recursal/Fanatic-Fandom huggingface dataset:
22平均律

音楽において、22平均律は22-tet、22-edo、22etと呼ばれ、オクターブを均等に22個のラージステップに分割したものである。各ステップは54.55セントとなる。

22ステップに分割する考えは、19世紀の音楽家、RHM Bosanquetに起源があるように思われる。Bosenquetはインド音楽理論の、オクターブを均等ではなく22個に分割することにインスピレーションを受けた。そして均等に分割したとき、まあまあ正確な5リミットの音楽になることを発見したのである。引き続いて20世紀に、理論家であるJosé Würschmidtが19平均律の次のステップの可能性であることに気がついた。J. Murray Barbourは古典的なチューニングの歴史の著書、『Tuning and Temperament』でのべている。

22平均律は実際、4セントのTEエラー内となる5リミットに近似する、12と19平均律に次ぐ3番目の均等分割である。ゼータ・ピークと少なくともみなせる整数や平均律ギャップはないけども。少なくともさらにその上、5リミットを超えて、12や19にはなく、3セントのエラーで7や11リミットにも近づくことができる。31平均律の場合さらによいとはいえ、22平均律はまだこれらの高いリミットハーモニーとして許容できる。そして実際、22平均律は一貫した11リミットを表現する、最小の平均律である。加えて、22平均律は12と19に似ておらず、ミーントーンシステムでもない。これらの効果により22平均律は、より小さな音楽領域類の探求を推進する。例えば、小さな適した楽器の制作などである。

22平均律はまた、11平均律の2.7.9.11.15.17に3と5の響きを加えたものとして扱うこともできる。より正確な2.3.5.7.11.17サブグループテンペラメントを作ることができるのである。私たちはまた、31倍音のわずか半セントの近似としても扱うことができる。

# 音程のネーミングシステム

22平均律の音程はおそらく、SuperpythとPorcupineテンペラメントの両方から検討されるシステムについて考えるのが最も良い。それゆえ、各テンペラメントのメジャーとマイナーとしてカテゴライズすることは筋が通っている。Sはsuperpythを示唆し、pはPorcupineを示唆する。なぜならpはprocupine、またはnot perfectを代表し、完全音程のPはもはやこのシステムでは使用しない。代わりにPを除いて数字で、または数か「Neutralで読み取られる。例えば、P5は5となり、N5 = Perfect fifthはNeutral fifthとなる。

# 22平均律の音程と近似値

各周波数比の大きさが16以内で表現される純正音程は以下のようになる。これはedjirulerを用いて、[number of equal divisions=22, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.3]というパラメータで生成したものである。「The “neighborhood” of JI」の一覧はこちら（huygens-fokker）を参照のこと。

]

# 22平均律の特徴

ひょっとしたら22平均律の最も顕著な特徴は、81/80のシントニックをテンパーアウトしないことである。それゆえ、ミーントーンテンペラメントのシステムではない。12平均律は区別しないが、22がピタゴラスと5リミットの音程の数を区別することを意味する。例えば9/8と10/9という2つの全音など。実際、これらの区別は大げさに5リミットJIにおいて、たくさんのより鋭い、34平均律や41平均律、53平均律のようなものと大げさに比較される。

ダイアトニックスケールはsuperpythテンペラメントから生成される。ミーントーン・ダイアトニックスケール（LLsLLLs, 5L2s）のように同じメロディーの構成を持つにもかかわらず。それは5/4と6/5よりも9/7と7/6に近い3度をもつ。

164セント・「小全音のフラット」は、22平均律において重要な音程である。なぜなら11リミットにおける3つの異なった調和の周波数比、10/9、11/10、12/11としての機能を持つからである。したがって、極端に曖昧で柔軟である。そのトレードオフは、とても12平均律ピアノのひずみとなり、それゆえほとんどの12平均律の聞き手は、聞き馴染みのあるものである。単純な5リミットの音楽を22平均律に移行させたとしても、よりコンプレックスをもったハーモニーが必ず生じ、とても異なった響きに聞こえる。

# ランク22テンペラメント

悪い22EDOランク2テンペラメントのリスト

複雑な22EDOランク2テンペラメントのリスト

平均律とは異なった22EDOランク2テンペラメントのリスト

# コンマをなだらかにする

22平均律を< 22 35 51 62 76 81 |ヴァルとみなしたとき、次のリストのコンマをテンパーアウトする。
229edo

The *229 equal division* divides the octave into 229 equal parts of 5.240 cents each. In the 5-limit it tempers out the wuerschmidt comma, 393216/390625, and in the 7-limit 3136/3125, 6144/6125 and 2401/2400, so that it supports wuerschmidt and hemiwuerschmidt temperament. 229 is the 50th prime EDO.
23-3reduced-limit

**3reduced-limits:**

* 1-3reduced-limit
* 2-3reduced-limit
* 4-3reduced-limit
* 5-3reduced-limit
* 7-3reduced-limit
* 8-3reduced-limit
* 10-3reduced-limit
* 11-3reduced-limit
* 13-3reduced-limit
* 14-3reduced-limit
* 16-3reduced-limit
* 17-3reduced-limit
* 19-3reduced-limit
* 20-3reduced-limit
* 22-3reduced-limit
* **23-3reduced-limit**
* 25-3reduced-limit
* 26-3reduced-limit
* 28-3reduced-limit
* 29-3reduced-limit
* 31-3reduced-limit
* 32-3reduced-limit
* 34-3reduced-limit
* 35-3reduced-limit
* 37-3reduced-limit
* 38-3reduced-limit
* 40-3reduced-limit
* 41-3reduced-limit
* 43-3reduced-limit
* 44-3reduced-limit
* 46-3reduced-limit
* 47-3reduced-limit
* 49-3reduced-limit
* 50-3reduced-limit
* 52-3reduced-limit

 **23-3reduced-limit** is the tritave version of 23-odd-limit. It contains all the wonders of 149edt.

In 23-odd-limit we have octave equivalence, and the following odd harmonics: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23

In 23-3reduced-limit we have tritave equivalence, and the following 3reduced harmonics: 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23

