A4 (or aug4): augmented fourth,
• d5 (or dim5): diminished fifth,
• P5 (or perf5): perfect fifth.

## Inversion

Major 13th (compound Major 6th) inverts to a minor 3rd by moving the bottom note up two octaves, the top note down two octaves, or both notes one octave

A simple interval (i.e., an interval smaller than or equal to an octave) may be inverted by raising the lower pitch an octave or lowering the upper pitch an octave. For example, the fourth from a lower C to a higher F may be inverted to make a fifth, from a lower F to a higher C.

There are two rules to determine the number and quality of the inversion of any simple interval:

1. The interval number and the number of its inversion always add up to nine (4 + 5 = 9, in the example just given).
2. The inversion of a major interval is a minor interval, and vice versa; the inversion of a perfect interval is also perfect; the inversion of an augmented interval is a diminished interval, and vice versa; the inversion of a doubly augmented interval is a doubly diminished interval, and vice versa.

For example, the interval from C to the E above it is a minor third. By the two rules just given, the interval from E to the C above it must be a major sixth.

Since compound intervals are larger than an octave, "the inversion of any compound interval is always the same as the inversion of the simple interval from which it is compounded."

For intervals identified by their ratio, the inversion is determined by reversing the ratio and multiplying the ratio by 2 until it is greater than 1. For example, the inversion of a 5:4 ratio is an 8:5 ratio.

For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12.

Since an interval class is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.

## Classification

Intervals can be described, classified, or compared with each other according to various criteria.

Melodic and harmonic intervals.

### Melodic and harmonic

An interval can be described as

• Vertical or harmonic if the two notes sound simultaneously
• Horizontal, linear, or melodic if they sound successively. Melodic intervals can be ascending (lower pitch precedes higher pitch) or descending.

### Diatonic and chromatic

In general,

Ascending and descending chromatic scale on C

The table above depicts the 56 diatonic intervals formed by the notes of the C major scale (a diatonic scale). Notice that these intervals, as well as any other diatonic interval, can be also formed by the notes of a chromatic scale.

The distinction between diatonic and chromatic intervals is controversial, as it is based on the definition of diatonic scale, which is variable in the literature. For example, the interval B–E (a diminished fourth, occurring in the harmonic C-minor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well. Otherwise, it is considered chromatic. For further details, see the main article.

By a commonly used definition of diatonic scale (which excludes the harmonic minor and melodic minor scales), all perfect, major and minor intervals are diatonic. Conversely, no augmented or diminished interval is diatonic, except for the augmented fourth and diminished fifth.

The distinction between diatonic and chromatic intervals may be also sensitive to context. The above-mentioned 56 intervals formed by the C-major scale are sometimes called diatonic to C major. All other intervals are called chromatic to C major. For instance, the perfect fifth A–E is chromatic to C major, because A and E are not contained in the C major scale. However, it is diatonic to others, such as the A major scale.

### Consonant and dissonant

Consonance and dissonance are relative terms that refer to the stability, or state of repose, of particular musical effects. Dissonant intervals are those that cause tension and desire to be resolved to consonant intervals.

These terms are relative to the usage of different compositional styles.

• In 15th- and 16th-century usage, perfect fifths and octaves, and major and minor thirds and sixths were considered harmonically consonant, and all other intervals dissonant, including the perfect fourth, which by 1473 was described (by Johannes Tinctoris) as dissonant, except between the upper parts of a vertical sonority—for example, with a supporting third below ("6-3 chords"). In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously considered dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice was still taught to beginning musicians throughout this period.
• <NAME> (1821–1894) theorised that dissonance was caused by the presence of beats. <NAME> further believed that the beating produced by the upper partials of harmonic sounds was the cause of dissonance for intervals too far apart to produce beating between the fundamentals. von Helmholtz then designated that two harmonic tones that shared common low partials would be more consonant, as they produced less beats. von Helmholtz disregarded partials above the seventh, as he believed that they were not audible enough to have significant effect. From this von Helmholtz categorises the octave, perfect fifth, perfect fourth, major sixth, major third, and minor third as consonant, in decreasing value, and other intervals as dissonant.
• <NAME> (1997) suggests the concept of interval strength, in which an interval's strength, consonance, or stability is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law and #Interval root

All of the above analyses refer to vertical (simultaneous) intervals.

### Simple and compound

Simple and compound major third

A simple interval is an interval spanning at most one octave (see Main intervals above). Intervals spanning more than one octave are called compound intervals, as they can be obtained by adding one or more octaves to a simple interval (see below for details).

### Steps and skips

Linear (melodic) intervals may be described as steps or skips. A step, or conjunct motion, is a linear interval between two consecutive notes of a scale. Any larger interval is called a skip (also called a leap), or disjunct motion. In the diatonic scale, a step is either a minor second (sometimes also called half step) or major second (sometimes also called whole step), with all intervals of a minor third or larger being skips.

For example, C to D (major second) is a step, whereas C to E (major third) is a skip.

More generally, a step is a smaller or narrower interval in a musical line, and a skip is a wider or larger interval, where the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.

Melodic motion in which the interval between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called stepwise or conjunct melodic motion, as opposed to skipwise or disjunct melodic motions, characterized by frequent skips.

### Enharmonic intervals

Enharmonic tritones: A4 = d5 on C

Two intervals are considered enharmonic, or enharmonically equivalent, if they both contain the same pitches spelled in different ways; that is, if the notes in the two intervals are themselves enharmonically equivalent. Enharmonic intervals span the same number of semitones.

For example, the four intervals listed in the table below are all enharmonically equivalent, because the notes F and G indicate the same pitch, and the same is true for A and B. All these intervals span four semitones.

Number
of semitones
Interval name Staff positions
1 2 3 4
4 major third F   A
4 major third   G   B
4 diminished fourth F     B
4 doubly augmented second   G A

When played as isolated chords on a piano keyboard, these intervals are indistinguishable to the ear, because they are all played with the same two keys. However, in a musical context, the diatonic function of the notes these intervals incorporate is very different.

The discussion above assumes the use of the prevalent tuning system, 12-tone equal temperament ("12-TET"). But in other historic meantone temperaments, the pitches of pairs of notes such as F and G may not necessarily coincide. These two notes are enharmonic in 12-TET, but may not be so in another tuning system. In such cases, the intervals they form would also not be enharmonic. For example, in quarter-comma meantone, all four intervals shown in the example above would be different.

## Minute intervals

Pythagorean comma on C; the note depicted as lower on the staff (B+++) is slightly higher in pitch (than C).

There are also a number of minute intervals not found in the chromatic scale or labeled with a diatonic function, which have names of their own. They may be described as microtones, and some of them can be also classified as commas, as they describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes. In the following list, the interval sizes in cents are approximate.

• A Pythagorean comma is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288 (23.5 cents).
• A syntonic comma is the difference between four justly tuned perfect fifths and two octaves plus a major third. It is expressed by the ratio 81:80 (21.5 cents).
• A septimal comma is 64:63 (27.3 cents), and is the difference between the Pythagorean or 3-limit "7th" and the "harmonic 7th".
• A diesis is generally used to mean the difference between three justly tuned major thirds and one octave. It is expressed by the ratio 128:125 (41.1 cents). However, it has been used to mean other small intervals: see diesis for details.
• A diaschisma is the difference between three octaves and four justly tuned perfect fifths plus two justly tuned major thirds. It is expressed by the ratio 2048:2025 (19.6 cents).
• A schisma (also skhisma) is the difference between five octaves and eight justly tuned fifths plus one justly tuned major third. It is expressed by the ratio 32805:32768 (2.0 cents). It is also the difference between the Pythagorean and syntonic commas. (A schismic major third is a schisma different from a just major third, eight fifths down and five octaves up, F in C.)
• A kleisma is the difference between six minor thirds and one tritave or perfect twelfth (an octave plus a perfect fifth), with a frequency ratio of 15625:15552 (8.1 cents) ( ).
• A septimal kleisma is the amount that two major thirds of 5:4 and a septimal major third, or supermajor third, of 9:7 exceeds the octave. Ratio 225:224 (7.7 cents).
• A quarter tone is half the width of a semitone, which is half the width of a whole tone. It is equal to exactly 50 cents.

## Compound intervals

Simple and compound major third

A compound interval is an interval spanning more than one octave. Conversely, intervals spanning at most one octave are called simple intervals (see Main intervals below).

In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called compound major third, spans one octave plus one major third.

Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a major seventeenth can be decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built by adding up four fifths.

