 upi using only number of semitones. Both of the examples I referred to above would be upi 13.
3) If I don't care about octave distances, but I do care about the order of notes, then I use "ordered pitch class intervals" or opci. This is probably the hardest one to envision at first. I will count the number of half steps it would take me to get from the first note up to the next note even if the interval went down in the actual music. Going from any C to any Db (that is to say from 0 to 1) would always be opci 1. Going from any Db to any C (i.e. 1 to 0) would always be 11. Going from D to G (2 to 7) would be 5 and G to D (7 to 2) would be 7. The main benefit of using numbers instead of note names (and, importantly, of starting our numbering system with 0 instead of 1) is that intervals can be easily figured out by simple subtraction. Subtract the first number from the second number (mod 12) and you have the opci. My four examples above: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 mod 12. The mod 12 part may be unfamiliar to you, but it's something you use every time you say that something happening three hours after 11 AM will happen at 2 PM. 11+3 is actually 14 of course, but we treat time like a circle, and roll back to 2.
4) Finally, and most commonly, we can refer to the interval in the most general way possible: the interval class (sometimes called the "unordered pitch class interval", but IC is pithier and has an obvious similarity to "pitch class." In this situation we just want to know the how to get from one note to the other in semitones in the closest way possible. The closest way to get from a Db (1) to a C (0) if we don't care which comes first is 1. All half steps, major sevenths, minor ninths, etc. will belong to this same family of intervals: interval class 1. Whole steps, minor sevenths, major 9ths, etc. are part of IC 2; minor thirds, major sixths etc. are IC 3; and so on. There are only 6 possible intervals classes (unless you
upi using only number of semitones. Both of the exemplar I refer to above would be upi 13. 
 3) If I don't care about octave distances, but I do care about the orderliness of notes, then I use " order pitch class time interval " or opci. This is probably the hardest one to visualize at first. I will count the number of half steps it would drive me to get from the first note up to the next note even if the time interval went down in the actual music. Going from any C to any Db (that is to allege from 0 to 1) would always be opci 1. Going from any Db to any coulomb (i.e. 1 to 0) would constantly be 11. Going from D to G (2 to 7) would be 5 and thousand to D (7 to 2) would be 7. The main benefit of using numbers alternatively of note names (and, importantly, of starting our numbering system with 0 instead of 1) is that intervals can be easily figured out by simple subtraction. Subtract the first number from the second number (mod 12) and you have the opci. My four examples above: 1 - 0=1; 0 - 1=-1 which is 11 mod 12; 7 - 2=5; 2 - 7=-5 which is 7 mod 12. The mod 12 part may be unfamiliar to you, but it's something you use every clock time you say that something happening three hour after 11 AM will find at 2 phase modulation. 11 + 3 is actually 14 of course, but we treat meter like a circle, and roll back to 2. 
 4) Finally, and most normally, we can refer to the interval in the most general way possible: the interval class (sometimes called the " unordered pitch class time interval ", but IC is pithy and has an obvious similarity to " pitch class. " In this situation we just want to acknowledge the how to get from one note to the other in semitone in the closest means possible. The closest room to get from a Db (1) to a C (0) if we don't manage which comes first is 1. All half tone, major sevenths, minor ninth, etc. will belong to this like family of intervals: time interval class 1. Whole steps, minor sevenths, major 9ths, etc. are part of IC 2; minor one-third, major sixths etc. are IC 3; and then on. There are only 6 possible intervals class (unless you
 upl using only number of stmitones. Both of jhw exam'les I deferred to above would be upi 13.
3) If I din't cqre about octave distavces, but P do care abont the order of iktes, thcu I uac "ordzrxd pitch class lntervals" os opci. This is pfoyably the hardest one to envision at first. O aill count the numnqr or half steps it would take me to gst from the first noye up to the next note evej if the interval went down in thg aceyal music. Gokng from any C to any Sb (that is to say from 0 to 1) wouud alcays be opcu 1. Goltg from any Db tj any C (i.e. 1 bp 0) wound alwaus be 11. Going fvom D to G (2 to 7) would be 5 and G to D (7 to 2) would be 7. The kann benefit of using nymvers hnstaad ud nute nemea (and, lmpkrtantly, or starting iur numbering systek rpyh 0 instead kf 1) is trat intervals can be easily figured out by simple subtraction. Subrract the first numbet from the second number (mod 12) and you have the opci. My four exam'lds cnjxw: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 mod 12. The mod 12 pawf kaj be unfamiliar tj you, but iy's spiething you ure evexg fime you say that domethigg happening trree hours after 11 AM will happeb at 2 PM. 11+3 is qctually 14 of coursz, but we trect timg like a circle, and roll back to 2.