* 24/23, 23/8
* 23/22, 66/23
* 22/21, 63/22
* 21/20, 20/7
* 20/19, 57/20
* 19/18, 54/19
* 18/17, 17/6
* 17/16, 48/17
* 16/15, 45/16
* 15/14, 14/5
* 14/13, 39/14
* 13/12, 36/13
* 12/11, 11/4
* 23/21, 63/23
* 11/10, 30/11
* 21/19, 19/7
* 10/9, 27/10
* 19/17, 51/19
* 9/8, 8/3
* 17/15, 45/17
* 8/7, 21/8
* 23/20, 60/23
* 15/13, 13/5
* 22/19, 57/22
* 7/6, 18/7
* 27/23, 23/9
* 20/17, 51/20
* 13/11, 33/13
* 19/16, 48/19
* 6/5, 5/2
* 23/19, 57/23
* 17/14, 42/17
* 11/9, 27/11
* 27/22, 22/9
* 16/13, 39/16
* 21/17, 17/7
* 5/4, 12/5
* 24/19, 19/8
* 19/15, 45/19
* 14/11, 33/14
* 23/18, 54/23
* 9/7, 7/3
* 22/17, 51/22
* 13/10, 30/13
* 30/23, 23/10
* 17/13, 39/17
* 21/16, 16/7
* 4/3, 9/4
* 27/20, 20/9
* 23/17, 51/23
* 19/14, 42/19
* 15/11, 11/5
* 11/8, 24/11
* 18/13, 13/6
* 7/5, 15/7
* 24/17, 17/8
* 17/12, 36/17
* 27/19, 19/9
* 10/7, 21/10
* 33/23, 23/11
* 23/16, 48/23
* 13/9, 27/13
* 16/11, 33/16
* 19/13, 39/19
* 22/15, 45/22
* 3/2, 2/1
* 23/15, 45/23
* 20/13, 39/20
* 17/11, 33/17
* 14/9, 27/14
* 36/23, 23/12
* 11/7, 21/11
* 30/19, 19/10
* 19/12, 36/19
* 27/17, 17/9
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 23/14, 42/23
* 33/20, 20/11
* 5/3, 9/5
* 27/16, 16/9
* 22/13, 39/22
* 39/23, 23/13
* 17/10, 30/17
* 12/7, 7/4
* 19/11, 33/19
1889edo

The 1889 division divides the octave into 1889 equal parts of 0.6353 cents each. It is strong in the 23-limit, though 1578, which among other things has a lower 23-limit relative error, rather puts it in the shade. It is distinctly consistent through the 27-limit, but not, unlike 1578, to the 29-limit.
174edo

Welcome to 174edo

## 174edo = 6edo × 29edo

Distinctively interesting choice for an edo, due to its divisibility by both 6 and 29. This means it can be used to play 6edo music and 29edo music at the same time, using the 174/29 and 174/6 vals respectively.

## 6.897

174edo has a step size of 6.8965517241379 cents. It is a commatonal EDO, because the step size represents commas like 256/255, 221/220, 170/169, 154/153.

## Meantone

This tuning provides a very close approximation to Quarter-comma meantone, with the tempered fifth located at 101\174. If this were to be the fifth, it would also follow septimal meantone, but neither undecimal meantone nor undecimal meanpop, unless the 11 is changed from the patent val as well.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 174edo (ordered by absolute error).
175edo

The *175 equal division* divides the octave into 175 equal parts of 6.857 cents each. It tempers out 1029/1024 and 225/224, so that it supports 7-limit miracle temperament, and in fact provides an excellent alternative to 72edo for 7-limit miracle. In the 11-limit, it tempers out 243/242, 385/384, 441/440 and 540/439, and supports 11-limit miracle. In the 13-limit, 175f tempers out 351/350 just as 72 does, and provides a tuning for benediction temperament very close to the POTE tuning.

# Music

http://www.archive.org/details/RachmaninoffPlaysBlackjack

play
19-3reduced-limit

**3reduced-limits:**

* 1-3reduced-limit
* 2-3reduced-limit
* 4-3reduced-limit
* 5-3reduced-limit
* 7-3reduced-limit
* 8-3reduced-limit
* 10-3reduced-limit
* 11-3reduced-limit
* 13-3reduced-limit
* 14-3reduced-limit
* 16-3reduced-limit
* 17-3reduced-limit
* **19-3reduced-limit**
* 20-3reduced-limit
* 22-3reduced-limit
* 23-3reduced-limit
* 25-3reduced-limit
* 26-3reduced-limit
* 28-3reduced-limit
* 29-3reduced-limit
* 31-3reduced-limit
* 32-3reduced-limit
* 34-3reduced-limit
* 35-3reduced-limit
* 37-3reduced-limit
* 38-3reduced-limit
* 40-3reduced-limit
* 41-3reduced-limit
* 43-3reduced-limit
* 44-3reduced-limit
* 46-3reduced-limit
* 47-3reduced-limit
* 49-3reduced-limit
* 50-3reduced-limit
* 52-3reduced-limit

 **19-3reduced-limit** is the tritave version of 19-odd-limit.

In 19-odd-limit we have octave equivalence, and the following odd harmonics: 3, 5, 7, 9, 11, 13, 15, 17, 19

In 19-3reduced-limit we have tritave equivalence, and the following 3reduced harmonics: 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19

* 19/18, 54/19
* 18/17, 17/6
* 17/16, 48/17
* 16/15, 45/16
* 15/14, 14/5
* 14/13, 39/14
* 13/12, 36/13
* 12/11, 11/4
* 11/10, 30/11
* 21/19, 19/7
* 10/9, 27/10
* 19/17, 51/19
* 9/8, 8/3
* 17/15, 45/17
* 8/7, 21/8
* 15/13, 13/5
* 7/6, 18/7
* 13/11, 33/13
* 19/16, 48/19
* 6/5, 5/2
* 17/14, 42/17
* 11/9, 27/11
* 16/13, 39/16
* 21/17, 17/7
* 5/4, 12/5
* 24/19, 19/8
* 19/15, 45/19
* 14/11, 33/14
* 9/7, 7/3
* 13/10, 30/13
* 17/13, 39/17
* 21/16, 16/7
* 4/3, 9/4
* 19/14, 42/19
* 15/11, 11/5
* 11/8, 24/11
* 18/13, 13/6
* 7/5, 15/7
* 24/17, 17/8
* 17/12, 36/17
* 27/19, 19/9
* 10/7, 21/10
* 13/9, 27/13
* 16/11, 33/16
* 19/13, 39/19
* 3/2, 2/1
* 17/11, 33/17
* 14/9, 27/14
* 11/7, 21/11
* 30/19, 19/10
* 19/12, 36/19
* 27/17, 17/9
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 5/3, 9/5
* 27/16, 16/9
* 17/10, 30/17
* 12/7, 7/4
* 19/11, 33/19
209edo

Welcome to 209edo

## 209edo = 11edo × 19edo

Distinctively interesting choice for an edo, due to its divisibility by both 11 and 19. This means it can be used to play 11edo music and 19edo music at the same time, using the 209/19 and 209/11 vals respectively.

## 5.742

209edo has a step size of 5.7416267942584 cents. It is a commatonal EDO, because the step size represents commas like 15625/15552, 1728/1715, 385/384, 169/168.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 209edo (ordered by absolute error).
230edo

Welcome to 230edo

## 230edo = 10edo × 23edo

Distinctively interesting choice for an edo, due to its divisibility by both 10 and 23. This means it can be used to play 10edo music and 23edo music at the same time, using the 230/23 and 230/10 vals respectively.

## 5.217

230edo has a step size of 5.2173913043478 cents. It is a commatonal EDO, because the step size represents commas like . It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 230edo (ordered by absolute error).
17ed5

# Division of the 5/1 into 17 tones

A hyperpyth tuning, 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ed5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ed5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.