The diatonic number DNc of a compound interval formed from n simple intervals with diatonic numbers DN1, DN2, ..., DNn, is determined by:

${\displaystyle DN_{c}=1+(DN_{1}-1)+(DN_{2}-1)+...+(DN_{n}-1),\ }$

which can also be written as:

${\displaystyle DN_{c}=DN_{1}+DN_{2}+...+DN_{n}-(n-1),\ }$

The quality of a compound interval is determined by the quality of the simple interval on which it is based. For instance, a compound major third is a major tenth (1+(8−1)+(3−1) = 10), or a major seventeenth (1+(8−1)+(8−1)+(3−1) = 17), and a compound perfect fifth is a perfect twelfth (1+(8−1)+(5−1) = 12) or a perfect nineteenth (1+(8−1)+(8−1)+(5−1) = 19). Notice that two octaves are a fifteenth, not a sixteenth (1+(8−1)+(8−1) = 15). Similarly, three octaves are a twenty-second (1+3×(8−1) = 22), and so on.

### Main compound intervals

Number of
semitones
Minor, major,
or perfect
intervals
Short Augmented or
diminished
intervals
Short
12 Diminished ninth d9
13 Minor ninth m9 Augmented octave A8
14 Major ninth M9 Diminished tenth d10
15 Minor tenth m10 Augmented ninth A9
16 Major tenth M10 Diminished eleventh d11
17 Perfect eleventh P11 Augmented tenth A10
18 Diminished twelfth d12
Augmented eleventh A11
19 Perfect twelfth or Tritave P12 Diminished thirteenth d13
20 Minor thirteenth m13 Augmented twelfth A12
21 Major thirteenth M13 Diminished fourteenth d14
22 Minor fourteenth m14 Augmented thirteenth A13
23 Major fourteenth M14 Diminished fifteenth d15
24 Perfect fifteenth or Double octave P15 Augmented fourteenth A14
25 Augmented fifteenth A15

It is also worth mentioning here the major seventeenth (28 semitones)—an interval larger than two octaves that can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 × 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom come up, most often being referred to by their compound names, for example "two octaves plus a fifth" rather than "a 19th".

## Intervals in chords

Chords are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the root of the chord. For instance a major triad is a chord containing three notes defined by the root and two intervals (major third and perfect fifth). Sometimes even a single interval (dyad) is considered a chord. Chords are classified based on the quality and number of the intervals that define them.

### Chord qualities and interval qualities

The main chord qualities are major, minor, augmented, diminished, half-diminished, and dominant. The symbols used for chord quality are similar to those used for interval quality (see above). In addition, + or aug is used for augmented, ° or dim for diminished, ø for half diminished, and dom for dominant (the symbol alone is not used for diminished).

### Deducing component intervals from chord names and symbols

The main rules to decode chord names or symbols are summarized below. Further details are given at Rules to decode chord names and symbols.

1. For 3-note chords (triads), major or minor always refer to the interval of the third above the root note, while augmented and diminished always refer to the interval of the fifth above root. The same is true for the corresponding symbols (e.g., Cm means Cm3, and C+ means C+5). Thus, the terms third and fifth and the corresponding symbols 3 and 5 are typically omitted. This rule can be generalized to all kinds of chords, provided the above-mentioned qualities appear immediately after the root note, or at the beginning of the chord name or symbol. For instance, in the chord symbols Cm and Cm7, m refers to the interval m3, and 3 is omitted. When these qualities do not appear immediately after the root note, or at the beginning of the name or symbol, they should be considered interval qualities, rather than chord qualities. For instance, in CmM7 (minor major seventh chord), m is the chord quality and refers to the m3 interval, while M refers to the M7 interval. When the number of an extra interval is specified immediately after chord quality, the quality of that interval may coincide with chord quality (e.g., CM7 = CMM7). However, this is not always true (e.g., Cm6 = CmM6, C+7 = C+m7, CM11 = CMP11). See main article for further details.
2. Without contrary information, a major third interval and a perfect fifth interval (major triad) are implied. For instance, a C chord is a C major triad, and the name C minor seventh (Cm7) implies a minor 3rd by rule 1, a perfect 5th by this rule, and a minor 7th by definition (see below). This rule has one exception (see next rule).
3. When the fifth interval is diminished, the third must be minor. This rule overrides rule 2. For instance, Cdim7 implies a diminished 5th by rule 1, a minor 3rd by this rule, and a diminished 7th by definition (see below).
4. Names and symbols that contain only a plain interval number (e.g., “seventh chord”) or the chord root and a number (e.g., “C seventh”, or C7) are interpreted as follows:
• If the number is 2, 4, 6, etc., the chord is a major added tone chord (e.g., C6 = CM6 = Cadd6) and contains, together with the implied major triad, an extra major 2nd, perfect 4th, or major 6th (see names and symbols for added tone chords).
• If the number is 7, 9, 11, 13, etc., the chord is dominant (e.g., C7 = Cdom7) and contains, together with the implied major triad, one or more of the following extra intervals: minor 7th, major 9th, perfect 11th, and major 13th (see names and symbols for seventh and extended chords).
• If the number is 5, the chord (technically not a chord in the traditional sense, but a dyad) is a power chord. Only the root, a perfect fifth and usually an octave are played.

The table shows the intervals contained in some of the main chords (component intervals), and some of the symbols used to denote them. The interval qualities or numbers in boldface font can be deduced from chord name or symbol by applying rule 1. In symbol examples, C is used as chord root.

Main chords Component intervals
Name Symbol examples Third Fifth Seventh
CM, or Cmaj M3 P5
Minor triad Cm, or Cmin m3 P5
Augmented triad C+, or Caug M3 A5
Diminished triad C°, or Cdim m3 d5
Dominant seventh chord C7, or Cdom7 M3 P5 m7
Minor seventh chord Cm7, or Cmin7 m3 P5 m7
Major seventh chord CM7, or Cmaj7 M3 P5 M7
Augmented seventh chord C+7, Caug7,
C75, or C7aug5
M3 A5 m7
Diminished seventh chord 7, or Cdim7 m3 d5 d7
Half-diminished seventh chord Cø7, Cm75, or Cm7dim5 m3 d5 m7

## Size of intervals used in different tuning systems

Number of
semitones
Name 5-limit tuning
(pitch ratio)
Comparison of interval width (in cents)
5-limit tuning Pythagorean
tuning
14-comma
meantone
Equal
temperament
0 Perfect unison 1:1 0 0 0 0
1 Minor second 16:15
27:25
112
133
90 117 100
2 Major second 9:8
10:9
204
182
204 193 200
3 Minor third 6:5
32:27
316
294
294
318
310
(wolf) 269
300
4 Major third 5:4 386 408
384
386
(wolf) 427
400
5 Perfect fourth 4:3
27:20
498
520
498
(wolf) 522
503
(wolf) 462
500
6 Augmented fourth
Diminished fifth
45:32
25:18
590
569
612
588
579
621
600
7 Perfect fifth 3:2
40:27
702
680
702
(wolf) 678
697
(wolf) 738
700
8 Minor sixth 8:5 814 792 814 800
9 Major sixth 5:3
27:16
884
906
906 890 900
10 Minor seventh 16:9
9:5
996
1018
996 1007 1000
11 Major seventh 15:8
50:27
1088
1067
1110 1083 1100
12 Perfect octave 2:1 1200 1200 1200 1200

In this table, the interval widths used in four different tuning systems are compared. To facilitate comparison, just intervals as provided by 5-limit tuning (see symmetric scale n.1) are shown in bold font, and the values in cents are rounded to integers. Notice that in each of the non-equal tuning systems, by definition the width of each type of interval (including the semitone) changes depending on the note that starts the interval. This is the art of just intonation. In equal temperament, the intervals are never precisely in tune with each other. This is the price of using equidistant intervals in a 12-tone scale. For simplicity, for some types of interval the table shows only one value (the most often observed one).

In 14-comma meantone, by definition 11 perfect fifths have a size of approximately 697 cents (700 − ε cents, where ε ≈ 3.42 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of about 738 cents (700 + 11ε, the wolf fifth or diminished sixth); 8 major thirds have size about 386 cents (400 − 4ε), 4 have size about 427 cents (400 + 8ε, actually diminished fourths), and their average size is 400 cents. In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε (the difference between the 14-comma meantone fifth and the average fifth). A more detailed analysis is provided at 14-comma meantone Size of intervals. Note that 14-comma meantone was designed to produce just major thirds, but only 8 of them are just (5:4, about 386 cents).

The Pythagorean tuning is characterized by smaller differences because they are multiples of a smaller ε (ε ≈ 1.96 cents, the difference between the Pythagorean fifth and the average fifth). Notice that here the fifth is wider than 700 cents, while in most meantone temperaments, including 14-comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided at Pythagorean tuning#Size of intervals.

The 5-limit tuning system uses just tones and semitones as building blocks, rather than a stack of perfect fifths, and this leads to even more varied intervals throughout the scale (each kind of interval has three or four different sizes). A more detailed analysis is provided at 5-limit tuning#Size of intervals. Note that 5-limit tuning was designed to maximize the number of just intervals, but even in this system some intervals are not just (e.g., 3 fifths, 5 major thirds and 6 minor thirds are not just; also, 3 major and 3 minor thirds are wolf intervals).

