31edo: Difference between revisions

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{{EDO intro|31}}

31edo's perfect fifth is flat of the just interval [[3/2]] (over five cents), as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo, and a zeta peak integer edo]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta edo]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 11-[[odd prime sum limit|odd-prime-sum-limit]].

One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.  

31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

{{Harmonics in equal|31|columns=12}}

{{Harmonics in equal|31|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}

31edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]].

{{Main|31edo/Individual degrees}}

{{See also|Table of 31edo intervals}}

{| class="wikitable center-all right-2 left-3"

! Degree

! colspan="3" | [[Ups and Downs Notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vd2)

! colspan="3" | [[SKULO interval names|SKULO]] notation (S or {{nowrap|U {{=}} 1}})

| 0.00

| 38.71

| 77.42

| 116.13

| 154.84

| 193.55

| 232.26

| 270.97

| 309.68

| 348.39

| 387.10

| 425.81

| 464.52

| 503.23

| 541.94

| 580.65

| 619.35

| 658.06

| 696.77

| 735.48

| 774.19

| 812.90

| 851.61

| 890.32

| 5/3, 42/25

| 929.03

| 967.74

| 7/4, 125/72

| 1006.45

| 16/9, 9/5, 34/19, 25/14

| 1045.16

| 1083.87

| 1122.58

| 1161.29

| 1200.00

Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:

! Monzo Format

| {{monzo|a b 0 1}}

| {{monzo|a b}} where {{nowrap|b &lt; &minus;1}}

| {{monzo|a b -1}}

| {{monzo|a b 0 0 1}}

| {{monzo|a b 0 0 -1}}

| {{monzo|a b 1}}

| {{monzo|a b}} where {{nowrap|b &gt; 1}}

| {{monzo|a b 0 -1}}

! JI Chord

! Notes of C Chord

| zo

| {{dash|C, E{{sesquiflat2}}, G|s=hair|d=med}}

| C subminor

| gu

| {{dash|C, E♭, G|s=hair|d=med}}

| ilo

| {{dash|C, E{{demiflat2}}, G|s=hair|d=med}}

| C neutral

| yo

| ru

| {{dash|C, E{{demisharp2}}, G|s=hair|d=med}}

| C supermajor

For a more complete list of chords, see [[31edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].

{{Sharpness-sharp2}}

! Name

=== Circle-of-fifths notation ===

[[Circle-of-fifths notation]] uses double sharps and double flats:

! Name

While using double sharp and double flat may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:

In 12edo, the enharmonic equivalences include C♯ = D♭, E♯ = F, and E = F♭. But in 31edo we have:

31edo can be notated with a seperate semi/sesqui sharp/flat chain (like [[17edo]]), with its own enharmonic circle of fifths.

[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]

=== Sagittal notation ===

The Revo flavor of Sagittal notation from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]]:

31edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 18\31 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 31edo’s [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.  

{{Q-odd-limit intervals|31}}

| +1.63

| 1.64

| 4.22

| +0.98

| 1.63

| 4.20

| +0.83

| 1.43

| 3.70

| +1.21

| 1.49

| 3.84

| +0.50

| 2.07

| 5.35

 

31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency #Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and the quince comma [[823543/819200]], thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]]. In the 11-limit, 31edo can be defined as the unique temperament that tempers out [[81/80]], [[99/98]], [[121/120]], and [[126/125]], and it supports [[orwell]], [[mohajira]], and the relatively high-accuracy temperament [[miracle]]. In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.

31edo [[tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 31 49 72 87 107 115 }}, comma values rounded to 5 significant digits.

| [[Ampersand]]

| [[Meter comma]]

| Schismina

* [[Syntonic&ndash;31 equivalence continuum]]

| [[1L 14s]], [[15L 1s]]

| [[1L 9s]], [[10L 1s]], [[10L 11s]]

| [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]]

| [[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]]

| [[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]]

| [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]]

| [[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]]

| [[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]]

| [[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]]

| [[3L 2s]], [[3L 5s]], [[3L 8s]], <br>[[3L 11s]], [[14L 3s]]

| [[3L 2s]], [[5L 3s]], <br>[[5L 8s]], [[13L 5s]]

| [[2L 3s]], [[5L 2s]], <br>[[7L 5s]], [[12L 7s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[2L 9s]], [[2L 11s]], [[2L 13s]], <br>[[2L 15s]], [[2L 17s]], [[2L 19s]], <br>[[2L 21s]], [[2L 23s]], [[2L 25s]], <br>[[2L 27s]]

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[File:31edo array kalimba.jpg|none|thumb|640x640px|31edo array kalimba built by Tristan Bay; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]]