But wait, an interesting pattern emerges:

22ed5 focuses on 9/5

27ed5 focuses on 13/5

29ed5 focuses on 17/5

(and 34=17\*2)

so: 22+27+29=78=39\*2

and behold, of the lot, 39ed5 offers the best balance between those intervals.
14L 3s

This MOS consists of three small steps separated by intervals of 5L-5L-4L. The temperament it belongs to is called Squares because because the number of generators it takes to get to each of the primes 3, 5 and 7 is a square (3=-4 generators, 5=-16, 7=-9). This puts the generator between 6/17edo (423.529) and 5/14edo (428.571 cents).
21/10

Interval information

Type: Utonal

In 7-limit Just Intonation, 21/10 is a septimal augmented octave measuring about 1284.5¢. It has a similar sound to its tritave inversion, 10/7, but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It is also an octave above 21/20.

*See also Gallery of Just Intervals*
256edo

`Welcome to 256edo. You might notice this EDO of a step size of 4.6875, aside from being commatonal and nanotonal, is a power of two, and the information about a note in a single octave fits within an 8-bit byte. Divisors are {{#rreplace: 256 128 64 32 16 8 4 2 1|/(\d+)(\D+)?/|\1\2}}. It therefore might not be unusual to see this tuning, or a multiple of it, in applications storing logarithmic frequencies. It is not consistent in the 5-odd-limit. It is contorted in 5-limit.`
238edo

Welcome to 238edo

## 238edo = 14edo × 17edo

Distinctively interesting choice for an edo, due to its divisibility by both 14 and 17. This means it can be used to play 14edo music and 17edo music at the same time, using the 238/17 and 238/14 vals respectively.

## 5.042

238edo has a step size of 5.0420168067227 cents. It is a commatonal EDO, because the step size represents commas like 32768/32805, 126/125. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 238edo (ordered by absolute error).
218edo

**218edo**, having a step size of 5.50458715596 cents, contains very accurate ratios, such as 7/4, 9/7, 8/7, 9/8, 10/9, 11/10 and 17/16 which are approximated within 0.55¢ deviation (10% the step size).

"218edo ~ 15601edo be NANOTONALS (Particulary I choose 218edo, because this EDO contains with much accurate the Ratios 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16 and very interesting accurate to the Pi Ratio [1981,7954 Cents Pi ; 1981,6514 Cents the 360\218edo]. 218edo contains a 'possible Comma' which is the Ratio 65207/65000). The size of 1\218 is 5,5046 Cents, but you can use the size of 5,5 Cents like a approximation too)."

The step size of 5.5045871559633 can be approximated as 5.5 cents, leading to a variation named 5.5cET (5.5 cent Equal-step Tuning) where the octave is reduced by 1 cent.

The following table shows the nearest matches for the interval, not the matches from the patent val. **Bold** numbers are off within less than 0.1 (10%) of the step size.

Suggested subgroups: 2.9.7.17 and 2.9.5.7.11.17.

Also explore 436edo.

Commas using the 13-limit patent val:

3-limit
1/1
5-limit
20000/19683
7-limit
4000/3969 65625/65536 245/243 2401/2400 60025/59049
11-limit
4000/3993 12005/11979 16384/16335 4375/4356 78125/77616 896/891 67228/66825 1375/1372 6875/6804 5632/5625 385/384 94325/93312 15488/15435 75625/75264 15488/15309 3388/3375 1331/1323 6655/6561 65219/64800 43923/43904 73205/72576
13-limit
28672/28561 86240/85683 20480/20449 5600/5577 16807/16731 25000/24843 6125/6084 86625/86528 68992/68445 58080/57967 96800/95823 847/845 41503/41067 33275/33124 65219/64896 29575/29403 4225/4224 21632/21609 676/675 33124/32805 9295/9261 46475/45927 13013/12960 28561/28512
192edo

**192edo** is twice the 96edo, uses a step size of 6.25 cents. Contorted in 5-limit. In 7-limit, it assumes 225/224 and 126/125 are both half syntonic comma. It is not consistent in 5-odd-limit.

7-limit: 6144/6125 3136/3125 2401/2400
11-limit: 59049/58564 65610/65219 243/242 540/539 6912/6875 441/440 5632/5625
13-limit: 88209/87880 91125/91091 46656/46475 59535/59488 41472/41405 21168/21125 6561/6500 16038/15925 13365/13312 81675/81536 3267/3250 53361/53248 15972/15925 65219/65000 14641/14625 6656/6655 28431/28160 3159/3136 3159/3125 51597/51200 19305/19208 30888/30625 1287/1280 63063/62500 1573/1568 3146/3125 77077/76800 86515/86436 17303/17280 3042/3025 2704/2695 4225/4224 8281/8250 21632/21609 676/675 43940/43923 19773/19600 79092/78125 32955/32768 24167/24010 24167/24000 85683/85184 28561/28512
18/13

**18/13**

563.3823 cents

File:Jid 18 13 pluck adu dr220.mp3 sound sample

In 13-limit Just Intonation, **18/13** is a low tritone measuring about 563.4¢. It represents the interval between the 13th and 18th overtones of the harmonic series. The low tritone in 17edo approximates 18/13 quite well, while 17edo's approximations of 13/8 and 9/8 are both a little sharp. The inversion of 18/13 is 13/9, a high tritone measuring about 636.6¢; the interval between these two tridecimal (13-based) tritones is 169/162, approximately 73.2¢.

Other just tritones include 7/5, 10/7, 45/32, 64/45, 24/17, 17/12, 23/16, 32/23, and, of course, infinitely many more intervals.

See: Gallery of Just Intervals
20203edo

The 20203 division divides the octave into 20203 equal parts of cents each. It is a very strong higher limit edo, with a lower 19-limit relative error than any edo until 128125. It is also distinctly consistent through the 45 limit, and has a lower relative error than any smaller distinctly consistent 41-limit patent val except 17461.
17/12

**17/12**

603.0004 cents

File:Jid 17 12 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 17/12 is the "second septendecimal tritone," measuring very nearly 603¢. Its inversion is the "first septendecimal tritone," 24/17, and the interval that separates them is the small comma 289/288, about 6¢. This difference is usually negligible, and tempering out this comma allows the 600¢ half-octave to function as both septendecimal tritones. Thus, every even-numbered EDO system contains a close approximation to these intervals.

17/12 is the mediant between the two septimal tritones 7/5 and 10/7.

See: Gallery of Just Intervals
190edo

*190edo* divides the octave into 190 equal parts of size 6.316 cents each. It is interesting because of the utility of its approximations; it tempers out 1029/1024, 4375/4374, 385/384, 441/440, 3025/3024 and 9801/9800. It provides the optimal patent val for both the 7 and 11 limit versions of unidec, the 118&190 temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank three 11-limit temperament portent, which tempers out 385/384 and 441/440, and gamelan, the rank three 7-limit temperament which tempers out 1029/1024, as well as slendric, the 2.3.7 subgroup temperament featured below. In the 13-limit, 190et tempers out 847/845, 625/624, 729/728, 1575/1573 and 1001/1000, and provides the optimal patent val for ekadash temperament and the rank three portentous temperament.

