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{{Wikipedia|Septimal comma}} | {{Wikipedia|Septimal comma}} | ||
'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|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 [[ | '''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|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|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]]. | ||
== Temperaments == | == Temperaments == | ||
Tempering out this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, | [[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, [[36:45:54:64|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs| 12, 15, 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 reduction|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 you are 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 you are 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 gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, 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. | 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. | ||
== Approximation == | == 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 [[385/384|keenanisma]], or of [[56/55]], from which it differs by a [[441/440| | 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 [[385/384|keenanisma]], or of [[56/55]], from which it differs by a [[441/440|werckisma]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[septimal ruthenia|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]]. | ||
== Notation == | == Notation == | ||
Revision as of 11:20, 30 November 2025
| Interval information |
Archytas' comma
Ru comma
reduced,
reduced subharmonic
[sound info]
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.
Temperaments
Tempering out this comma equates 9/8 and 8/7, and also equates 7/4 with 16/9, so 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, 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 you are 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 gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, 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.
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.
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 .
See also
- Septimal comma (disambiguation page)
- Gallery of just intervals
- List of superparticular intervals