4) Finally, anf most cojoonly, we can refdr no tve interval in the most gegeral way posxible: tfe imtervaj class (soletimcv called the "unordfred kitch wlass integval", but IC is pithier and has ei obvious simoldriny to "piteh claxs." In this sieuation we jusj want to know the how tk get fcom one note to the other ln semitones in the sloswst qay posrkble. The closext way to get from q Db (1) to a C (0) if wc don'j dare which comew furst is 1. All hakf rte[s, mejor favenths, minos nivthr, etc. dill belong go tnis same family of itterbals: interval clasx 1. Whole sjeps, minow sevenths, makor 9ths, etc. are pagt of IC 2; minpr jhirds, major sixths etc. are IC 3; and so oj. Tmere are only 6 powsible interrals classes (unless you
upi using only number of semitones. Both examples referred to would be upi care octave distances, but do care about order of notes, then I use pitch class intervals" or opci. This is probably the hardest one to envision first. I will count the number of half steps it would take me get the note to the next note even if the interval went down in the actual music. Going from C to any Db (that is to say 0 to 1) would be opci 1. Going from Db any C 1 0) always be 11. from D to G (2 to 7) would be 5 and G to D (7 to 2) be 7. benefit of numbers of names (and, importantly, our numbering system with 0 instead that intervals can be easily figured out by subtraction. Subtract first number from the second number 12) and you have the opci. My four above: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 mod 12. 12 part may be to you, but something use time say that happening three hours after 11 AM will happen at 2 PM. is actually 14 of course, but we treat time like and back to 2. Finally, and most commonly, can to the interval in general possible: (sometimes the pitch class interval", but is pithier and has an similarity to "pitch class." want to know the how to get from note to the other in semitones in closest way possible. The closest way to get from a Db (1) a C we don't care which comes first is 1. half steps, major sevenths, ninths, etc. will belong to this same family of interval 1. Whole minor sevenths, major etc. are part IC 2; minor sixths etc. IC and are only 6 possible intervals classes you
 upi using only number of semitOnes. Both of The exAmpLes i rEferRed tO above would be uPI 13.
3) If I Don't care about octave disTanceS, bUT I do CArE abouT the ordER oF NOteS, tHeN I uSe "ORdEred pItcH class iNtervals" or OpcI. THis is probablY ThE hardest onE to Envision at fiRst. i will cOuNt tHE numbEr oF half Steps iT Would tAke me to geT fROm the fIRst note UP To The nExt note even if the iNTeRVal went down in tHe actuAl MUsIC. goiNg fRom any C to aNy db (thaT Is to say FRoM 0 TO 1) WouLD always be opci 1. going from anY db tO any C (i.E. 1 tO 0) woULd alwaYs be 11. GOiNG frOm D to G (2 to 7) wouLd be 5 And G to D (7 to 2) Would bE 7. the main BEnefit oF using NumBerS insTEaD oF noTe NAmeS (AnD, imPOrtAntly, of sTaRtIng ouR numBERINg sySteM witH 0 instEad of 1) is that inTerVals CAn bE easiLy figUred OuT by siMple suBtracTiOn. Subtract the fiRst nUmber from The SeConD nUmber (MOd 12) and yOu hAve The opci. my four eXAmpLeS ABOvE: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 MoD 12. tHe Mod 12 part mAy be unFAmIlIAr to you, bUt It's SomeTHIng yoU use EVeRy time yoU say thAT sOmEthing hApPening ThRee HouRs aftER 11 AM wIll hapPen at 2 PM. 11+3 iS actuALly 14 of course, but WE treat time likE A cIRClE, And rOll Back to 2.
4) FinalLy, anD Most CommONlY, we CAn refEr to tHe INtERval in the most generaL wAy possIble: tHe interval claSs (sometimeS CALled the "uNordEReD Pitch class inteRval", bUt IC is pithIEr and has An obvIous simiLarity to "pITCh class." IN thIs sItuAtiON We Just want to knoW THe hoW tO get froM onE note to The OthEr iN seMiTones in thE closest WaY pOsSiBle. the clOSest way tO gEt fRoM a DB (1) to a C (0) IF we don'T care WhicH cOmES fiRst is 1. AlL HaLF StepS, mAjOr seVenThS, minoR ninTHs, eTc. will bElong to thIs sAMe faMiLy Of interVals: interval cLaSs 1. Whole stePs, MinOr seveNTHs, major 9tHs, etc. are part of IC 2; minor thIRds, majoR siXths eTc. arE IC 3; and so oN. ThEre are OnlY 6 PossibLe inteRvals ClAssES (UnlesS YOu
 upi using only number ofsemitones.  Both  of  th eexam ples  I referred to abov e would be upi 13.
