recursal/Fanatic-Fandom huggingface dataset:
15平均律

15 japanese.png

Wikipediaから

音楽において、15 equal temperamentは15TETや15EDO、または15ETと呼ばれる。15平均律はオクターブを15の均等なステップに分割し、テンパーされた音階である。各ステップは周波数比2^(1/15)、または80セントである。5平均律が3つの音階、または3平均律が5つの音階と解釈することが可能である。

15平均律は7リミットにもとづく音程に近いいくつかの音程を持つため、7リミットテンペラメントとして見なすことが可能である。しかしまた、明らかに11リミット音程に近い音程もいくつか含んでいる。ゆえに、11リミットテンペラメントとして述べることが合理的である。しかし12平均律の3度と9度の響きからは離れているため、15平均律に近いJIを探すことは、要求されているそれらのコードを避けることがベストだと考えられる。もしくは、注意深く扱う必要がある。15平均律はまた、明確なシントニックコンマを持つ5アドリミットインターバル（3/2、5/4、6/5）の表現と、異なっていると認識できる最も小さい平均律である。このことは注目に値する。

15平均律のシステムにおいて、メジャーサードは2（1200cent）を完全に分割できず、そして一般の3全音（600セント）の欠如と結びつく。このチューニングは、はじめ方向感覚を失わせる。しかしながら、ギターは12音システムギターと異なり、Eからe、または6弦から1弦のように対照的にチューニングできる。カーブを学ぶことはとても管理しやすい。すべてのコードはどこでも同じ転調のようにみえ、マイナーアルペジオは垂直に積み重ねられ、とても簡単に演奏できる。15音はたぶん誰もがより優れたハーモニーとゼンハーモニー、音数の管理のしやすさ、5平均律の重なりによる音の特徴に興味を持つ、有望な駆け出しとなる。

# 15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=15, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.25]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。

# イメージ

15edo wheel.png15edo wheel 02.png15edo wheel 03.png

# 12平均律の音程と近似値

15平均律は6/5や5/3といった周波数比5の中で、12平均律以上にいくつかの小さな改善点を提案する。そしてよりよい7倍音や11倍音の響きの近似を持つ。しかし3倍音からは離れる。この近似から抜け出すための特別な方法を行うと、バグと同様にたくさんの特徴をもつ、5L5sMOS音階を構築する。5L5sMOS音階の各音は7リミットオトーナルとウトーナルテトラのためのルートとして役立つ。同様に5リミットのメジャーまたはマイナーセブンスコードとしても役立つ。ウトーナルとはいくつかの周波数の最大公倍数のリミットを差し、オトーナルとは最大値をさす。たとえば10:12:15のコードは5リミットのウトーナリティであり、15リミットのオトーナリティである（otonality and utonality）。5L5sMOS音階はBlackwood temperamentとして知らており、Easley Blackwood, Jrから名づけられている。彼は初めてこれを文章で残した。この文章はまた、Igliashon Jonesにより、「Five is Not an Odd Number」で広範囲にわたり書かれている。15平均律の深いハーモニーの扱い方は、基本的にこのテンペラメントと7と11リミットの拡張にもとづく。Harmony in 15edo Blacksmith[10]もまた参照のこと。

# 記譜法

15平均律の記譜法にはさまざまな方法がある。そして記譜法の選択は、作曲家が重点を置く点が、ランク2テンペラメントなのか、MOS音階なのかに大きく依存している。

# Blackwoodの記譜法

10進数バージョン：1から0（0は10という意味）という言葉を使い、5平均律の3つのチェーンの1つは奇数によってあらわされる。偶数によって2番目が表される。そして3番目は臨時記号のついた数で表される。3番目の臨時記号は奇数にシャープをつけたものでも、偶数にフラットをつけたものでも構わない。

