Some symbols used for a quarter-tone flat are a downward-pointing arrow ↓, or a flat combined with an upward-pointing arrow ↑. Similarly, for a quarter-tone sharp, an upward-pointing arrow may be used, or else a sharp with a downward-pointing arrow. Three-quarter flat and sharp symbols are formed similarly. A further modern notation involves reversed flat signs for quarter-flat, so that an enharmonic tetrachord may be represented: D E F G , or A B C D . The double-flat symbol () is used for modern notation of the third tone in the tetrachord to keep scale notes in letter sequence, and to remind the reader that the third tone in an enharmonic tetrachord (say F, shown above) was not tuned quite the same as the second note in a diatonic or chromatic scale (the E expected instead of F). Scale Like the diatonic scale, the ancient Greek enharmonic scale also had seven notes to the octave (assuming alternating conjunct and disjunct tetrachords), not 24 as one might imagine by analogy to the modern chromatic scale. A scale generated from two disjunct enharmonic tetrachords is: D E F G || A B C D or, in music notation starting on E: , with the corresponding conjunct tetrachords forming A B C |D| E F G or, transposed to E like the previous example: . Tunings The precise ancient Pythagorean tuning of the enharmonic genus is not known. Aristoxenus believed that the pyknon evolved from an originally pentatonic trichord in which a perfect fourth was divided by a single "infix"—an additional note dividing the fourth into a semitone plus a major third (e.g., E, F, A, where F is the infix dividing the fourth E–A). Such a division of a fourth necessarily produces a scale of the type called pentatonic, because compounding two such segments into an octave produces a scale with just five steps. This became an enharmonic tetrachord by the division of the semitone into two quarter tones (E, E↑, F, A). Archytas, according to Ptolemy, Harmonics, ii.14—for no original writings by him survive—used 9:7, as in all three of his genera; here it is the mediant of 4:3 and 5:4, as (4+5):(3+4) = 9:7: hypate parhypate lichanos mese 4:3 9:7 5:4 1:1 | 28:27 |36:35| 5:4 | -498 -435 -386 0 cents Also according to Ptolemy, Didymus uses the same major third (5:4) but divides the pyknon with the