64/63: Difference between revisions

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== Temperaments ==

[[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 }}.

[[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, 17, 22, 27, 37, 49 and 59 }}.

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]].

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 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.

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.

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 ==

@@ Line 13: / Line 13: @@
 == Temperaments ==
-[[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 }}.
+[[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, 17, 22, 27, 37, 49 and 59 }}.
-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]].
+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 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.
+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.
-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 ==