64/63: Difference between revisions
Cleanup and amend on problematic statements |
expanded temperaments section, + comma pumps section, swap notation and approximation sections |
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== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]] | [[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. Equal temperaments tempering out 64/63 include {{EDOs| 12, 15, 17, 22, 27, 37, 49 and 59 }}. | ||
Archytas' comma is similar to Didymus' | Archytas' comma is similar to Didymus' 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 | 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. | ||
The archy temperament contains a version of the [[5L 2s|diatonic]] scale where the major third represents 9/7, and the minor third represents [[7/6]]. Tertian harmony can thus be used, with the major and minor triads representing [[~]][[14:18:21]] and ~[[6:7:9]] respectively, rather than the ~[[4:5:6]] and ~[[10:12:15]] triads in meantone. Since the fifth is sharp in archy rather than flat in [[meantone]], the large steps (major seconds) are wider, being around 212-222[[cent|{{c}}]] depending on the tuning. The small steps (minor seconds) are thus narrower, being around 44-71{{c}}. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to use milder tunings like [[17edo]] to avoid the small step being too narrow. | |||
In the full [[7-limit]], 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. This simplifies harmonic logic in temperaments like [[pajara]]. | |||
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. | |||
{{todo|add sound example}} | |||
== Notation == | == Notation == | ||
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=== Sagittal notation === | === Sagittal notation === | ||
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{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{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}. | In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{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{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}. | ||
== 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|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]]. | |||
== See also == | == See also == | ||