The above-mentioned symmetric scale 1, defined in the 5-limit tuning system, is not the only method to obtain just intonation. It is possible to construct juster intervals or just intervals closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular, the asymmetric version of the 5-limit tuning scale provides a juster value for the minor seventh (9:5, rather than 16:9). Moreover, the tritone (augmented fourth or diminished fifth), could have other just ratios; for instance, 7:5 (about 583 cents) or 17:12 (about 603 cents) are possible alternatives for the augmented fourth (the latter is fairly common, as it is closer to the equal-tempered value of 600 cents). The 7:4 interval (about 969 cents), also known as the harmonic seventh, has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. For further details about reference ratios, see 5-limit tuning#The justest ratios.

In the diatonic system, every interval has one or more enharmonic equivalents, such as augmented second for minor third.

## Interval root

Intervals in the harmonic series.

Although intervals are usually designated in relation to their lower note, <NAME> and Hindemith both suggest the concept of interval root. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its top note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the interval root of its strongest interval.

As to its usefulness, Cope provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" (added sixth chords by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According to the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

## Interval cycles

Interval cycles, "unfold [i.e., repeat] a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by <NAME> using the letter "C", for cycle, with an interval-class integer to distinguish the interval. Thus the diminished-seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0–11 to indicate the lowest pitch class in the cycle.

## Alternative interval naming conventions

As shown below, some of the above-mentioned intervals have alternative names, and some of them take a specific alternative name in Pythagorean tuning, five-limit tuning, or meantone temperament tuning systems such as quarter-comma meantone. All the intervals with prefix sesqui- are justly tuned, and their frequency ratio, shown in the table, is a superparticular number (or epimoric ratio). The same is true for the octave.

Typically, a comma is a diminished second, but this is not always true (for more details, see Alternative definitions of comma). For instance, in Pythagorean tuning the diminished second is a descending interval (524288:531441, or about −23.5 cents), and the Pythagorean comma is its opposite (531441:524288, or about 23.5 cents). 5-limit tuning defines four kinds of comma, three of which meet the definition of diminished second, and hence are listed in the table below. The fourth one, called syntonic comma (81:80) can neither be regarded as a diminished second, nor as its opposite. See Diminished seconds in 5-limit tuning for further details.

Number of
semitones
Generic names Specific names
Quality and number Other naming convention Pythagorean tuning 5-limit tuning 14-comma
meantone
Full Short
0 perfect unison
or perfect prime
P1
diminished second d2 descending
Pythagorean comma
(524288:531441)
lesser diesis (128:125)
diaschisma (2048:2025)
greater diesis (648:625)
1 minor second m2 semitone,
half tone,
half step
diatonic semitone,
major semitone
limma (256:243)
augmented unison
or augmented prime
A1 chromatic semitone,
minor semitone
apotome (2187:2048)
2 major second M2 tone, whole tone, whole step sesquioctavum (9:8)
3 minor third m3 sesquiquintum (6:5)
4 major third M3 sesquiquartum (5:4)
5 perfect fourth P4 sesquitertium (4:3)
6 diminished fifth d5 tritone
augmented fourth A4
7 perfect fifth P5 sesquialterum (3:2)
12 perfect octave P8 duplex (2:1)

Additionally, some cultures around the world have their own names for intervals found in their music. For instance, 22 kinds of intervals, called shrutis, are canonically defined in Indian classical music.

### Latin nomenclature

Up to the end of the 18th century, Latin was used as an official language throughout Europe for scientific and music textbooks. In music, many English terms are derived from Latin. For instance, semitone is from Latin semitonus.

The prefix semi- is typically used herein to mean "shorter", rather than "half". Namely, a semitonus, semiditonus, semidiatessaron, semidiapente, semihexachordum, semiheptachordum, or semidiapason, is shorter by one semitone than the corresponding whole interval. For instance, a semiditonus (3 semitones, or about 300 cents) is not half of a ditonus (4 semitones, or about 400 cents), but a ditonus shortened by one semitone. Moreover, in Pythagorean tuning (the most commonly used tuning system up to the 16th century), a semitritonus (d5) is smaller than a tritonus (A4) by one Pythagorean comma (about a quarter of a semitone).

Number of
semitones
Quality and number Short Latin
nomenclature
0 Perfect unison P1 unisonus
1 Minor second m2 semitonus
Augmented unison A1 unisonus superflua
2 Major second M2 tonus
Diminished third d3
3 Minor third m3 semiditonus
Augmented second A2 tonus superflua
4 Major third M3 ditonus
Diminished fourth d4 semidiatessaron
5 Perfect fourth P4 diatessaron
Augmented third A3 ditonus superflua
6 Diminished fifth d5 semidiapente, semitritonus
Augmented fourth A4 tritonus
7 Perfect fifth P5 diapente
Diminished sixth d6 semihexachordum
8 Minor sixth m6 hexachordum minus, semitonus maius cum diapente, tetratonus
Augmented fifth A5 diapente superflua
9 Major sixth M6 hexachordum maius, tonus cum diapente
Diminished seventh d7 semiheptachordum
10 Minor seventh m7 heptachordum minus, semiditonus cum diapente, pentatonus
Augmented sixth A6 hexachordum superflua
11 Major seventh M7 heptachordum maius, ditonus cum diapente
Diminished octave d8 semidiapason
12 Perfect octave P8 diapason
Augmented seventh A7 heptachordum superflua

## Pitch-class intervals

In post-tonal or atonal theory, originally developed for equal-tempered European classical music written using the twelve-tone technique or serialism, integer notation is often used, most prominently in musical set theory. In this system, intervals are named according to the number of half steps, from 0 to 11, the largest interval class being 6.

In atonal or musical set theory, there are numerous types of intervals, the first being the ordered pitch interval, the distance between two pitches upward or downward. For instance, the interval from C upward to G is 7, and the interval from G downward to C is −7. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory.

The interval between pitch classes may be measured with ordered and unordered pitch-class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. For unordered pitch-class intervals, see interval class.

## Generic and specific intervals

In diatonic set theory, specific and generic intervals are distinguished. Specific intervals are the interval class or number of semitones between scale steps or collection members, and generic intervals are the number of diatonic scale steps (or staff positions) between notes of a collection or scale.

Notice that staff positions, when used to determine the conventional interval number (second, third, fourth, etc.), are counted including the position of the lower note of the interval, while generic interval numbers are counted excluding that position. Thus, generic interval numbers are smaller by 1, with respect to the conventional interval numbers.

### Comparison

Specific interval Generic interval Diatonic name
Number of semitones Interval class
0 0 0 Perfect unison
1 1 1 Minor second
2 2 1 Major second
3 3 2 Minor third
4 4 2 Major third
5 5 3 Perfect fourth
6 6 3
4
Augmented fourth
Diminished fifth
7 5 4 Perfect fifth
8 4 5 Minor sixth
9 3 5 Major sixth
10 2 6 Minor seventh
11 1 6 Major seventh
12 0 7 Perfect octave

## Generalizations and non-pitch uses

Division of the measure/chromatic scale, followed by pitch/time-point series

The term "interval" can also be generalized to other music elements besides pitch. David Lewin's Generalized Musical Intervals and Transformations uses interval as a generic measure of distance between time points, timbres, or more abstract musical phenomena.

For example, an interval between two bell-like sounds, which have no pitch salience, is still perceptible. When two tones have similar acoustic spectra (sets of partials), the interval is just the distance of the shift of a tone spectrum along the frequency axis, so linking to pitches as reference points is not necessary. The same principle naturally applies to pitched tones (with similar harmonic spectra), which means that intervals can be perceived "directly" without pitch recognition. This explains in particular the predominance of interval hearing over absolute pitch hearing.


text:
# why does a step up transformer cause lower currents?

The usual answer I hear is because power needs to be conserved, but I dont find that answer satisfying. Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit). But if i have a 2x transformer (which AFAIK is made of inductors which are linear elements), then the output of the transformer will have twice the voltage and half the current. What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

• Vin * Iin= Vout * Iout (+ losses) – <NAME> EE75 Jul 23 '18 at 0:11
• The input side of the transformer has a quarter of the resistance. – immibis Jul 23 '18 at 2:44
• en.wikipedia.org/wiki/Conservation_of_energy – Jules Jul 23 '18 at 8:13
• It's called "impedance transformation". – <NAME> Jul 23 '18 at 16:24
• Oh and by the way, Ohm's Law describes resistor. Plenty of things "violate Ohm's Law" because they aren't resistors. Capacitors, for example. Or microprocessors. – immibis Jul 24 '18 at 5:13

Ohm's law is not violated because a transformer is not a resistor. The transformer simply changes the source's view of the load impedance. So if you have a 1:2 transformer connected to a source and a load resistor, then the source side would look like one quarter of the load resistance.