## Music

The 11th Slendric Fanfare

16 Slendric Virgins

by Chris Vaisvil
2000edo

The 2000 equal division divides the octave into 2000 equal parts of exactly 0.6 cents each. It is distinctly consistent through the 29 limit and a strong 29-limit system; the only smaller edo with a smaller 29-limit relative error being 1578. The only ones to beat it in the 23-limit are 1578 and 1889, and in the 19-limit, nothing smaller defeats it, the first edo to do so being 2460.

2000 = 2^4 \* 5^3; some of its divisors are 10, 16, 25, 50, 80, 100, 125 and 200. Also there is the 1000 division of millioctaves, where it might be argued that cutting these in half makes for a better system.
248edo

The *248 equal division* divides the octave into 248 equal parts of 4.389 cents each. It tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440 and 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It supports bischismic temperament, providing the optimal patent val for 11-limit bischismic, and excellent tunings in the 7 and 13-limits. It also provides the optimal patent val for essence temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords. 248 has divisors 2, 4, 8, 31, 62, and 124.
17-odd-limit

**Odd limits:**

* 1-odd-limit
* 3-odd-limit
* 5-odd-limit
* 7-odd-limit
* 9-odd-limit
* 11-odd-limit
* 13-odd-limit
* 15-odd-limit
* **17-odd-limit**
* 19-odd-limit
* 21-odd-limit
* 23-odd-limit
* 25-odd-limit
* 27-odd-limit
* 29-odd-limit
* 31-odd-limit
* 33-odd-limit
* 35-odd-limit
* 37-odd-limit
* 39-odd-limit
* 41-odd-limit
* 43-odd-limit
* 45-odd-limit
* 47-odd-limit
* 49-odd-limit
* 51-odd-limit

 This is a list of **17-odd-limit** intervals. Also see 17-limit.

* 18/17, 17/9
* 17/16, 32/17
* 16/15, 15/8
* 15/14, 28/15
* 14/13, 13/7
* 13/12, 24/13
* 12/11, 11/6
* 11/10, 20/11
* 10/9, 9/5
* 9/8, 16/9
* 17/15, 30/17
* 8/7, 7/4
* 15/13, 26/15
* 7/6, 12/7
* 20/17, 17/10
* 13/11, 22/13
* 6/5, 5/3
* 17/14, 28/17
* 11/9, 18/11
* 16/13, 13/8
* 5/4, 8/5
* 14/11, 11/7
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 17/13, 26/17
* 4/3, 3/2
* 15/11, 22/15
* 11/8, 16/11
* 18/13, 13/9
* 7/5, 10/7
* 24/17, 17/12
23平均律

23tET、または23EDOは、オクターブを23分割する、テンパーされたシステムであり、各ステップはおおよそ52.173913セントであり、それはまた新造語としてIcositriphony（Icositrifonia）と呼ばれる。5/3と11/7、そして13と17に近似し、2.5/3.11/7.13.17純正調サブグループとみなされる。もしこのサブグループが17リミット46平均律のコンマを加えられたなら、より大きな17リミット2\*23サブグループ、2.9.15.21.33.13.17が得られる。これは17リミット46平均律のようなコンマと同じ23がもつチューニングの中で、最も大きいサブグループである、そしておそらく今までのところ、23平均律のハーモニーの分析にもとづくとみなされ、純正音程の近似とされる。23平均律は9番目の、19平均律の後であり、29平均律の前の平均律である。

# 23平均律の音程と近似値

各周波数比の大きさが16以内で表現される純正音程は以下のようになる。これはedjirulerを用いて、[number of equal divisions=23, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.3]というパラメータで生成したものである。「The “neighborhood” of JI」の一覧はこちら（huygens-fokker）を参照のこと。

下に示すチャートは、23平均律で利用できるMavilaのMOS音階を示す。主にペンタトニック、アンチダイアトニック、9と16のMOSである。これは23平均律自身の個々のステップを示すouter ringであり、16と9、7と5音のMOSを示す。

23平均律は、民族音楽学者であるErich von Hornbostelによって提案された、678セントの膨れた5度の結果である。この膨れた5度は、bamboo pipeを強く吹くことにより生まれると彼は主張している。

23平均律はまた、3、5、7、11倍音に20セント以内で近似しない、最も大きい平均律、という点で重要な意味をもつ。それは一般的な微分音理論家から非常に外れた領域のハーモニーの探求に適している。加えるなら、これらのハーモニーに近似するのに失敗した事実にもかかわらず、5/3、7/3、11/3、7/5、11/7、そして11/5の間の音程には非常に近似する。その最も低いハーモニーは23平均律によって良く13、17、21、そして23に近似する。より詳細なものはここを見ていただきたい。

9、16、25平均律のように、23平均律を扱う手法はペロギックテンペラメント、つまり135/128のコンマをテンパーアウトし、5/1（Armodueシステムと関連する）とともに、3つの「鋭い4/3」が3つ分と同じとみなすことである。この意味は、「3/2」を23平均律の13degreeでマッピングし、その結果としてAnti-diatonic scale （3 3 4 3 3 3 4）の7音となり、Superdiatonic scale（3 3 3 1 3 3 3 3 1）の9音に拡大される。Armodueシステムを使い、23平均律の記譜できる。しかしダイアトニックシステム類をもつ17平均律の記譜法のように、フラットはエンハーモニックシャープより低くなる。なぜなら23平均律は、「Armodue 6th」が16平均律よりシャープされ、17平均律のダイアトニック5thのように12平均律よりシャープされるからである。言い換えれば、2bは1#より低くなり、17平均律のように、EbはD#より低い。
20ed5

20th root of 5 "Hieronymus' Tuning"

A harmonic entropy minimum, that has better approximations of a variety of just intervals than Bohlen Pierce (of course, not the same intervals) among which are 13/12, 7/6, 14/11, 11/8, 3/2, 13/8, 7/4, 21/11, 33/32, ~9/4, 39/32, 21/16, 10/7, 20/13, 10/3 ... etc. In terms of strict 5/1 equivalence and high-limit harmony, it also approximates the harmonics and their pentave reductions: ‎8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents. Note that there are (at most) ~4.3 pentaves within human hearing range; imagine if that were the case with octaves (yes that is a helpful analogy).

One way of looking at it comes by constructing it via four tempered 3/2 (meantone without octaves) each of which is divided into five tones, which in turn approximate 11/8 13/8 7/6 etc., and themselves end up on the "pentave", 5/1, wherein the scale repeats itself. By analogy to common practice, this is familiar extended meantone but *turned entirely inside-out*. Interestingly, while Hieronymus does not repeat at the octave or even approximate it well, factors of 2 are nevertheless important to its perception and structure; it might even be helpful to think of the 3/2 intervals as a cellular structure of sorts.