ギターバージョン：15平均律ギターにおいて、5平均律からうまれる「完全4度」のため、すべての開放弦は完全4度でチューニングすることが可能である。そして間隔は正確に2オクターブのままである。もしBから4度圏を始めるなら（B-E-A-D-G-(B)）、その時開放弦は通常（E-A-D-G-B-E）として記録される。しかしながら、4度圏は5回で閉じてしまう。15平均律の残り10音に続かないのである。臨時記号を用いて5平均律の2つのチェーンを記録する必要がある。この記譜法は5平均律ではない、臨時記号を使う5線譜にもとづくものとしても、特別なものではない。しかしながらギターにおける15平均律の直感的アプローチが反映されているため、便利である。なぜなら5平均律は、フレットボードをナビゲートするのに便利な3リミットのランドマーク（完全4度や完全5度）のセットという利便性を提供するためである。コードチャートを書くのに特に便利で、そこでは独創的な臨時記号を表す記号は無視できる。

# ランク2テンペラメント

悪い15ETランク2テンペラメントのリスト

平均律とは異なった15ET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

15平均律を< 15 24 35 42 52 56 |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。

# 理論

The 15-Tone Scale System by Ivor Darreg Permalink

The Pentadecaphonic System

15-EDO Tutorial by Brent Carson Permalink

# 実践理論と本

Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009. 15平均律において知られている、すべての音階とテンペラメントファミリーのレポジトリ。300以上の例とコード進行がおさめられている。
10/9

Interval information

In 5-limit Just Intonation, **10/9** is a small whole tone of about 182.4¢. It is a superparticular interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is 9/8 (about 203.9¢), which is 81/80 (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to 12edo's single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, 19edo, 31edo and other meantone systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

*See also Gallery of Just Intervals*
135/128

**135/128**

92.1787 cents

File:Jid 135 128 pluck adu dr220.mp3 sound sample

The 5-limit interval 135/128, about 92.2 cents in size, is called the *pelogic comma*, *major limma* or *major chroma*.

As a comma it represents the difference between three perfect fourths and a just major third (plus an octave).

See: Gallery of Just Intervals
14348edo

The 14348 division divides the octave into 14348 equal parts of 0.083635 cents each. It is a strong 17-limit system, with a lower 17-limit relative error than any smaller edo aside from 7033. It is also distinctly consistent in the 29 limit, and has a lower 23-limit relative error than any lower division aside from 2460, 8269, 8539 and 11664. Besides all that it is a zeta peak edo. It factors as 2^2 \* 17 \* 211, so 17, 34, 68 and 422 are all divisors.
107edo

107edo has 11.214953271 cents as its step size.

Tempers out:

File:1.png 1/1 in the 3-limit

File:2.png 3125/3072 in the 5-limit

File:3.png 3125/3087 2240/2187 1029/1024 12005/11664 in the 7-limit

File:4.png 14700/14641 9375/9317 16384/16335 10000/9801 5488/5445 100/99 78125/76032 8575/8448 1375/1372 4125/4096 1232/1215 48125/46656 26411/25920 12100/11907 3388/3375 21175/20736 1331/1323 9317/9216 in the 11-limit

File:5.png 28672/28561 87500/85683 28812/28561 28875/28561 24500/24167 15625/15379 2205/2197 61875/61516 60368/59319 60500/59319 6655/6591 20480/20449 62500/61347 20580/20449 1875/1859 14000/13689 38416/38025 1540/1521 75625/74529 847/845 33275/33124 17500/17303 78125/77077 1575/1573 28125/28028 81920/81081 3920/3861 84035/82368 4096/4095 2500/2457 196/195 30625/29952 87808/85293 16807/16640 275/273 88000/85293 13475/13312 9680/9477 53361/53248 5324/5265 14641/14625 36400/35937 364/363 31213/30976 1625/1617 11375/11264 40768/40095 5200/5103 10192/10125 15925/15552 572/567 7007/6912 77077/76800 86515/86436 86515/86016 84500/83853 2704/2673 57967/57024 84500/83349 676/675 21125/20736 57967/57600 9295/9261 9295/9216 43940/43923 43940/43659 76895/76032 15379/15360 in the 13-limit
128edo

The 128 equal division divides the octave into 128 equal parts of exactly 9.375 cents each. It is the optimal patent val for 7-limit rodan temperament. It tempers out 2109375/2097152 in the 5-limit; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

# Scales

radon5

radon11

radon16

128 notes per octave on Alto Saxophon - Philipp Gerschlauer
138edo

Welcome to 138edo

## 138edo = 6edo × 23edo

Distinctively interesting choice for an edo, due to its divisibility by both 6 and 23. This means it can be used to play 6edo music and 23edo music at the same time, using the 138/23 and 138/6 vals respectively.