Transformers don't work at DC and hence understanding what's going on requires a bit more math and physics. Let's take a simple example circuit:

simulate this circuit – Schematic created using CircuitLab

In this case, we have a source (V1), a load (R1) and a 1:2 step-up transformer (N = 0.5). Some current will flow on the primary side of the transformer, $i_p(t)$. The current in the primary will induce a magnetic field in the transformer core that's proportional to the current and to the number of turns in the primary:

$$B_p(t) = \mu_0 k_1 N_p i_p(t)$$

Don't worry about the $k_n$ constants, they're dependent on the construction of the transformer and they'll cancel out at the end, assuming the transformer is ideal. This results in a magnetic flux on the secondary:

$$\phi_s(t) = k_2 B_p(t) = \mu_0 k_1 k_2 N_p i_p(t)$$

The oscillating magnetic flux will induce a voltage in the secondary of the transformer that's proportional to the derivative of the magnetic flux and to the number of turns in the secondary:

$$v_s(t) = N_s \frac{d}{dt} \phi_s(t) = \mu_0 k_1 k_2 N_p N_s \frac{d}{dt}i_p(t)$$

The same thing can be calculated in the other direction:

$$v_p(t) = N_p \frac{d}{dt} \phi_p(t) = \mu_0 k_1 k_2 N_p N_s \frac{d}{dt}i_s(t)$$

If we assume an ideal transformer with 100% coupling between the primary and secondary, then $\phi_s(t) = \phi_p(t)$. If $\phi_s(t) = \phi_p(t)$, then $\frac{d}{dt}\phi_s(t) = \frac{d}{dt}\phi_p(t)$. This means we can write:

$$\phi_s(t) = \phi_p(t) = \mu_0 k_1 k_2 N_p i_p(t) = \mu_0 k_1 k_2 N_s i_s(t)$$

and

$$\frac{d}{dt}\phi_s(t) = \frac{d}{dt}\phi_p(t) = \frac{v_p(t)}{N_p} = \frac{v_s(t)}{N_s}$$

from which you can write

$$i_s(t) = \frac{N_p}{N_s}i_p(t)$$

and

$$v_s(t) = \frac{N_s}{N_p}v_p(t)$$

If $N_s = 2 N_p$ as in the example circuit, then $v_s(t) = 2v_p(t)$ and $i_s(t) = \frac{1}{2}i_p(t)$. This makes sense from a standpoint of conservation of energy; $P = v_s i_s = 2 v_p \frac{1}{2} i_p = v_p i_p$.

Ohm's law on the load resistor indicates that $v_s(t) = 100\ \Omega\cdot\ i_s(t)$. By plugging in both current and voltage relations, we get $v_p(t) = 25\ \Omega\cdot\ i_p(t)$. In effect, the transformer makes the 100 Ω resistor 'look like' a 25 Ω resistor from the standpoint of the source. Again, this makes sense - half the voltage and double the current is 1/4 the resistance. In general a transformer will 'transform' the load impedance by a factor of $N_p^2/N_s^2$

• You saved me trying to write something. The only thing I intended to add was some verbiage designed to point out that if a 1:2 step-up doubled the voltage at the secondary and also doubled the current (according to the OP's Ohm's Law malperspective), then we'd have an infinite supply of energy from using a simple transformer. And that doesn't happen in nature. +1 – jonk Jul 23 '18 at 4:36

The usual answer I hear is because power needs to be conserved, ...

This is correct.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

That would be correct if you didn't change the load resistance.

But if i have a 2x transformer ... then the output of the transformer will have twice the voltage and half the current.

This is true for a given output power.

simulate this circuit – Schematic created using CircuitLab

Figure 1. (a) Reference setup. (b) Doubling the voltage quadruples the power into the same resistance. (c) By quadrupling the load resistance the power is the same for double the voltage.

To maintain output power for a doubling of output voltage the load resistance would be increased by a factor of 4 (the step-up ratio squared).

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

Yes, but only if the load resistance stays the same.

But if i have a 2x transformer (which AFAIK is made of inductors which are linear elements), then the output of the transformer will have twice the voltage and half the current.

Yes, but only if you increase the resistance by a factor of 4.

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing, because it's not. That is, Ohm's law holds. It's just that, with a higher voltage, for the same power you increase the resistance so as to get less current. Once you've done that, the transformer is supplying the same power but at less current. For the same size wiring, with the load current halved and the load resistance being the dominant term in setting the system current, the power lost in the wiring (assuming it is unchanged) drops by a factor of 4 and the system overall becomes more efficient.

Nothing to do with Ohms Law: the secondary side of the transformer is a separate circuit, electrically unconnected to the first.

In this case the relevant equation is P=VI combined with conservation of energy. The output power of the transformer cannot exceed the input power.

The behavior of a transformer requires a negative-feedback behavior inside the core, where primary-flux is almost perfectly cancelled by the secondary-flux.

If secondary currents increase, the direction of secondary flux change causes a drop in total summed core flux, and the energy source driving the primary is allowed to provide more current into the primary and thus into the secondary.

KEY to this behavior is a very small resistance between the energy source and the ideal primary winding mathematics model.

With FLUX defining the transformer behavior, a doubling of turns in either primary or in secondary is all that is needed to permit a precise halving of the current in that winding.

We call this type of circuit a magnetic circuit. It's equation is similar to Ohms law.

F=Φ*R

Where F = magnetomotive force, Φ = magnetic flux, R = magnetic reluctance

Magnetic flux is analogous to current.

Reluctance is analogous to resistance.

Magnetomotive force is analogous to electromotive force (voltage).

From Ampere's law we know that an inductor will produce currents perpendicular to the magnetic field lines. The strength of the current will be proportional to the number of coils in the solenoid.

From Faraday's law we know that a conductor in a changing magnetic field will produce a current in it. This current will be in the direction to oppose the forces of the changing magnetic field (Lentz's law).

So in any transformer, we have a conductor wrapped around each side of a metal core. They are not electrically connected, only magnetically coupled.

An alternating current in the coil on the primary side, induces a flow of magnetic flux circulating in the transformer core. The magnetic flux is a changing magnetic field which flows in the transformer core.

There is a another conductor wrapped around the secondary side of the transformer core. Since this secondary side conductor will be stationary in a changing magnetic field (induced by the primary side) it will produce a current in the secondary conductor. This current opposes the magnetic field which created it.

The primary and secondary "turns ratio" or number of coils or loops in the windings will control the voltage and current outputs relative to the input values.

As you can see above, the primary and secondary turns are directly proportional to the primary and secondary voltage.

However, the primary and secondary voltage (and turns) ratio is indirectly proportional to the current ratio.

format: plain


text:
Uniformly Accelerated Motion

Uniformly Accelerated Motion

Objects on the earth surface are accelerated towards the centre of the earth at a rate of 9.8 m/ sec².  If we raise an object above the surface of the earth and then drop it and the object will start from rest, its velocity will be increased by 9.8 m/ sec². For each second it is falling towards the earth surface until the object strike the ground.

What is Uniformly Accelerated Motion?

If the acceleration always remains constant, then that acceleration is called uniform acceleration. A movement with uniformly increasing or decreasing speed is called Uniformly Accelerated Motion. An object is said to moving with uniform acceleration, if equal change in velocity takes place in equal intervals of time, however small these intervals may be. In uniform acceleration, velocity changes with a uniform rate.

The acceleration of a freely falling object due to the gravitation is uniform acceleration. Equal forces act on an object of uniform acceleration, both magnitude and direction of uniform acceleration remains constant. So, it occurs when the speed of an object changes at a constant rate.

In the above figure uniform acceleration is shown by successive change of velocity with time along a straight line. The uniform acceleration of a body is 9 m/ sec means that the velocity of the body changes in each second by 9 m/ sec in the same direction.

How to find the Uniformly Accelerated Motion?

Problem: A car was travelling at a speed of 4.9 m/ sec, the driver saw a cat on the road and slammed on breaks. After 3 seconds the car came to stop, how far did the car travel from the point where the brakes were first passed to the point where the car stopped?

Solution: Given,

Initial Velocity (u) = 4.9 m/ sec

Final Velocity (V) = 0 m/ sec [∵ After some time car came to stop],

Time (t) = 6 sec

Distance (s) =?

Using Kinematic Relation:

$$Dis\tan ce\,\,(s)=\frac{(V+u)}{2}\times t$$.

$$Dis\tan ce\,\,(s)=\frac{(4.9+0)}{2}\times 3=7.35m$$.

