31edo: Difference between revisions

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'''31 equal divisions of the octave''' ('''31edo'''), or '''31(-tone) equal temperament''', '''tricesimoprimal meantone temperament''' ('''31tet''', '''31et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 31 [[equal]]ly large steps. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.7 [[cent]]s. The term ''tricesimoprimal'' was first used by [[Adriaan Fokker]].

== Basic theory ==

31edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, ~~31-et~~ 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]]. (Another way 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.)

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]]. Another way 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.

One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (such as [[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.

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== Notations ==

From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:

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== Just approximation ==

=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

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== Relationship to 12-edo ==

Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

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31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next ETs in those subgroups are 72, 72, 41, and 46, respectively.

31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next ETs doing better in those subgroups are 72, 72, 41, and 46, respectively.

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 ~~{{monzo|-15 0 -2 7}} (~~[[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.

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.

=== Commas ===

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

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

{| class="commatable wikitable center-all left-3 right-4 left-6"

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== Music ==

* [[31-edo compositions|An alphabetical list of Tricesimoprimal Compositions]].

* [http://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 Aire #2 in 31-equal temperament] by [[Jon Lyle Smith]] {{dead link}}

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=== By [[Cam Taylor]] ===

* [https://soundcloud.com/camtaylor-1/enharmonic-melody-for-guitar Enharmonic melody for guitar]

* [https://soundcloud.com/camtaylor-1/what-use-is-a-boy What use is a boy]

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=== By Johann alias [[Circular17]] ===

* [https://d.tube/v/circular17/QmbyDumQJNH3MYZvphivVPSELW3XSkQPDt2CtWqdD6giTm Mystère et tolérance]

* [https://d.tube/v/circular17/QmfGU62ozp4GcuGVbSx9QnmkJLtnHvZjtyJ91B3un447Hf Wave from the past]

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=== By [[Zhea Erose]] ===

* [https://www.youtube.com/watch?v=wvLLV5qle48 Glitterdance]

* [https://www.youtube.com/watch?v=8Ol2gzkcprE Divinate]

== See ~~Also~~ ==

== See also ==

=== Pedagogy ===

The [[MicroPedagogyCollective]] is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.

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=== Books ===

{{External image| http://ronsword.com/images/TSG_sm.jpg {{dead link}} }}

[http://www.ronsword.com/books.html Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009.] {{dead link}} - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.

=== Articles ===

* [[Wikipedia: 31 equal temperament]]

* [http://www.huygens-fokker.org/docs/beerart.html de Beer, Anton, ''The Development of 31-tone Music''] [http://www.webcitation.org/5xeFzBM9b Permalink]

* [http://www.huygens-fokker.org/docs/fokkerorg.html Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ''] [http://www.webcitation.org/5xeG6Tmli Permalink]

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=== Videos ===

* [https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' – Youtube] by Stephen Weigel – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.

=== Software ===

* [http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]

* [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano — Mini synthesizer in Pixilang]

@@ Line 13: / Line 13: @@
 | Augmented 1sn = 2\31 = 77¢
 }}
+{{Wikipedia| 31 equal temperament }}
 '''31 equal divisions of the octave''' ('''31edo'''), or '''31(-tone) equal temperament''', '''tricesimoprimal meantone temperament''' ('''31tet''', '''31et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 31 [[equal]]ly large steps. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.7 [[cent]]s. The term ''tricesimoprimal'' was first used by [[Adriaan Fokker]].
 == Basic theory ==
-{{Odd harmonics in edo|edo=31}}
+{{Primes in edo|31}}
 edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).
-Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et 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]]. (Another way 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.)
+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]]. Another way 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.
 One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (such as [[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.
@@ Line 454: / Line 455: @@
 == Notations ==
 From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:
@@ Line 461: / Line 461: @@
 == Just approximation ==
 === 15-odd-limit interval mappings ===
 The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
@@ Line 626: / Line 625: @@
 == Relationship to 12-edo ==
 Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
@@ Line 822: / Line 820: @@
 |}
-et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next ETs in those subgroups are 72, 72, 41, and 46, respectively.
+et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next ETs doing better in those subgroups are 72, 72, 41, and 46, respectively.
-edo 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 {{monzo|-15 0 -2 7}} ([[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.
+edo 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.
 === Commas ===
-EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 31 49 72 87 107 115 }}, comma values rounded to 5 significant digits.)
+edo [[tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 31 49 72 87 107 115 }}, comma values rounded to 5 significant digits.
 {| class="commatable wikitable center-all left-3 right-4 left-6"
@@ Line 1,145: / Line 1,143: @@
 == Music ==
 * [[31-edo compositions|An alphabetical list of Tricesimoprimal Compositions]].
 * [http://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 Aire #2 in 31-equal temperament] by [[Jon Lyle Smith]] {{dead link}}
@@ Line 1,161: / Line 1,158: @@
 === By [[Cam Taylor]] ===
 * [https://soundcloud.com/camtaylor-1/enharmonic-melody-for-guitar Enharmonic melody for guitar]
 * [https://soundcloud.com/camtaylor-1/what-use-is-a-boy What use is a boy]
@@ Line 1,167: / Line 1,163: @@
 === By Johann alias [[Circular17]] ===
 * [https://d.tube/v/circular17/QmbyDumQJNH3MYZvphivVPSELW3XSkQPDt2CtWqdD6giTm Mystère et tolérance]
 * [https://d.tube/v/circular17/QmfGU62ozp4GcuGVbSx9QnmkJLtnHvZjtyJ91B3un447Hf Wave from the past]
@@ Line 1,176: / Line 1,171: @@
 === By [[Zhea Erose]] ===
 * [https://www.youtube.com/watch?v=wvLLV5qle48 Glitterdance]
 * [https://www.youtube.com/watch?v=8Ol2gzkcprE Divinate]
-== See Also  ==
+== See also  ==
 === Pedagogy ===
 The [[MicroPedagogyCollective]] is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.
@@ Line 1,188: / Line 1,181: @@
 === Books ===
 {{External image| http://ronsword.com/images/TSG_sm.jpg {{dead link}} }}
 [http://www.ronsword.com/books.html Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009.] {{dead link}} - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.
-=== Articles  ===
+=== Articles ===
-* [[Wikipedia: 31 equal temperament]]
 * [http://www.huygens-fokker.org/docs/beerart.html de Beer, Anton, ''The Development of 31-tone Music''] [http://www.webcitation.org/5xeFzBM9b Permalink]
 * [http://www.huygens-fokker.org/docs/fokkerorg.html Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ''] [http://www.webcitation.org/5xeG6Tmli Permalink]
@@ Line 1,204: / Line 1,194: @@
 === Videos ===
 * [https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' – Youtube] by Stephen Weigel – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.
-=== Software  ===
+=== Software ===
 * [http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]
 * [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano — Mini synthesizer in Pixilang]