64/63
64/63, the septimal comma (also Archytas' comma, or more simply and systematically the archytas comma or archy comma), is a small 7-limit superparticular comma which separates 9/8 and 8/7 and has the eighth square number as a numerator. It can be considered the 2.3.7-subgroup equivalent of the syntonic comma, and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between 32/27 and 7/6, and the difference between 81/64 and 9/7. Since its numerator is a power of 2, it is a Mersenne comma.
| Interval information |
Archytas' comma
rM, ruma
reduced,
reduced subharmonic
[sound info]
Temperaments
Tempering out this comma leads to superpyth temperament (sometimes called archy in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates 7/4 with 16/9. This means that the just dominant seventh chord, 1–5/4–3/2–16/9, and the harmonic seventh chord, 1–5/4–3/2–7/4, are equated to the same chord. Equal temperaments tempering out 64/63 include 12, 15, 17, 22, 27, 37, 49 and 59.
Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths octave reduced equal a relatively consonant major third. In the case of 81/80, the major third is 5/4, while with Archytas' comma, the major third is 9/7.
If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process.
See Archytas family for the family of rank-3 temperaments where it is tempered out. See Archytas clan for the clan of rank-2 temperaments where it is tempered out.
Comma pumps
The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord.
Notation
This interval is significant in the Functional Just System and Helmholtz–Ellis notation as the septimal formal comma which translates a Pythagorean interval to a nearby septimal interval.
Sagittal notation
In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal and is called the 7 comma, or 7C for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F . The upward version is called 1/7C or 7C up and is represented by .
Approximation
If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of 55/54, from which it differs by a keenanisma, or of 56/55, from which it differs by a werckisma. In addition, its incredible proximity to 1/44th of the octave – to the point where the 44-64/63 comma is tempered out in edos as large as tens of thousands – enables the tuning of ruthenium temperament. As a result, the major second of 22edo is a good approximation to 17/15, due to it being the mediant of 9/8 and 8/7, so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out 2025/2023.