Therefore, the car stopped after 7.35m.


text:
# All Questions

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text:
. This article discusses its representation in different coordinate systems i.e. It is mandatory to procure user consent prior to running these cookies on your website. Let R be a region in xyz space with surface This is an open surface - the divergence theorem, however, only applies to closed surfaces. We compute the two integrals of the divergence theorem. The volume integral of the divergence of a vector field over the volume enclosed by surface S isequal to the flux of that vector field taken over that surface S.” Hence, this theorem is used to convert volume integral into surface integral. to cylindrical coordinates. on the entire surface is Example1 Let V be a spherical ball of radius 2, centered at the origin, with a concentric ball of radius 1 removed. Must Evaluate Symmetry PPT. divergence theorem, which we really derived from Green's theorem, told us that the flux across our boundary of this region-- so let me write that out. The Divergence Theorem relates relates volume Now, let us compute the volume integral. of F. There are various Properties and Applications of Surface Integrals. negative direction, then the surface integral equals -128*pi. Aldi Hemp Seeds Australia, Bounty Hunter Training Academy Nc, Cohiba Habana Cuba, Shirdi Airport Helpline Number, Nurses Tv Series 2020 Season 2, Dark Souls 2 Earthen Peak Jump, " />

Divergence Theorem Statement The Gauss divergence theorem states that the vector’s outward flux through a closed surface is equal to the volume integral of the divergence over the area within the surface. It compares the surface integral with the volume integral. The above equation says that the integral of a quantity is 0. <NAME>. (2018) Electromagnetics, Vol. The Divergence Theorem can be also written in coordinate form as is. Using the Divergence Theorem, we can write: ${I }={ \iint\limits_S {{x^3}dydz + {y^3}dxdz }}+{{ {z^3}dxdy} }= {\iiint\limits_G {\left( {3{x^2} + 3{y^2} + 3{z^2}} \right)dxdydz} }= {3\iiint\limits_G {\left( {{x^2} + {y^2} + {z^2}} \right)dxdydz}}$, By changing to spherical coordinates, we have, ${I }={ 3\iiint\limits_G {\left( {{x^2} + {y^2} + {z^2}} \right)dxdydz} }= {3\iiint\limits_G {{r^2} \cdot {r^2}\sin \theta drd\psi d\theta } }= {3\int\limits_0^{2\pi } {d\psi } \int\limits_0^\pi {\sin \theta d\theta } \int\limits_0^a {{r^4}dr} }= {3 \cdot 2\pi \cdot \left[ {\left. The Divergence Theorem In other words, it equates the flux of a vector field through a closed surface to a volume of the divergence of that same vector field. Notice that the divergence theorem equates a surface integral with a triple integral over the volume inside the surface. Click or tap a problem to see the solution. Divergence of a vector field is the measure of “Outgoingness” of the field at a given point. The divergence of Let E E be a simple solid region and S S is the boundary surface of E E with positive orientation. The middle integral with respect to r circle x^2+y^2=16. This depends on finding a vector field whose divergence is equal to the given function. of Mathematics, Oregon State In this article, you will learn the divergence theorem statement, proof, Gauss divergence theorem, and examples in detail. However, the z integral must be done before the r integral.) The Divergence Theorem states: normal vector has x component and y component equal to 0. The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa. Use the Divergence Theorem to evaluate ∬ S →F ⋅d→S ∬ S F → ⋅ d S → where →F = 2xz→i +(1−4xy2) →j +(2z −z2) →k F → = 2 x z i → + (1 − 4 x y 2) j → + (2 z − z 2) k → and S S is the surface of the solid bounded by z =6 −2x2 −2y2 z = 6 − 2 x 2 − 2 y 2 and the plane z =0 z = 0. The one dimensional fundamental theorem in effect converts the v in the integrand to an n v on the boundary, where n is the outward directed unit vector normal to it. However, the divergence of F is nice: div. computation in is described by the inequalities 0<=z<=16-x^2-y^2 and Divergence Theorem Let $$Σ$$ be a closed surface in $$\mathbb{R}^ 3$$ which bounds a solid $$S$$, and let $$\textbf{f}(x, y, z) = f_1(x, y, z)\textbf{i}+ f_2(x, y, z)\textbf{j}+ f_3(x, y, z)\textbf{k}$$ be a vector field defined on some subset of $$\mathbb{R}^ 3$$ that contains $$Σ$$. }$, By switching to cylindrical coordinates, we have, ${I = 3\iiint\limits_G {dxdydz} }= {3\int\limits_{ – 1}^1 {dz} \int\limits_0^{2\pi } {d\varphi } \int\limits_0^a {rdr} }= {3 \cdot 2 \cdot 2\pi \cdot \left[ {\left. University. This website uses cookies to improve your experience while you navigate through the website. ∬ S F ⋅ d S = ∭ B div. Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr 3 /3 and r = 3units.Thus we get 180π. Via Gauss’s theorem (also known as the divergence theorem), we can relate the ﬂux of any The outward normal vector ... Use the divergence theorem to convert the surface integration term into a volume integration term: Continuity … 1 Gauss' law in differential form involves the divergence of the electric field: -2 Use the divergence theorem to convert the differential form of Gauss' law into the integral form. In cylindrical coordinates, we have 0<=z<=16-r^2, 0<=r<=4, and Copyright © 1996 Department The sum of all sources subtracted by the sum of every sink will result in the net flow of an area. Another way to say the same thing is: the flux integral of v over a bounding surface is the integral of its divergence over the interior . F = 3 + 2 y + x. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV Div (3x i + 2y j) = 3 + 2 = 5. integral of the vector field is. The \[{\iint\limits_S {\mathbf{F} \cdot d\mathbf{S}} }={ \iiint\limits_G {\left( {\nabla \cdot \mathbf{F}} \right)dV} ,}$, ${\nabla \cdot \mathbf{F} }={ \frac{{\partial P}}{{\partial x}} + \frac{{\partial Q}}{{\partial y}} + \frac{{\partial R}}{{\partial z}}}$. The theorem Take the derivative here, you just get 2. {\left( {\frac{{{r^5}}}{5}} \right)} \right|_0^a} \right] }= {\frac{{12\pi {a^5}}}{5}. We now use the divergence theorem to justify the special case of this law in which the electrostatic field is generated by a stationary point charge at the origin. coordinates the region R This website uses cookies to improve your experience. This theorem is used to solve many tough integral problems. Yep. This is the same as the surface integral! The Divergence Theorem relates surface integrals of vector fields to volume integrals. Divergence Theorem states: Here div F is the divergence The Divergence Theorem can be also written in coordinate form as, ${\iint\limits_S {Pdydz + Qdxdz }}+{{ Rdxdy} }= {\iiint\limits_G {\left( {\frac{{\partial P}}{{\partial x}} + \frac{{\partial Q}}{{\partial y}} }\right.}+{\left. { \frac{{\partial R}}{{\partial z}}} \right)dxdydz}}$, In a particular case, by setting $$P = x,$$ $$Q = y,$$ $$R = z,$$ we obtain a formula for the volume of solid $$G:$$, ${V \text{ = }}\kern0pt{ \frac{1}{3}\Big| {\iint\limits_S {xdydz }}+{{ ydxdz }}+{{ zdxdy} } \Big|}$. A plot of the paraboloid is z=g(x,y)=16-x^2-y^2 for z>=0 is shown on contact us. Theorem 15.4.2 gives the Divergence Theorem in the plane, which states that the flux of a vector field across a closed curve equals the sum of the divergences over the region enclosed by the curve. So this is going to be equal to 2x-- let me do that same color-- it's going to be equal to 2x times-- let me get this right, let me go into that pink color-- 2x times 2z. On the surface S_2, the normal vector points in the negative Hence we have proved the Divergence Theorem for any region formed by pasting together regions that can be smoothly parameterized by rectangular solids. The divergence theorem can also be used to evaluate triple integrals by turning them into surface integrals. The function does this very thing, so the 0-divergence function in the direction is Exercise 17.2 Notice that the divergence of (x, y, 0) otherwise known as r or as r ur is 2. which we denote by S_2, is the disk x^2+y^2<=16. If you have questions or comments, don't hestitate to xy plane. The second operation is the divergence, which relates the electric ﬁeld to the charge density: divE~ = 4πρ . in the Let F(x,y,z)= be a vector The surface field whose components P, Q, and R have continuous partial derivatives. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. Let R be a region in xyz space with surface S. Let n denote the unit normal vector to S pointing in the outward direction. EXAMPLE 4 Find a vector field whose divergence is the given F function .0 Ba b consists of two pieces. Stokes' theorem is a vast generalization of this theorem in the following sense. 0<=theta<= 2*pi. {\left( { – \cos \theta } \right)} \right|_0^\pi } \right] \cdot}\kern0pt{ \left[ {\left. The surface integral of F In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. . Hence, the surface integral on S_2 is 0. . So the Divergence Theorem for Vfollows from the Divergence Theorem for V1 and V2. The surface of the region R technical restrictions on the region R and the surface S; see the Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. z direction. Remember that Green's theorem applies only for … It follows that that the bottom of R, But opting out of some of these cookies may affect your browsing experience. Gauss’s Law – Field Between Parallel Conducting Plates. We use the divergence theorem to convert the surface integral into a triple integral. In general $n$-dimensional space, Stokes' theorem relates a 1-dimensional line integral, two a 2-dimensional surface integral. In cartesian Because the only quantity for which the integral is 0, is 0 itself, the expression in the integrand can be set to 0. 0<=x^2+y^2<=16. is valid for regions bounded The above volume integral where the unit normal vector n points away from the region You take the derivative, you get negative z. Show your work a. b. becomes: (There are several ways in the which the integrals can be ordered. Divergence Theorem. It means that it gives the relation between the two. So that's right. inner integral is, (Note that r is held constant). satisfying 0<=z<=16-x^2-y^2 and F=. This category only includes cookies that ensures basic functionalities and security features of the website. The intersection of the parabaloid with the z plane is the We will convert Maxwell's four equations from integral form to differential form by using both the Divergence Theorem and Stokes' Theorem. The flux across the boundary, so the flux is … the flux web page shows that if the normal vector points in the by ellipsoids, spheres, and rectangular boxes, for example. 1. Let $$G$$ be a three-dimensional solid bounded by a piecewise smooth closed surface $$S$$ that has orientation pointing out of $$G$$ and let, ${\mathbf{F}\left( {x,y,z} \right) } = { \Big( {P\left( {x,y,z} \right),}}\kern0pt{{Q\left( {x,y,z} \right), }}\kern0pt{{R\left( {x,y,z} \right)} \Big)}$. Contributors and Attributions. Use outward normal n. Solution: Given the ugly nature of the vector field, it would be hard to compute this integral directly. Mathematically, ʃʃʃ V div A dv = ʃ ʃʃ V (∆ .A) d v = ʃ ʃ S A .ds Let F(x,y,z)= be a vector field whose components P, Q, and R have continuous partial derivatives. On S_2, F==, since S_2 is We'll assume you're ok with this, but you can opt-out if you wish. Consider a small volume of space, where the divergence of the electric field is positive. The Divergence Theorem relates surface integrals of vector fields to volume integrals. be a vector field whose components have continuous partial derivatives. R. Let us first consider the surface S_1. Then, Let’s see an example of how to use this theorem. It converts the electric potential into the electric ﬁeld: E~ = −gradφ = −∇~ φ . The circular symmetry of the region R suggests we convert These cookies will be stored in your browser only with your consent. (which is also 1 and 2 dimensional if $n=2$) $\endgroup$ – mlk May 30 '17 at 12:59 integrals to surface Our previous Introduction. Let →F F → be a vector field whose components have continuous first order partial derivatives. references for the details. For a normal vector pointing in the positive z direction the is the divergence of the vector field $$\mathbf{F}$$ (it’s also denoted $$\text{div}\,\mathbf{F}$$) and the surface integral is taken over a closed surface. the left in the figure above. We also use third-party cookies that help us analyze and understand how you use this website. \textbf{0} = 0 .\] The following theorem shows that this will be the case in general: Recall that the flux was measured via a line integral, and the sum of the divergences was measured through a … In fact, since S_2 lies in the plane z=0, the unit Verify the Divergence Theorem in the case that R is the region These cookies do not store any personal information. In this way, it is analogous to Green's theorem, which equates a line integral with a double integral over the region inside the curve. … S. Let n denote the unit normal vector to S pointing in the outward The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus. F== is, It follows that the relevant volume integral is. Hence, 128*pi. Necessary cookies are absolutely essential for the website to function properly. points in the positive z direction. Example 16.9.2 Let ${\bf F}=\langle 2x,3y,z^2\rangle$, and consider the three-dimensional volume inside the cube with faces parallel to the principal planes and opposite corners at $(0,0,0)$ and $(1,1,1)$. The partial derivatives with respect to x, y and z are converted into the ones with respect to ρ, φ and z. integrals of vector fields. It follows that. Convert the equation to differential form. A multiplier which will convert its divergence to 0 must therefore have, by the product theorem, a gradient that is multiplied by itself. If (x, y, z) (x, y, z) is a point in space, then the distance from the point to the origin is r = x 2 + y 2 + z 2. r = x 2 + y 2 + z 2. i. The fundamental theorem of calculus states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: ∫ = − (). Cartesian, Cylindrical and Spherical along with an intuitive explanation. { \frac{\partial }{{\partial z}}\left( z \right)} \right]dxdydz} }= {\iiint\limits_G {\left( {1 + 1 + 1} \right)dxdydz} }= {3\iiint\limits_G {dxdydz} . F r = 1 r 2 〈 x r, y r, z r 〉. direction. You also have the option to opt-out of these cookies. ∇⋅F = ∂P ∂x + ∂Q ∂y + ∂R ∂z is the divergence of the vector field F (it’s also denoted divF) and the surface integral is taken over a closed surface. On the other hand, the divergence theorem relates an $n-1$-dimensional "hypersurface"-integral to an $n$-dimensional volume integral. Let F r F r denote radial vector field F r = 1 r 2 〈 x r, y r, z r 〉. The The principal utility of the Divergence Theorem is to convert problems that are defined in terms of quantities known throughout a volume into problems that are defined in terms of quantities known over the bounding surface and vice-versa. {\left( {\frac{{{r^2}}}{2}} \right)} \right|_0^a} \right] }= {6\pi {a^2}.}\]. Displaying divergence theorem PowerPoint Presentations. In order to use the divergence theorem, we need to close off the surface by inserting the region on the xy-plane "inside" the paraboloid, which we will call . Use the divergence theorem to rewrite the left side as a volume integral. }\], \[{\iint\limits_S {\mathbf{F} \cdot d\mathbf{S}} }= {\iiint\limits_G {\left( {\nabla \cdot \mathbf{F}} \right)dV} }= {\iiint\limits_G {\left[ {\frac{\partial }{{\partial x}}\left( x \right) + \frac{\partial }{{\partial y}}\left( y \right) }\right.}}+{{\left. The Divergence Theorem relates relates volume integrals to surface integrals of vector fields. Let us denote the paraboloid by S_1. In particular, let be a … 2z, and then minus z squared over 2. In this approach, you start with the divergence formula in Cartesian then convert each of its element into the cylindrical using proper conversion formulas. surface integral equals 128*pi. The Divergence Theorem The Divergence Theorem relates flux of a vector field through the boundary of a region to a triple integral over the region. <0,0,-1>. This article discusses its representation in different coordinate systems i.e. It is mandatory to procure user consent prior to running these cookies on your website. Let R be a region in xyz space with surface This is an open surface - the divergence theorem, however, only applies to closed surfaces. We compute the two integrals of the divergence theorem. The volume integral of the divergence of a vector field over the volume enclosed by surface S isequal to the flux of that vector field taken over that surface S.” Hence, this theorem is used to convert volume integral into surface integral. to cylindrical coordinates. on the entire surface is Example1 Let V be a spherical ball of radius 2, centered at the origin, with a concentric ball of radius 1 removed. Must Evaluate Symmetry PPT. divergence theorem, which we really derived from Green's theorem, told us that the flux across our boundary of this region-- so let me write that out. The Divergence Theorem relates relates volume Now, let us compute the volume integral. of F. There are various Properties and Applications of Surface Integrals. negative direction, then the surface integral equals -128*pi.