Adding octaves makes it jerome temperament, with generator a meantone fifth divided in five, and Hieronymus is the generator chain of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, or even higher limit system.
133edo

Welcome to 133edo

## 133edo = 7edo × 19edo

Distinctively interesting choice for an edo, due to its divisibility by both 7 and 19. This means it can be used to play 7edo music and 19edo music at the same time, using the 133/19 and 133/7 vals respectively.

## 9.023

133edo has a step size of 9.0225563909774 cents. It is a commatonal EDO, because the step size represents commas like 2048/2025, 126/125, 121/120, 91/90.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 133edo (ordered by absolute error).
197edo

The 197 equal division gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and catakleismic, the temperament tempering out both 225/224 and 4375/4374, which has wedgie <<6 5 22 -6 18 37||. Among patent vals, in fact, it gives the best results for both. In fact, the 11-limit patent val <197 312 457 553 682| has a comma basis [225/224, 4375/4374, 441/440, 65536/65219], so taking 225/224 and 441/440 together (prodigy temperament) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.

If we use 197e, the <197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit marvel, we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that.

197edo is the 45th prime edo.

## Scales

* prismarv
* marveldene
* pump12\_1

Pump1 by Chris Vaisvil

* pump12\_1
* pump13
* pump14
* pump15
* pump16
* pump17
* pump18
26/17

**26/17**

735.5723 cents

File:Jid 26 17 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 26/17 is the "septendecimal super-fifth," measuring about 735.6¢. It differs from a 3/2 perfect fifth by the comma 52/51, about 33.6¢. Although this difference is considerable, 26/17 may be used as a wide perfect fifth, thus allowing septendecimal versions of root-3rd-P5 chords -- in particular, 17:20:26, 17:21:26, 17:22:26.

26/17 is the mediant between 3/2 and 23/15. Its inversion is 17/13, the "septendecimal sub-fourth".

See: Gallery of Just Intervals
232edo

The *232 equal division* divides the octave into 232 equal parts of 5.172 cents each. It provides the optimal patent val for 13-limit mystery temperament, the rank three pele temperament and the rank three trimyna temperament and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank five temperament. Aside from its patent val, the 232d val is worth considering. Both temper out the wuerschmidt comma, 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 3125/3097 and 245/243, supporting bohpier temperament. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 441/440 and 676/675, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports variants of octoid.

Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows minthmic chords, and because it tempers out 847/845 it allows the cuthbert triad, making it a very flexible harmonic system.
260edo

Welcome to 260edo

## 260edo = 13edo × 20edo

Distinctively interesting choice for an edo, due to its divisibility by both 13 and 20. This means it can be used to play 13edo music and 20edo music at the same time, using the 260/20 and 260/13 vals respectively.

## 4.615

260edo has a step size of 4.6153846153846 cents. It is a commatonal EDO, because the step size represents commas like 176/175. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 260edo (ordered by absolute error).
203edo

Welcome to 203edo

## 203edo = 7edo × 29edo

Distinctively interesting choice for an edo, due to its divisibility by both 7 and 29. This means it can be used to play 7edo music and 29edo music at the same time, using the 203/29 and 203/7 vals respectively.

## 5.911

203edo has a step size of 5.9113300492611 cents. It is a commatonal EDO, because the step size represents commas like 20000/19683, 225/224, 169/168.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 203edo (ordered by absolute error).
252edo

**252edo** uses a step size of 4.7619047619048 cents, and is a commatonal and nanotonal EDO. It can be thought of as fourteen sets of 18edo. Divisors are {{#rreplace: 252 126 84 63 42 36 28 21 18 14 12 9 7 6 4 3 2 1|/(\d+)(\D+)?/|\1\2}}. It is consistent in the 7-odd-limit. It is contorted in 5-limit.
2513edo

The 2513 division divides the octave into 2513 equal parts of 0.4775 cents each. It is a very strong 5-limit system, with a lower 5-limit relative error than any edo until we reach the cosmically excellent 4296edo. A basis for its 5-limit commas is senior, |-17 62 -35> and fortune, |-107 47 14>; it also tempers out pirate, |-90 -15 49>. It is uniquely consistent through to the 11-limit, and tempers out 420175/419904 in the 7-limit and 151263/151250 and 234375/234256 in the 11-limit.
25edo

# 25 tone equal temperament

25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the Blackwood temperament, which takes the very sharp fifths of 5EDO as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (5/4) and 7 (7/4). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not consistent. It therefore makes sense to use it as a 2.5.7 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is 50EDO. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the 2\*25 subgroup 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

# Music

*Study in Fives* by Paul Rapoport

Fantasy for Piano in 25 Note per Octave Tuning *play* by Chris Vaisvil

*Flat fourth blues* by Fabrizio Fulvio Fausto Fiale

File:25edochorale.mid 25edochorale.mid Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")

File:25 edo prelude largo.mid 25 edo prelude largo.mid Peter Kosmorsky (2011, Blackwood)

# Intervals

* based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.

File:25ed2-001.svg

25ed2-001.svg

# Relationship to Armodue

Like 16-EDO and 23-EDO, 25-EDO contains the 9-note "Superdiatonic" scale of 7L2s (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9-EDO or 666.67 cents, to 4\7-EDO or 685.71 cents). The 25-EDO generator for this scale is the 672-cent interval. This allows 25-EDO to be used with the Armodue notation system in much the same way that 19-EDO is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25-EDO Armodue 6th is flatter than that of 16-EDO (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.

# Commas

25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)
23-limit

In 23-limit Just Intonation, all ratios contain no prime factors higher than 23.

Ratios of 23 in the 23-odd limit include:

24/23 .. 73.681¢

23/22 .. 76.956¢

23/21 .. 157.493¢

26/23 .. 212.253¢

23/20 .. 241.961¢

23/19 .. 330.761¢

28/23 .. 340.552¢

23/18 .. 424.364¢

30/23 .. 459.994¢

23/17 .. 523.319¢

23/16 .. 628.274¢

34/23 .. 676.681¢

32/23 .. 571.726¢

23/15 .. 740.006¢

36/23 .. 775.636¢

23/14 .. 859.448¢

38/23 .. 869.239¢

40/23 .. 958.039¢

23/13 .. 987.747¢

42/23 ..1042.507¢

44/23 .. 1123.044¢

23/12 .. 1126.391¢

94edo is the first EDO to be consistent in the 23-limit. The smallest EDO where the 23-limit is distinctly consistent, meaning each element of the tonality diamond is distinguished, is 282edo.

See: Harmonic Limit, 19-limit, 17-limit
17-limit

In 17-limit Just Intonation, all ratios in the system will contain no primes higher than 17. The 17-limit adds to the 13-limit a "minor ninth" of about 105¢ -- 17/16 -- and several other intervals between the 17th overtone and the lower ones. See also 17-odd-limit and 17-3reduced-limit.