## 8.696

138edo has a step size of 8.695652173913 cents. It is a commatonal EDO, because the step size represents commas like 3125/3072, 225/224, 100/99.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 138edo (ordered by absolute error).
17tppp4 flash

Embedded streaming recordings of 17tppp4:

Paul Kotheimer: *Three Short Songs* File:17tppp4 3shortsongsI.mp3 File:17tppp4 3shortsongsII.mp3 File:17tppp4 3shortsongsIII.mp3

K.:*Needs A Title* File:17tppp4 Needs A Title.mp3

Ian Dicke: *These Things Don't Happen Here* File:17tppp4 These Things.mp3

Dan Stearns: *Cirrus and Pileus* File:17tppp4 Cirrus.mp3

Jacob Barton: *Utopian Rounds* File:17tppp4 Multiverse.mp3

Chris Bailey: *Lost and Found Things, #2* File:17tppp4 lostfound.mp3

Paul Kotheimer: *One Dollar* File:17tppp4 One Dollar.mp3

Paul Rapoport: "Goodbye (Apple)" and "Bananas" from *Songs of Fruits and Vegetables*

Andrew Heathwaite: *Two Sinner* File:17tppp4 Heathwaite.mp3

Travis Weller: *Dirt, dust, and pollen, paralleled by signal bells* [In two parts because of file size restrictions] File:17tppp4 OwlI.mp3 File:17tppp4 OwlII.mp3

Nathan Brock: *Song* File:17tppp4 Brock Song.mp3

Elaine Walker: *Love Song* File:17tppp4 Walker Love Song.mp3

Tommy Scheurich: Confrontation and Finale from *Story Worth Telling* File:17tppp4 confinale.mp3
15-limit tonality diamond

The **15-limit tonality diamond** has the following notes:

## Symmetry group

The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:

* Transformation *R*: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
* Transformation *S*: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
* Transformation *S'*: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
* Transformation *T*: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13

These generators have the relations *R*² = *S*² = *T*² = *S**² = I, (*****SS**)³ = I, *RS* = *SR*, *RS **=*** **S***R*, and *T* commutes with the three other generators. Thus the symmetry group is isomorphic to *S*₃ × *C*₂².

### Orbits and Invariant Subsets

The Hendrix diamond is invariant under action by *R,* *S'* and *T*, and the images of the action of *S* and *S*² on the Hendrix diamond are the 11-Hendrix diamond and 13-Hendrix diamond respectively.

Two other interesting invariant subsets are the 5-limit tonality diamond and the tonality diamond constructed from the harmonics 1, 3, 5, 9 and 15.
105edo

**105edo** is the equal division of the octave into 105 equal parts of 11.429 cents each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, tempering out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the optimal patent val for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though 74edo is in that case the optimal patent val. 105 is highly composite, being the product 3\*5\*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all.

## 105edo close-up

```
C . . Dbb B## . . C# . . Db . . . C## . . D
```
1448edo

The 1448 division divides the octave into 1448 equal parts of 0.8287 cents each. It is a strong 13-limit system, and if you don't care about 17, a terrific 2.3.5.7.11.13.19.23 system. It is a zeta peak edo. A basis for the 13-limit commas is 3025/3024, 4225/4224, 4375/4374, 140625/140608 and 823680/823543.
14平均律

# 14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=14, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.25]というパラメータで生成したものである。

# ハーモニー

14平均律の特徴は、5:7:9:11:17:19を除いて低いリミットのJIアプローチを探すことには向いていないということである。しかしながら、7/5、7/6、9/7、10/7、10/9、11/7、11/9、そして11/10は近似であると認識することができる。もし14平均律がこれらの近似であると認識できるのであれば、高い損傷を負い本ページ最後のコンマのリストをテンパーアウトする11リミットテンペラメントは、複雑ではなく、シンプルなテンペラメントであるという結論になるだろう。