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# Basic Geometry : How to find circumference

## Example Questions

### Example Question #21 : How To Find Circumference

Find the circumference of a circle with a radius of .

Explanation:

Recall the formula for finding the circumference of a circle:

We can substitute in the value for the radius in order to find the circumference of the circle in question.

Solve.

Simplify.

### Example Question #21 : How To Find Circumference

Find the circumference of a circle with a radius of .

Explanation:

Recall the formula for finding the circumference of a circle:

We can substitute in the value for the radius in order to find the circumference of the circle in question.

Solve.

Simplify.

### Example Question #22 : How To Find Circumference

Find the circumference of a circle with a radius of

Explanation:

Recall the formula for finding the circumference of a circle:

We can substitute in the value for the radius in order to find the circumference of the circle in question.

Solve.

Simplify.

### Example Question #23 : How To Find Circumference

Find the circumference of a circle with a radius of .

Explanation:

Recall the formula for finding the circumference of a circle:

We can substitute in the value for the radius in order to find the circumference of the circle in question.

Solve.

Simplify.

### Example Question #24 : How To Find Circumference

Find the circumference of a circle with a radius of .

Explanation:

Recall the formula for finding the circumference of a circle:

We can substitute in the value for the radius in order to find the circumference of the circle in question.

Solve.

Simplify.

### Example Question #214 : Basic Geometry

What is the circumference of a circle whose radius is .

Explanation:

To find circumference, you must use the follow equation.

### Example Question #25 : How To Find Circumference

Find the circumference of a circle that is inscribed in a square with side lengths of .