## 17-limit Intervals

see Harmonic Limit, seventeen limit tetrads
224edo

The *224 equal temperament* divides the octave into 224 equal parts of 5.357 cents each. It is a very strong 13-limit system, tempering out 32805/32768 in the 5-limit; 4375/4374, 16875/16807 and 65625/65536 in the 7-limit; 540/530, 1375/1372 and 4000/3993 in the 11-limit; and 625/624, 729/728, 1575/1573 and 2200/2197 in the 13-limit, leading to an abundance of precisely-tuned essentially tempered chords. It defines the optimal patent val for octoid temperament in the 7-, 11- and 13-limit, and for mirkwai, the 7-limit planar temperament tempering out 16875/16807. It also provides an excellent tuning for indra and shibi temperaments. It is the twelfth zeta integral edo.

224 = 32 \* 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.
193edo

# 193 tone equal temperament

The *193-EDO* divides the octave into 193 equal parts of 6.21762 cents each. It provides the optimal patent val for sqrtphi temperament in the 13-, 17- and 19-limits, and for 13-limit minos and vish temperaments.

Approximation of the intervals:

Square root of Pi: **159\193** (988.60104 cents), and

Phi: **134\193** (833.16062 cents), both inside in the superdiatonic scale: 25 25 25 9 25 25 25 25 9

193edo is the 44th prime EDO.
19-odd-limit

**Odd limits:**

* 1-odd-limit
* 3-odd-limit
* 5-odd-limit
* 7-odd-limit
* 9-odd-limit
* 11-odd-limit
* 13-odd-limit
* 15-odd-limit
* 17-odd-limit
* **19-odd-limit**
* 21-odd-limit
* 23-odd-limit
* 25-odd-limit
* 27-odd-limit
* 29-odd-limit
* 31-odd-limit
* 33-odd-limit
* 35-odd-limit
* 37-odd-limit
* 39-odd-limit
* 41-odd-limit
* 43-odd-limit
* 45-odd-limit
* 47-odd-limit
* 49-odd-limit
* 51-odd-limit

 This is a list of **19-odd-limit** intervals. Also see 19-limit.

* 20/19, 19/10
* 19/18, 36/19
* 18/17, 17/9
* 17/16, 32/17
* 16/15, 15/8
* 15/14, 28/15
* 14/13, 13/7
* 13/12, 24/13
* 12/11, 11/6
* 11/10, 20/11
* 10/9, 9/5
* 19/17, 34/19
* 9/8, 16/9
* 17/15, 30/17
* 8/7, 7/4
* 15/13, 26/15
* 22/19, 19/11
* 7/6, 12/7
* 20/17, 17/10
* 13/11, 22/13
* 19/16, 32/19
* 6/5, 5/3
* 17/14, 28/17
* 11/9, 18/11
* 16/13, 13/8
* 5/4, 8/5
* 24/19, 19/12
* 19/15, 30/19
* 14/11, 11/7
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 17/13, 26/17
* 4/3, 3/2
* 19/14, 28/19
* 15/11, 22/15
* 26/19, 19/13
* 11/8, 16/11
* 18/13, 13/9
* 7/5, 10/7
* 24/17, 17/12
17/11

**17/11**

753.6375 cents

File:Jid 17 11 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 17/11 is the "septendecimal subminor sixth," measuring about 753.6¢. It is the mediant between 3/2 and 14/9. Its inversion is 22/17, the "septendecimal supermajor third." It is sometimes known as the major fifth in systems such as 16 in which 3/2 is split into two very distinct intervals.

See: Gallery of Just Intervals
24平均律

四分音（Wikipedia）

# 基本原則

24平均律システムは、オクターブを各50セントで24パートに分割したものである。クオータートーンチューニングと知られており、12ETをそれぞれ均等に2分割したものである。クオータートーンは一般に、12音を含む最小のmicrotonal equal temperamentであるため、加えて理論12音の理論が使えるため、12音類の保持のためにmicrotonal tuningで使用される。そしてまた、時々アラビア音楽の実践のために使用される。このチューニングに飛び込むこと、一般的な12Equalのソフトウェアや楽器を使用しmicrotonal musicを作成することは容易である。またこのページを参照のこと。

# テンペラメントとしての24平均律

24平均律の中で5リミットの近似となるのは、12平均律と同じであり、すなわち24平均律は新しい5リミットを提供しない。それらは異なった音程だけれども、7倍音にもとづく音程は（7:4、7:5、7:6）が12平均律の中同様24平均律でも現れる。満足のいく7倍音の近似値のレベルを達成する、12平均律の12音が要求するハイレベルチューニングは、72、84、156、平均律で生成される。

11倍音は、（11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11）で提供され、24平均律で非常に良く提供される。24平均律における550セントの音程は、1.3セント11:8からフラットされており、ほぼ区別できない。加えて、11:9の音程は7ステップで正確に完全5度の半分が示される。良いいくつかの24平均律のコードは以下である。

・0-4-8-11-14（9:8と11:8のメジャーコード）、反行形は0-3-6-10-14（マイナー）

・0-7-14（中立）

・0-5-10（もう1つの中立種、4度を2つに分ける。0-5-10はペンタトニックスケールに向け拡大させられる。0-5-10-14-19-24（godzilla）、それはペンタトニックではないいくつかのインドネシアのスレンドロに近い。似た手法として、0-7-14を0-4-7-11-14-18-21-24 （mohajira）への拡大する、いくつかのアラビア音階の8音音階に近い）

# 24平均律の音程と近似値

各周波数比の大きさが16以内で表現される純正音程は以下のようになる。これはedjirulerを用いて、[number of equal divisions=24, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.3]というパラメータで生成したものである。「The “neighborhood” of JI」の一覧はこちら（huygens-fokker）を参照のこと。

# 音階

24平均律は5つのカテゴリーの2つのセットの音程を分割する。InfraとMinor、Neutral、Major、Ultraにより、2度、3度、6度、7度を分割し、加えてnarrow、perfect、wide、augmentに4度、5度、1度、8度を分割する。他の奇妙なエンハーモニックのために、wideとnarrowは、augmentとdiminish音程とともに結合され使用される。550セントはnarrow diminished fifthと呼ばれ、850セントはwide augmented fifthと呼ばれるように。
17/8

**17/8**

1304.9554095004 cents

In 17-limit Just Intonation, 17/8 is the "first septendecimal diminished ninth" measuring very nearly 1305¢. It is the mediant between 2/1 and 15/7, the two septendecimal tritones are each 3¢ away from the 600¢ half-octave, and so they are well-represented in all even-numbered EDO systems, including 12edo. Indeed, the latter system, containing good approximations of the 3rd and 17th harmonics, can use the half-octave as 24/17 and 17/12 in close approximations to chords such as 8:12:17 and 16:17:24. 22edo is another good EDO system for using the half-octave in this way.

See: Gallery of Just Intervals
1edo

One note repeated in octaves is an example of a trivial temperament.

The first piece of Musica ricercata by György Ligeti simulates 1edo by playing only one note throughout different octaves on the piano, though the piano is not itself tuned to 1edo. The last note of the piece breaks the pattern.