14平均律は非常にわずかなゼンハーモニックアピールを持ち、これは17平均律に似ている。これは12TETでは通常2つであるのに対し、3つのタイプの3度と3つのタイプの6度を持つためである。14平均律はまた4度と5度と認識できるものも持つため、これは音を足すのではない、3和音の響きの代わりの探求によい。5L4sのトライアド・リッチ9音MOS音階をもち、そこで9音のうち7音がサブマイナー、スーパーメジャー、中立3和音のためのトニックをもつ。

# 記譜法

Ivor Darregがこの記事を書いた。

14音階は、7音平均律の音に加え、通常のシャープとフラットを使うという新しい状況を提供する。このことは7音の5度圏（明らかに歪んでいるが）がすでにF C G D A E Bというように記譜され名づけられているため、混同してしまうようなミスリードを引き起こしそうである。しかし14音の5度圏は存在しない。そこには単純に、7つの5度が円の中に交わることなく2セット存在する。したがってBb、B、またはBナチュラルとFシャープのようなものは、14音システム音程の中では5度では「ない」のだろう。B Fのような音も、7や14の中ではCGやAEが引き起こすようなとても歪んだ5度の響きがする。私たちが提案するのは新しい14音の呼び方であり、F\* C\* G\* D\* A\* E\* B\*という7つのセットを2つ目とすることである。アスタリスクは5線譜に書きたいと思うような、望むものでよく、矢印などでもよい。

下の図はTDWによって作成されたものであり、提案されたものと同じ推薦される「スタンダード・ノーテーション」である。

# コンマをなだらかにする

14平均律を< 14 22 33 39 48 52 |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。

# モード

6 6 2 - MOS of 2L1s

5 5 4 - MOS of 2L1s

5 4 5 - MOS of 2L1s

4 4 4 2 - MOS of 3L1s

4 2 4 4 - MOS of 3L1s

4 1 4 4 1 - MOS of 3L2s

4 1 4 1 4 - MOS of 3L2s

3 3 3 3 2 - MOS of 4L1s

3 2 3 3 3 - MOS of 4L1s

3 2 2 2 2 3 - MOS of 2L4s

2 2 3 2 2 3 - MOS of 2L4s

**3 3 1 3 3 1 -** MOS of 4L2s

3 1 3 3 1 3 - MOS of 4L2s

3 1 3 1 3 3 - MOS of 4L2s

2 2 2 2 2 2 2 = 7edo

2 2 1 2 2 2 2 1 - MOS of 6L2s

2 2 2 1 2 2 2 1 - MOS of 6L2s

**2 2 2 2 1 2 2 1 -** MOS of 6L2s

2 1 2 2 1 2 2 2 - MOS of 6L2s

2 1 2 1 2 1 2 1 2 - MOS of 5L4s

2 1 2 1 2 1 2 2 1 - MOS of 5L4s

2 1 2 1 2 2 1 2 1 - MOS of 5L4s

2 1 1 2 1 2 1 1 2 1 - MOS of 4L6s

2 1 1 1 2 1 1 2 1 1 1 - MOS of 3L8s

**1 1 2 1 1 1 2 1 1 1 2** - MOS of 3L8s

# 本

Libro\_Tetradecafónico.PNG

*Sword, Ron. "Tetradecaphonic Scales for Guitar" IAAA Press. First Ed: June 2009.*
239edo

The *239 equal division* divides the octave into 239 equal parts of 5.021 cents each. In the 7-limit, it tempers out 10976/10935, 5120/5103 and 2401/2400 and supports hemififths temperament, providing an excellent tuning. It also supports and provides a good tuning for quasiorwell temperament. 239 is the 52nd prime EDO.
126edo

The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit srutal temperament. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.
13-limit

The 13-prime-limit refers to a constraint on selecting just intonation intervals such that the highest prime number in all ratios is 13. Thus, 40/39 would be allowable, since 40 is 2\*2\*2\*5 and 39 is 3\*13, but 34/33 would not be allowable, since 34 is 2\*17, and 17 is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, 3/2 is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. 23/13 is not within the 13-limit, since 23 is a prime number higher than 13).