Explanation:

Notice that when a circle is inscribed in a square, the side length of the square is also the diameter of the circle.

Recall how to find the circumference of a circle:

Plug in the given diameter to find the circumference.

### Example Question #26 : How To Find Circumference

Find the circumference of a circle that is inscribed in a square that has side lengths of .

Explanation:

Notice that when a circle is inscribed in a square, the side length of the square is also the diameter of the circle.

Recall how to find the circumference of a circle:

Plug in the given diameter to find the circumference.

### Example Question #21 : How To Find Circumference

Find the circumference of a circle that is inscribed in a square that has side lengths of .

Explanation:

Notice that when a circle is inscribed in a square, the side length of the square is also the diameter of the circle.

Recall how to find the circumference of a circle:

Plug in the given diameter to find the circumference.

### Example Question #25 : How To Find Circumference

Find the circumference of a circle that is inscribed in a square that has side lengths of .


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# Can $F=qvB$ be applied to a solenoid and a parallel moving magnet?

If you take a scenario like this,

Charge gathers at either end, and if you could somehow form a complete circuit without there being a canceling effect, there would be a current flowing. This makes sense to me, the charges are moving in a B field with constant velocity, so F=qvB the electrons are forced down the rod.

But then, my textbook states for working out the emf across the conductor can be modelled using Faradays law, this doesn't make sense to me because there is not changing magnitude of flux in the conductor.

The derivation they use is as follows:

Displacement, $$s=v\Delta t$$

Area of flux cut, $$A=lv\Delta t$$

So $$\Delta \Phi = BA = Blv\Delta t$$

Faraday's law gives $$\epsilon = \frac{\Delta \Phi}{\Delta t} = \frac{Blv\Delta t}{\Delta t} = Blv$$

In then the converse fashion, in the situation where there is a solenoid moving relative to a permanent magnet:

Then here, Faradays law makes sense to me, as an emf is only induced when there is some change of flux over time, not constant relative motion, but then how can I relate this F=qvB Because the electrons are moving parallel to the magnetic field (roughly) so I assume there would be no significant motion of electrons.

My overall question is how are these two processes related? The first scenario makes sense to me physically, but not in relation to Faraday's law but for the second the converse is true. I understand the relation to Faraday's law but not how the magnetic field is moving electrons in the wire. I fear the second is to do with special relativity effects, and I will not understand the calculus but if that is the case, at least I know it is.

• Have you heard of the Biot-Savart law? It might help. – <NAME> Apr 7 at 14:41

This question certainly made me think a lot about magnetic fields, the magnetic force $$\vec{F}=q\vec{v}x\vec{B}$$, coils moving in (uniform and non-uniform) magnetic fields and relativity!
I think relativity is not so important here (maybe "a bit" in the second example) because it only tells us the $$\vec{B}$$-field is actually an electrical field.


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# Parallel and perpendicular lines in linear functions

### Parallel and perpendicular lines in linear functions

Parallel lines are lines with identical slope. In other words, these lines will never cross each other. Perpendicular lines will always pass through each other and form right angles at the interception. In this lesson, we will learn how to use information such as, points in lines and their slopes, to determine whether the lines are parallel, perpendicular or neither.

#### Lessons

Parallel
lines
$-$identical slope so they never intersect each other, unless overlapped.

Perpendicular
lines
$-$two lines form right angles to each other when they intersect. If the slope of first line is ${a \over b}$, the slope of perpendicular line is the slope of perpendicular line is $- {b \over a}$. The product of the two slopes is -1.
• 1.
Determine whether the three points A (-2,-1), B(0,4) & C(2,9) all lie on the same line.

• 2.
Determine the following slopes are parallel, perpendicular, or neither.
i) $m_1 = {2 \over 5}, m_2= {2 \over 5}$

ii) $m_1 = {1 \over5} , m_2 = - {5 \over 1}$

iii) $m_1 = {4 \over 7}, m_2 = {12 \over 21}$

iv) $m_1 =$undefined, $m_2 = 0$

v) $m_1 =mn^{-1}; m_2 =-m^{-1}b$

• 3.
Given the points of two lines, determine when the lines are parallel, perpendicular or neither.
a)
Line 1: (3,2) & (1,4); Line 2: (-1,-2) & (-3,-4)

b)
Line 1: (5,6) & (7,8); Line 2: (-5,-6) & (-7,-8)

• 4.
Show that the points A(-3,0), B(1,2) and C(3,-2) are the vertices of a right triangle.

• 5.
Show that the points A(-1,-1), B(3,0), C(2,4) and D(-2,3) are the vertices of a square.


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# Which mathematical criteria can I use to check either a surface is homeomorphic to a disk?

I'm currently working on 3D mesh segmentation.

And I want the segmented parts obtained to be homeomorphic to a disk (I mean easily unfoldable, so that we can map it as a 2D part without any points overlaping).

I have read a lot of papers on the subject, most of them talk about this homeomorphism to 2D disk, but none of them show how to prove it, or how to do it.

SO I really don't know what can I use on this surface to check if it is conformal to what I want.

Is there any formula using the points, the erea or the normals, or any criteria that shows that a given surface is topologycaly equivalent to disk?

Thank you all.

-
Doesn't it suffice to check that your mesh has only one boundary curve? – lhf Apr 28 '11 at 19:16
1 boundary component + Euler characteristic equal to $1$ – user8268 Apr 28 '11 at 19:23
How about connected and Euler-Characteristic 1 – <NAME> Apr 28 '11 at 19:26
thanks all of you guys ;) – Gatsu Apr 28 '11 at 20:27

## 1 Answer

The comments are right... this is easy. You first need to find the (directed) boundary edges of the mesh (which can be done by iterating over all faces of the mesh and canceling out those edges that appear twice in opposite directions, the remaining ones will be on the boundary). Secondly, you search for closed loops formed by boundary edges (build a lookup table that maps every boundary vertex to its successor, based on the list of boundary edges you already have, and follow this chain until you meet a vertex that you've encountered before). If there is only one loop, the patch is disk-like.

-
Thanks a lot guys, for your answers. I tough I got it at one moment, but I still need to fix a last point. For example, if we consider a sphere built with meshes, and a (somehow) small hole on the surface of the sphere. We can say that there is only one boundary curve on this structure, and it will be the contour of the hole. Can we still say that this mesh is homeomorphic to disk? Because I think to map it, I'll need to cut it anyway. Thank you again ;) – Gatsu Apr 28 '11 at 20:16
Doesn't a torus with a hole in it also have only one boundary component? I think you need to check Euler characteristic as well. @Gatsu: Yes, a sphere with a hole in it is homeomorphic to a disk. – MartianInvader Apr 28 '11 at 20:20
Ok, but since the example you gave might occur, I'd better to check the Euler Characteristic as well, right? And this last has to be equal to 1 to be topologically equal to a disk. Thank you a lot, I think it will be enough for my problem. – Gatsu Apr 28 '11 at 20:26


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The paper considers the estimation of welfare measures when the functional form of demand is unknown. An adaptation of an argument of Gallant (1987) is used to show that welfare estimators based on a Fourier functional form for demand will be consistent under weak assumptions. Simulation evidence is presented for equivalent variation. True demand is a generalized Box–Cox function, estimated demand is a Fourier form, and equivalent variation is estimated by applying Vartia's (1983) algorithm to the estimated demand function. The estimator of equivalent variation has small asymptotic bias in the case of the assumed family of data generating processes.

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#### Distributed Detection of Cycles

##### <NAME>, <NAME>

Distributed property testing in networks has been introduced by Brakerski and Patt-Shamir (2011), with the objective of detecting the presence of large dense sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016) have shown how to detect 3-cycles in a constant number of rounds by a distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown how to detect 4-cycles in a constant number of rounds as well. However, the techniques in these latter works were shown not to generalize to larger cycles $C_k$ with $k\geq 5$. In this paper, we completely settle the problem of cycle detection, by establishing the following result. For every $k\geq 3$, there exists a distributed property testing algorithm for $C_k$-freeness, performing in a constant number of rounds. All these results hold in the classical CONGEST model for distributed network computing. Our algorithm is 1-sided error. Its round-complexity is $O(1/\epsilon)$ where $\epsilon\in(0,1)$ is the property testing parameter measuring the gap between legal and illegal instances.

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# Is there any main method for finding norm of function in $L_1$ space?