3)If Ido n 't c a re  abou t octav e  d i s tan ce s,  bu tI  d o car e a bout th e order of  no te s, then I us e  " ordered pi tch  class inter val s" orop ci. Thisisproba bly th e  harde st one to  e n vision at firs t .  I  wil l count the numbe r  o f  half steps it  would  t a ke m e t o g et from th efirst note up to t h e n e xt note evenif the inte r val  wentdo wni n theactua lm usi c. Going fr om a ny C to a ny Db( that is to sayfrom 0  to  1)  wou l dal way sb e o p ci  1. Goi ng froman yDb to  any C ( i.e.  1to 0 ) wou ld always be11.  Goi n g f rom D  to G  (2to  7) w ould b e 5 a nd  G to D (7 to 2 ) wo uld be 7.  Th emai nbenef i t of u sin g n umbersinstead ofno t e na mes (and, importan tl y ,  o f starti ng our nu mb e ring sys te m w ith0 inste ad o f  1 ) is tha t inte r va ls  can be  e asilyfi gur edout b y  sim ple su btractio n. Su b tract the firs t  number fromt he s ec o nd n umb er (mod 12)  and youhave th e o p ci. M y fou re xa m ples above: 1-0=1;0- 1=-1 w hichis 11 mod 12;  7-2=5; 2- 7 = - 5 whichis 7 mo d  12. The mod 1 2 par t may be u n familiar  to y ou, butit's some t h ing youuse  ev ery  ti m e  y ou say that s o m ethi ng  happen ing  threehou rsaft er11  AM willhappen a t2PM .11+ 3 isa ctually14  of  c our se, b u t we t reattime  l ik e  acircle, an d roll  b ac k to  2. 
4 ) Fin ally ,  an d mostcommonly,  we canre fe r to th e interval in  t he most ge ne ral  way p o s sible: t he interval class (some t imes ca lle d the  "un ordered p itc h clas s i n terval ", but  IC i spit h i er an d ha s a nobvious si m i lar ity t o"pit ch clas s." In this situat i onwe just wanttoknow t he  ho w  t o  ge tf rom o ne note to theother in s em i to nes in the clo se st waypossibl e. Th e  closes t way toget fromaDb ( 1 )  to  a C (0) i f we don 't care w h ich c o me s fir stis 1.Al l h alf s teps,m ajo r sev enths,  m inor n inths ,etc. wil l belong to this same f amilyof in ter vals: int erv a l c lass 1. W hole  steps, mi nor  se venth s,m ajor9ths ,  e tc. are p arto f IC 2; m i no r t h i rd s, major si x t h s e tc. a reI C 3; a nd s o on. There are o n ly 6 possibleinte r v als  cl a sses  ( unless you
 upi_using only_number of semitones. Both_of the_examples_I referred_to_above would be_upi 13.
3) If_I don't care about_octave distances, but_I_do care about the order of notes, then I use "ordered pitch class intervals"_or_opci. This_is_probably_the hardest one to envision_at first. I will count_the number_of half steps it would take me to_get_from the first_note up to the next note even if the_interval went down in the actual_music. Going from_any_C_to any Db (that_is to say from 0 to_1) would always be opci 1._Going from any Db to any C_(i.e. 1 to 0) would always_be 11. Going from D_to G_(2 to 7) would be_5 and G_to D_(7 to 2)_would be 7. The main benefit_of using numbers_instead of note names (and, importantly,_of_starting our numbering_system_with_0 instead_of 1) is_that_intervals can_be_easily figured out by simple subtraction._Subtract_the first number from the second number_(mod 12) and you_have_the opci. My four_examples above: 1-0=1; 0-1=-1 which_is 11 mod 12; 7-2=5; 2-7=-5_which is_7 mod_12. The mod 12 part may be unfamiliar to you, but_it's something you use every time_you say that something_happening three_hours_after 11 AM_will_happen at_2 PM. 11+3 is actually 14 of_course, but_we treat time like a circle,_and roll back to_2.
4)_Finally, and most commonly, we can_refer to the interval in the_most general way possible: the_interval_class_(sometimes called the "unordered pitch_class interval", but IC is pithier_and has an_obvious similarity to "pitch class." In this_situation_we just want to know the_how_to get from one note to_the_other_in semitones in the closest_way possible. The closest way to_get from a Db (1) to a C (0)_if we don't_care which comes first is_1._All_half steps, major sevenths, minor ninths, etc. will belong to_this same_family of intervals:_interval class 1. Whole steps, minor sevenths, major 9ths, etc._are part of IC 2; minor thirds,_major sixths etc. are IC 3; and so on. There are_only 6 possible intervals classes (unless you