The 13-prime-limit can be modeled in a 5-dimensional lattice, with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a sixth dimension is need.

Edos good for 13-limit are 1, 2, 3, 4, 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 183, 207, 217, 224, 270, 494, 851, 1075, 1282, 1578, 2159, 2190, 2684, 3265, 3535, 4573, 5004, 5585, 6079, 8269, 8539, 13854, 14124, 16808, 20203, 22887, 28742, 32007, 37011, 50434, 50928, 51629, 54624, 56202, 59467, 64471, 65052, ... .

.

# Music

Venusian Cataclysms play by Dave Hill

Chord Progression on the Harmonic Overtone Series play by Dave Hill
109edo

**109edo** is the equal division of the octave into 109 parts of 11.009 cents each. It tempers out 20000/19683 in the 5-limit; 245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the optimal patent val for 7-limit octacot temperament, and 11 and 13 limit leapweek; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th prime edo.
16/9

**16/9**

996.09000 cents

File:Jid 16 9 pluck adu dr220.mp3 sound sample

In 3-limit Just Intonation, 16/9 is the Pythagorean minor seventh, at about 996.1 cents. It is equal to two perfect fourths, or (4/3)\*(4/3). It differs from the nearby 5-limit minor seventh 9/5 (~1017.6 cents) by the syntonic comma of 81/80 (~21.5 cents).

See: Gallery of Just Intervals
121edo

The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374 and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the optimal patent val for 13-limit grendel temperament. It is consistent through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows swetismic chords, because it tempers out 325/324 it allows marveltwin chords, because it tempers out 640/637 it allows huntmic chords, because it tempers out 352/351 it allows minthmic chords, because it tempers out 676/675 it allows island chords and because it tempers out 1575/1573 it allows the nicolic tetrad. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

# 13-limit detempering of 121et

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
16edX

This is the scale which occurs as the dominant major edX.

## Intervals

By a surprising coincidence, neutral 16edX turns out to be a false Pelogic temperament with a very pure 5:4 (mistuned by no more than 3.596 cents), and tuning 5:3 pure creates a tenth almost exactly equal to 38/29edo, and tuning 4:3 pure creates an almost exact 43/74edo fifth.
13/8

**13/8**

840.52766 cents

File:Jid 13 8 pluck adu dr220.mp3 sound sample

13/8 is a neutral sixth measuring about 840.5¢, falling between the categories of minor sixth and major sixth. In 13-limit Just Intonation, 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the superparticular ratios 13/12 and 14/13, respectively. It differs from the 11-limit neutral sixth 18/11 by 144/143, about 12.1¢.

See: Gallery of Just Intervals
20/17

Interval information

In 17-limit Just Intonation, **20/17** is the "septendecimal minor third", measuring about 281.4¢. It is the mediant between 7/6 and 13/11, as it is (7+13)/(6+11). It is found in the harmonic series between the 17th and 20th harmonics.

See:

* 4\17 (282.353 cents)
* 17/10 the octave complement
* 51/20 the tritave complement
* Gallery of Just Intervals
13-3reduced-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

 **13-3reduced-limit** is the tritave version of 13-odd-limit.

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

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

* 13/12, 36/13
* 12/11, 11/4
* 11/10, 30/11
* 10/9, 27/10
* 9/8, 8/3
* 8/7, 21/8
* 15/13, 13/5
* 7/6, 18/7
* 13/11, 33/13
* 6/5, 5/2
* 11/9, 27/11
* 5/4, 12/5
* 9/7, 7/3
* 13/10, 30/13
* 4/3, 9/4
* 15/11, 11/5
* 11/8, 24/11
* 18/13, 13/6
* 7/5, 15/7
* 10/7, 21/10
* 13/9, 27/13
* 3/2, 2/1
* 11/7, 21/11
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 5/3, 9/5
* 12/7, 7/4
12edo comma pump examples

The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: Meantone comma pump examples. These work in any meantone temperament, including 19edo, 26edo, 31edo, etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by 12edo are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