Is there any main method for finding norm of function in $L_1$ space? For example :

$f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$

-
Welcome to Math.SE! Please edit your question to make it more precise: you say functional but give an example of a function. Which one are you asking about? –  user53153 Jan 6 '13 at 18:16
i am sorry,i typed wrong. i meant function –  Javidan Jan 6 '13 at 18:17
Direct evaluation works for your example. $\|\sin\|_1 = \int_{-\pi}^\pi |\sin x| dx = 2 \int_0^\pi \sin x dx = 4$. –  copper.hat Jan 6 '13 at 18:21
then for finding norm of function on $L_1[a,b]$ space, i need to integrate it from a to b ? –  Javidan Jan 6 '13 at 18:31
@Javidan: Using the definition of the norm doesn't hurt. In some contexts there may be other methods. Make sure that you integrate the absolute value of the function, i.e., use the definition of the $L_1$ norm. –  <NAME> Jan 6 '13 at 18:37

By definition, if $f \in L^p(X)$, then $$\| f \|_p := \left(\int_X |f|^pd\mu\right)^{1/p}.$$
In this specific case case $p=1, X=[-\pi,\pi], f=\sin$, and $\mu$ is the Lebesgue measure we have (as @copper.hat points out) $$\|\sin||_1 = \int_{-\pi}^{\pi}|\sin(x)|dx = 2\int_0^\pi \sin x dx = 4.$$


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# Counting operations for a matrix power

edited January 22 in Problems

This is a bit of a variation on problem 1.7.2 from our text.

Let $A \in \mathbb R^{n\times n}$. How many multiplications are required to compute $A^8$?

$$\frac{(n^3)-n}{3}, n = 8$$
• @dan Hmm... $n$ refers to the dimension of the matrix, not the matrix power. It's not 8; rather; it's unspecified so that your final answer should depend upon $n$.


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# Predicting the mass of hydrogen

After spinning around the atomic mass equation for calculating neutron separation energies, I have run into somewhat of a conundrum. It appears that the mass of the simplest of the examples, hydrogen is not correctly produced, yet I cannot put my finger on what is the reason for this. Let me elaborate: the atomic mass of some nucleus $$^A _Z X_N$$ is defined as

$$m(A,Z) = Z\cdot (m_p + m_e)+ (A-Z)\cdot m_n -B/c^2,$$

where subscripts $$m_p$$, e and n mean proton, electron and the neutron masses, respectively. And where B is the net binding energy of a nucleus. If I take the hydrogen atom, and try to predict its mass this way, I find, unsurprisingly

$$m(^1 H) = m_p + m_e - B/c^2$$

However, as there is only one nucleon in hydrogen, the contributions to B do not feature a any term from nuclear binding, only the binding energy of the single atomic state in hydrogen, i.e. the ionization energy of hydrogen is featured. The calculation does not add up, however.

Proton mass is 938.272 MeV/$$c^2$$, electron mass is 511 keV/$$c^2$$ and the ionization energy of hydrogen is 14 eV. Using these values gives an atomic mass of 938.8 MeV/$$c^2$$ for hydrogen, but the measured atomic mass of hydrogen is 939.0 MeV$$/c^2$$. Is there an explanation for the surprising 200 keV/$$c^2$$ discrepancy?

• What is your source for the atomic mass being 939.0 MeV/$c^2$? As far as I know, the accepted value is 1.00784 amu = 938.8 MeV/$c^2$. – Thorondor Sep 30 '20 at 13:19
• The place to find the latest news in atomic mass is the Atomic Mass Evaluation 2012 papers, easily found via google or through the original in Chinese Physics C 36(12) 1287-1602 (part I) and 1603-2014 (part II with the mass tables). – <NAME> Sep 30 '20 at 13:43

Without knowing the source of your numbers, I suspect that you have the atomic mass of elemental hydrogen, which is a mixture of naturally-occurring isotopes of hydrogen, primarily $$^1H$$ and $$^2H$$, with a trace of $$^3H$$.
You need to find the mass of the isotope $$^1H$$. That will fix the discrepancy.


text:
# How do you graph y=tan((1/2)x)?

Aug 4, 2018

$\text{ }$

#### Explanation:

$\text{ }$
color(red)(y=f(x)=[tan(x/2)]

Let us look at the standard form:

color(blue)(y=f(x)=a*tan(bx-c)+d

color(green)(a=1, b=1/2, c=0 and d=0

Period : color(red)(pi/b, with $b = \frac{1}{2}$

$\Rightarrow 2 \pi$

x-scale: $\textcolor{b l u e}{\frac{\text{Period}}{2}}$

$\Rightarrow \frac{2 \pi}{2} = \pi$

Let us look at the data table, with constraint $- 2 \pi < x < 2 \pi$

For the sake of focus and clarity contraint is used.

Graph the given trigonometric function:

Note:

1. Constraint $- 2 \pi < x < 2 \pi$

2. Parent graph of $y = f \left(x\right) = \tan \left(x\right)$ is also available in color $\textcolor{red}{\text{RED}}$ for comparison

3. Graph of the given function $y = f \left(x\right) = \left[\tan \left(\frac{x}{2}\right)\right]$ is in color(blue)("BLUE"

4. x-intercepts : They happen within the periods of $2 \pi$
i.e., $\left(- 4 \pi , - 2 \pi , 0 , 2 \pi , 4 \pi\right)$ etc

Hope you find this useful.


text:
# Qualitative Evaluation of Interest-Point-Detectors and Descriptors in Computer Vision

I'm writing on a qualitative evaluation study of interest-point-detectors and descriptors. I've read all Mikolajczyk papers as well as most of the surves from Datta et al. etc. Now I'm implementing an evaluation-tool in OpenCV. I'll take to images. One referred as source- and one as comparison-image.

1.) Detectors: Mikolajczyk uses the repeatability-criterion, correspondence count, matching score and an other metric to evaluate the performance of the detector. I would use repeatability and a correspondence count for matched regions.

2.) Descriptors: Here I would use the widely used Recall and Precision on the matched regions to describe the performance of the descriptor.

My question so far: Are these both good metrics for evaluation?

Now I'm trying to implement this is OpenCV and need a good eye from somebody to tell me if this could do. Here I use SURF-Detector and -Descriptor for testing the metrics. Recall and Precision are not implemented yet, but there is a function calculating this.

#include <opencv.hpp>
#include "opencv2/core/core.hpp"
#include "opencv2/features2d/features2d.hpp"
#include "opencv2/nonfree/features2d.hpp"
#include "opencv2/highgui/highgui.hpp"
#include <stdio.h>
#include <stdlib.h>
#include <iostream>

int startSURF() {

std::cout << "Starting " << std::endl;

if (!sourceImage.data) {
std::cout << "Source-Image empty" << std::endl;
return -1;
} else if (!comparisonImage.data) {
std::cout << "Comparison-Image empty" << std::endl;
return -1;
}

//Detect keypoint with SURF
int minHessian = 400;
cv::Mat sourceMatchedImage, comparisonMatchedImage;
std::vector<cv::KeyPoint> sourceKeypoints, comparisonKeypoints;

cv::SurfFeatureDetector surfDetect(minHessian);
surfDetect.detect(sourceImage, sourceKeypoints);
surfDetect.detect(comparisonImage, comparisonKeypoints);

//Calculate the SURF-Descriptor
cv::SurfDescriptorExtractor surfExtractor;
surfExtractor.compute(sourceImage, sourceKeypoints, sourceMatchedImage);
surfExtractor.compute(comparisonImage, comparisonKeypoints,
comparisonMatchedImage);

//Flann-Matching
cv::FlannBasedMatcher flann;
std::vector<cv::DMatch> matches;
flann.match(sourceMatchedImage, comparisonMatchedImage, matches);

//Repeatability and Correspondence-Counter
float repeatability;
int corrCounter;
cv::Mat h12;

std::vector<cv::Point2f> srcK;
std::vector<cv::Point2f> refK;

for (int i = 0; i < matches.size(); i++) {
srcK.push_back(sourceKeypoints[matches[i].queryIdx].pt);
refK.push_back(comparisonKeypoints[matches[i].queryIdx].pt);
}

std::cout << "< Computing homography via RANSAC. Treshold-default is 3" << std::endl;
h12 = cv::findHomography( srcK,refK, CV_RANSAC, 1 );

cv::evaluateFeatureDetector(sourceImage, comparisonImage, h12,
&sourceKeypoints, &comparisonKeypoints, repeatability, corrCounter);

std::cout << "repeatability = " << repeatability << std::endl;
std::cout << "correspCount = " << corrCounter << std::endl;
std::cout << ">" << std::endl;

std::cout << "Done. " << std::endl;
return 0;
}

I'm uncertain if this code works because SURF gets bad repeatability (e.g. 0.00471577) for my testing images with a rotation of almost 45°. Does anybody see a problem with the code?

Is there a way to evaluate the detector without RANSAC? I did not find a yet implemented method for this. Is the default of 3 a good threshold? I could overwrite it but the problem is that a good threshold can only be determined by experimental results. But I need a robust default-value for all detectors.

I think I definitively need the homography. But I never found a way to set the localisation-error ε for the repeatability-measure. Is there a way to set ε?

A lot of questions. I hope I was specific enough but if there is a question please comment and I will do my best to answer in order get help here. I wish you <NAME> Christmas!