## Links

* Musica ricercata - Wikipedia: *I. Sostenuto – Misurato – Prestissimo*
* Ligeti - Musica ricercata: I. Sostenuto - Misurato - Prestissimo by Eliza McCarthy | Free Listening on SoundCloud

es:1-EDO
17/15

**17/15**

216.6867 cents

File:Jid 17 15 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 17/15 is the "septendecimal whole tone" measuring about 216.7¢. It is the mediant between 9/8 and 8/7, as it is (9+8)/(8+7). It is found in the harmonic series between the 17th and 15th overtones. 11edo's second degree, measuring approximately 218.2¢, is close in size to 17/15 -- indeed, the 11edo system has excellent approximations of the 15th and 17th harmonics, and so this harmonic function is plausible in 11edo.

√2/(17/15) is three cents flat of a 5/4 major third, and this or 17/15 itself can be used for a tuning for wizard and its various relatives (lizard, gizzard, etc.).

See: Gallery of Just Intervals
207edo

The 207 equal division of the octave, 207edo, divides the octave into 207 equal parts of size 5.797 cents each. It tempers out 32805/32768 in the 5-limit, 6144/6125 and 19683/19600 in the 7-limit, 441/440 and 43923/43904 in the 11-limit, and 351/350, 847/845, 676/675, 729/728, 1716/1715 in the 13-limit. It serves as the patent val in the 11- and 13-limits for swetneus temperament. It is significantly more accurate on the 2.3.7.11.13 subgroup, a favorite of many people, and one which contains both 729/728 and 10648/10647, which it tempers out.
195edo

Welcome to 195edo

## 195edo = 13edo × 15edo

Distinctively interesting choice for an edo, due to its divisibility by both 13 and 15. This means it can be used to play 13edo music and 15edo music at the same time, using the 195/15 and 195/13 vals respectively.

## 6.154

195edo has a step size of 6.1538461538462 cents. It is a commatonal EDO, because the step size represents commas like 126/125.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 195edo (ordered by absolute error).
183edo

*183edo* divides the octave into 183 equal parts of 6.557 cents each. It is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the schisma, 32805/32768, in the 5-limit. In the 7-limit, it tempers out porwell, 6144/6125, cataharry, 19683/19600 and mirkwai, 16875/16807. In the 11-limit it tempers out 540/539 and 3025/3024; in the 13-limit, 351/350 and 676/675; in the 17-limit 442/441, 561/560 and 715/714; and in the 19-limit 456/455. It is the optimal patent val for 13-, 17- and 19-limit mirkat temperament, the 72&183 temperament, and an excellent tuning for the rank-three temperaments madagascar and borneo.

As a no-sevens temperament, it tempers out 32805/32768, 5632/5625, 8019/8000, 676/675, 4425/4424, 6656/6655, 936/935, 1089/1088, and 1377/1375.
17/14

**17/14**

336.1295 cents

File:Jid 17 14 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 17/14 is the "septendecimal supraminor third," measuring about 336.1¢. It is the mediant between 6/5 and 11/9, as it is (6+11)/(5+9). A 14:17:21 root-3rd-P5 triad can be built with 17/14 as the bottom third and 21/17 as the top third. This may thus represent a septendecimal "shading" of a minor triad.

See: Gallery of Just Intervals
2187/2048

**2187/2048**

113.6850 cents

File:Jid 2187 2048 pluck adu dr220.mp3 sound sample

The *apotome*, also known as the Pythagorean chromatic semitone or the Pythagorean major semitone, is the interval 3^7/2^11 = 2187/2048 which is the chromatic semitone in the Pythagorean (3-limit) version of the diatonic scale. Unlike the situation in meantone tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of 256/243. It measures about 113.7¢ and can be generated by stacking seven 3/2 perfect fifths and octave-reducing the resulting interval.

See: Gallery of Just Intervals, comma
215edo

The *215 equal temperament* divides the octave into 215 equal parts of 5.581 cents each. It tempers out 4000/3969 and 65625/65536, and the patent val provides the optimal patent val for the 3&53 temperament tempering them both out, and the rank three temperament tempering 4000/3969 out. The 215c val tempers out 2401/2400, 5120/5103, and supports hemififths temperament.
179edo

The 179 equal division divides the octave into 179 steps of 6.704 cents each. It tempers out the parakleisma, |8 14 -13>, in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit parkleismic temperament. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit ulmo temperament. 179 is the 41st prime number, with all of the advantages and disadvantages that brings.
264edo

**264edo** uses a step size of 4.5454545454545 cents, and is a commatonal and nanotonal EDO. It can be thought of as twelve sets of 22edo, or as 88edo with a three times smaller step size. Divisors are {{#rreplace: 264 132 88 66 44 33 24 22 12 11 8 6 4 3 2 1|/(\d+)(\D+)?/|\1\2}}. In 7-limit, it has the same 1/1—225/224—126/125—81/80 split as 171edo does, but not as accurate. It is consistent and uniquely consistent in the 7-odd-limit.
25-3reduced-limit

**3reduced-limits:**

* 1-3reduced-limit
* 2-3reduced-limit
* 4-3reduced-limit
* 5-3reduced-limit
* 7-3reduced-limit
* 8-3reduced-limit
* 10-3reduced-limit
* 11-3reduced-limit
* 13-3reduced-limit
* 14-3reduced-limit
* 16-3reduced-limit
* 17-3reduced-limit
* 19-3reduced-limit
* 20-3reduced-limit
* 22-3reduced-limit
* 23-3reduced-limit
* **25-3reduced-limit**
* 26-3reduced-limit
* 28-3reduced-limit
* 29-3reduced-limit
* 31-3reduced-limit
* 32-3reduced-limit
* 34-3reduced-limit
* 35-3reduced-limit
* 37-3reduced-limit
* 38-3reduced-limit
* 40-3reduced-limit
* 41-3reduced-limit
* 43-3reduced-limit
* 44-3reduced-limit
* 46-3reduced-limit
* 47-3reduced-limit
* 49-3reduced-limit
* 50-3reduced-limit
* 52-3reduced-limit

 **25-3reduced-limit** is the tritave version of 25-odd-limit.

In 25-odd-limit we have octave equivalence, and the following odd harmonics: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

In 25-3reduced-limit we have tritave equivalence, and the following 3reduced harmonics: 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25