# Augmented (128/125) comma pumps

* *Looney Tunes* theme song, c. 1940. *http://www.youtube.com/watch?v=F9S7yhD5M9A*
* *I Wish*, Skee-Lo, 1995. *http://www.youtube.com/watch?v=Zyf0YwUJcqk*
* *Giant Steps*, John Coltrane 1959 http://en.wikipedia.org/wiki/Giant\_Steps\_%28composition%29 https://www.youtube.com/watch?v=2kotK9FNEYU
	+ **Mike Battaglia note:** you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.
15/8

**15/8**

1088.2687 cents

File:Jid 15 8 pluck adu dr220.mp3 sound sample

In 5-limit Just Intonation, 15/8 is a major seventh of about 1088.3¢. It is also the 15th overtone (octave-reduced), and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15 is 3\*5, it can be seen as a perfect fifth above a major third or vice versa, and this understanding is compatible with the 1100¢ interval of 12edo.

*Since 15 is a perfect fifth above 10 (15/10 = 3/2), root-3rd-P5 triads can be formed with the 10th harmonic as root and 15th harmonic as perfect fifth. The simplest and most familiar example is the classic minor triad 10:12:15 -- a 6/5 with a 5/4 stacked on top of it. Another is the Barbados triad, 10:13:15 -- a 13/10 on bottom and a 15/13 on top. And a particularly uncommon but mentionable example is the 23-limit inframinor triad 20:23:30.*

See: Gallery of Just Intervals
11edt

11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of 7edo, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11\_of\_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

Mozart's sonata #11 in A Major K331 in 11 EDT (using a 11 => 12 key mapping so octaves become tritaves)

Frozen Time Occupies Wall Street by Chris Vaisvil =>information about the piece

Molly's Playground by Chris Vaisvil => information about the piece
13/10

**13/10**

454.21395 cents

File:Jid 13 10 pluck adu dr220.mp3 sound sample

In 13-limit Just Intonation, 13/10 is an interseptimal interval measuring about 454.2¢. It falls in an ambiguous zone between a wide major third such as 9/7 and a flat perfect fourth such as 21/16. The descriptor "interseptimal" comes from Margo Schulter, and indicates its position between those two septimal (7-based) extremes. 13/10 appears between the 10th and 13th overtones of the harmonic series and appears in such chords as 8:10:13, a quasi-augmented triad. 13/10 also appears in the relatively-simple 10:13:15 triad, which consists of an interseptimal ultramajor third (13/10) and an interseptimal inframinor third (15/13) which stack to make a 3/2 perfect fifth. It is well-approximated in 24edo, 29edo, 37edo, and of course, infinitely many other EDO systems.

See: Gallery of Just Intervals, List of root-3rd-P5 triads in JI
113edo

**113edo** is the 30th prime EDO. It has a step size of 10.619469026549 cents. It is a commatonal EDO. It is consistent and uniquely consistent in the 13-odd-limit and it is marvel, making it an interesting EDO choice, and quite possibly an alternative to 94edo for uniquely consistent in the 13-odd-limit marvel.
104edo

*104edo* divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, <104 165 241|, tempers out 3125/3072, and supports magic temperament. The sharp major third val, <104 165 242|, tempers out 2048/2025 and supports diaschismic temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the optimal patent val for 11-limit magic and the 13-limit magic extension necromancy. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

**17-limit Regular Temperaments**
17edt

# Properties

17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21.

17edt is the sixth zeta peak tritave division.

# Discussion

17edt is closely related to 13edt, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents),

leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .3 cents flat (in addition to equaling 256).

# Intervals

* Notes named so that C D E F G H J A B C = Lambda mode

It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).

# Z function

Below is a plot of the no-twos Z function in the vicinity of 17edt.