* 25/24, 72/25
* 24/23, 23/8
* 23/22, 66/23
* 22/21, 63/22
* 21/20, 20/7
* 20/19, 57/20
* 19/18, 54/19
* 18/17, 17/6
* 17/16, 48/17
* 16/15, 45/16
* 15/14, 14/5
* 14/13, 39/14
* 27/25, 25/9
* 13/12, 36/13
* 25/23, 69/25
* 12/11, 11/4
* 23/21, 63/23
* 11/10, 30/11
* 21/19, 19/7
* 10/9, 27/10
* 19/17, 51/19
* 9/8, 8/3
* 17/15, 45/17
* 25/22, 66/25
* 8/7, 21/8
* 23/20, 60/23
* 15/13, 13/5
* 22/19, 57/22
* 7/6, 18/7
* 27/23, 23/9
* 20/17, 51/20
* 13/11, 33/13
* 19/16, 48/19
* 25/21, 63/25
* 6/5, 5/2
* 23/19, 57/23
* 17/14, 42/17
* 11/9, 27/11
* 27/22, 22/9
* 16/13, 39/16
* 21/17, 17/7
* 5/4, 12/5
* 24/19, 19/8
* 19/15, 45/19
* 14/11, 33/14
* 23/18, 54/23
* 9/7, 7/3
* 22/17, 51/22
* 13/10, 30/13
* 30/23, 23/10
* 17/13, 39/17
* 21/16, 16/7
* 25/19, 57/25
* 33/25, 25/11
* 4/3, 9/4
* 27/20, 20/9
* 23/17, 51/23
* 19/14, 42/19
* 15/11, 11/5
* 11/8, 24/11
* 18/13, 13/6
* 25/18, 54/25
* 7/5, 15/7
* 24/17, 17/8
* 17/12, 36/17
* 27/19, 19/9
* 10/7, 21/10
* 33/23, 23/11
* 23/16, 48/23
* 36/25, 25/12
* 13/9, 27/13
* 16/11, 33/16
* 19/13, 39/19
* 22/15, 45/22
* 25/17, 51/25
* 3/2, 2/1
* 23/15, 45/23
* 20/13, 39/20
* 17/11, 33/17
* 14/9, 27/14
* 39/25, 25/13
* 25/16, 48/25
* 36/23, 23/12
* 11/7, 21/11
* 30/19, 19/10
* 19/12, 36/19
* 27/17, 17/9
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 23/14, 42/23
* 33/20, 20/11
* 5/3, 9/5
* 42/25, 25/14
* 27/16, 16/9
* 22/13, 39/22
* 39/23, 23/13
* 17/10, 30/17
* 12/7, 7/4
* 19/11, 33/19
18平均律

“3音システム”

# 基本的な特徴

18平均律はオクターブを、それぞれ約66.667セントで18個のパートに分割したものである。30セントのエラー（純正音程とのずれ）を許容しない限り、12平均律の3度、5度、7度の響きには全く近くない。しかしながら、9/8、7/6、21/16、15/11、12/7、16/9、そして13/7のチューニングには最も近づく。それはまた、36/35をテンパーアウトしない、5:6:7のハーモニックコードシリーズに近似する、最も小さい平均律である。それゆえ、6/5にも7/6にも近似するインターバルを使用しない。

実際に利用できる素晴らしい協和音にアクセスするため、とても「一般的に実践的ではない」アプローチをしなければならない。これは、通常のクローズボイシング「根音-3度-5度」コードタイプと、より圧縮された、またはより拡大され代理で使われるコードを避けることを意味する。18平均律はたぶん、17リミット4\*18サブグループの、2.9.75.21.55.39.51純正律サブグループとして扱われる。このサブグループ上では、完全な17リミット上の72として、同様のコンマを正確にテンパーアウトする。そして正確に同様のチューニングをする。サブグループは1つのコードにたどり着く。例えば、32:36:39:42:51:55:64:75。18平均律の場合、0-3-5-7-12-14-18-22。コードのトランスポーズや転回、またそのサブコードはたくさんの協和源を提供する。

しかしながら、使用するときは少し低い。18平均律は28/27（62.96セント）をテンパーアウトする、そして3つの5度、3/2（701.96）より高い（11*18=733.33cent**）。したがって、9/8は非常に7/6に似ている。9/8（203.91）として200セント（3*18）に比較的近く、それは8/7（231.17）とも認識できる。この扱いは大きい5度によって生成された音階に当てはまる。これはFatherテンペラメントとして知られる。もし本当にこのようにするならば、全てシャープされた、3/2と半オクターブからなる音階で生成される。600セントは11/8（551.32セント）、そして866セントを13/8（840.53セント）と呼ぶ。もしそれが可能ならば、たくさんの人はmavilaテンペラメントを好むだろう。

18平均律はサブ平均律2、3、6、そして9をもつ。そして18平均律自体は36平均律と72平均律の4分の1である。そのことはとてもフラットされた4度をもつ点と、良いサブマイナーサードをもつ点で、13平均律と類似点を生む。11平均律においては短3度がとてもシャープされており、その2つでとてもフラットされた5度が、16平均律とはシャープされた4度とフラットされた5度が、17平均律と19平均律とは狭い半音、そしてその半音3つが包む全音が類似点である。一般的な慣習から大きく逸れた素晴らしいチューニングである。

# 18平均律の音程と近似値

各周波数比の大きさが16以内で表現される純正音程は以下のようになる。これはedjirulerを用いて、[number of equal divisions=18, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.3]というパラメータで生成したものである。「The “neighborhood” of JI」の一覧はこちら（huygens-fokker）を参照のこと。

## ヘクサトニック（6音音階）：

6音均等の全音音階：3 3 3 3 3 3

4L2sのBicycle：4 4 1 4 4 1

2L4sのRiceヘクサトニック：2 5 2 2 5 2

## オクタトニック（8音音階）：

5L3sのFatherオクタトニック：3 1 3 3 1 3 3 1

2L6sのRiceオクタトニック：2 2 3 2 2 2 3 2

# ギターアプリケーション

18平均律は、初めてフレットをし直すときに理想的な音階である。なぜなら12平均律からすべての偶数の数を保持できるからである。つまり1/3の仕事が終わっているのである。

「Fatherオクタトニック」音階は、6限ギターにおいて、4つの466.667セント音程と、2弦と3弦の間に生まれる1つの533.333セント音程による「逆スタンダード」チューニングでとても簡単にマップされる。そして14や16、21のような平均律より、やわからな学習曲線を作る。すべての平均律はほとんどすべて一様に、解放に4度のシャープされたものと短3度、中立3度のシリーズがチューニングされている。そしてそれらの音階はしばしば、手動で大きく拡大され位置が移動される。

# コンマをなだらかにする

18平均律を< 18 29 42 51 62 67 |ヴァルとみなした時、次のリストのコンマをテンパーアウトする。

# 曲を聴く

* 18ETPrelude by Aaron Andrew Hunt
* Prelude in 18et by Chris Vaisvil => composer notes
* Flippertronics by Chris Vaisvil
* Gerbils at the Wheel of Government by Chris Vaisvil (in 9 and 18 edo simultaneously)
* Do Androids Dream Of 18ED2? by Carlo Serafini (blog entry)
250edo

Welcome to 250edo. 250edo factors into {{#rreplace: 1 2 5 10 25 50 125 250|/(\d+)(\D+)?/|\1\2}}. This means it can be used to play some of the harmonic content of 50edo, but can use the five times more divisions to reduce harmonic trash. It has the same accurate 10/9 and 9/5 as 125edo, because it is contorted in 5-limit and 7-limit.

## 4.800

250edo has a step size of 4.8 cents. It is a commatonal EDO, because the step size represents commas like 385/384, 325/324. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 250edo (ordered by absolute error).