17edt.png
126/125

Deutsch

The **starling comma** or **septimal semicomma**, 126/125 (about 13.8 cents), is the only superparticular 7-limit comma which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which 12/7 falls short of three 6/5 minor thirds. It is also the amount by which two 5/3 major sixths (octave-reduced) exceed the 7/5 tritone, and the amount by which three 5/3s (octave-reduced) fall short of the 7/6 septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to starling temperament, and three minor thirds plus a 7/6 subminor third, when 126/125 is tempered out, gives the starling tetrad or septimal semicomma diminished seventh chord.

http://en.wikipedia.org/wiki/Septimal\_semicomma
159edo

The *159 equal division* divides the octave into 159 equal parts of 7.547 cents each. A salient fact about it is that 159 = 3\*53, so that it shares the same 5-limit thirds and fifths with 53edo. In the 7-limit it tempers out 1029/1024 and 10976/19935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for guiron and tritikleismic temperaments. It has a very accurate 11, and in the 11-limit tempers out 385/384, 441/440 and 4000/3993. In the 13-limit it tempers out 325/324 and 364/363, in the 17-limit 273/272 and 375/374, and in the 19-limit 343/342 and 361/360. It also provides the optimal patent val for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament portending.

Another and notable temperament supported by 159 is yarman temperament, with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.

Yarman temperament has MOS of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for arabic/turkish/persian music.

# Articles

* Ozan Yarman's dissertation
* paper, also by Ozan Yarman, gives a summary.
* Letter to Ozan Yarman by Margo Schulter Permalink
18edt

18edt means the division of the tritave into 18 equal parts of size 105.664 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a neutral sixth. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more of a 13/8, though this is allegedly a no-twos tuning. With octaves added, it also has a minor third and a major tenth which are both excellent as well as a minor thirteenth and major seventeenth which are still decent even though it skips actual octaves (in fact it is the non-octave semitone scale of 34edo). On the 3.7.13 subgroup it tempers out 351/343 and 2197/2187.

As the double of 9edt, it is the analog of 14edo insofar as it has a doubled harmonic chain. However, it, like 8edt, is not schismatic because 3:5:7 is a redundant chord. As a multiple of 9edt, it is the widest variety of 'White-Extraterrestrial-Tree' temperament.

0: 1/1

1: 105.664 cents 16/15

2: 211.328 cents 9/8

3: 316.993 cents 6/5

4: 422.657 cents 9/7

5: 528.321 cents 27/20

6: 633.985 cents 13/9

7: 739.649 cents 17/13

8: 845.313 cents 5/3

9: 950.9775 cents 19/11

10: 1056.642 cents 9/5

11: 1162.306 cents 49/25

12: 1267.970 cents 27/13

13: 1373.634 cents 20/9

14: 1479.298 cents 7/3

15: 1584.963 cents 5/2

16: 1690.627 cents 8/3

17: 1806.2905 cents 45/16

18: 3/1
22edt

**22edt** is the equal division of the third harmonic into 22 tones, each 86.453 cents in size.

22edt has good approximations of the 7th, 11th, 19th and 20th harmonics. It also has the 4L+5s MOS with L=3 and s=2 approximating 5/3 somewhat fuzzily.

Like 11edt, both the octave and minor whole tone 10:9 are about 10c off (sharp and flat respectively) dissonant but recognizable. Like 16edt and Blackwood, admitting the octave induces an interpretation into a tritave-based version of Whitewood temperament.

# **Intervals**

86.4525

172.905

259.3575

345.81

432.2625

518.715

605.1675

691.62

778.0725

864.525

950.9775

1037.43

1123.8825

1210.335

1296.7875

1383.24

1469.6925

1556.145

1642.5975

1729.05

1815.5025

3/1

# Compositions

http://www.archive.org/details/TuneIn22Edt - Kosmorsky

22 edt piano improvisation by Chris Vaisvil
17edo Solfege

Expanding the "movable-do" solfege system for 17edo: an aid for microtonal singing; a way of wrapping yr brain around a new sound system.

Ordinary solfege syllables work very well in 17:

Note that syllables usually treated as enharmonic equivalents become distinct pitches in 17 (eg. "ra" & "di"). Spelling matters here.

This system could work just fine on its own, but what if I want to respell the pitches to put them into another pitch class? For example, what if I want to treat D# as a kind of third (a neutral third) instead of a kind of second (an augmented second)? For pitches, we have up-down arrows (thank you Sagittal) that can allow us to respell like this:

So, for example, we can respell our augmented second, D#, as a neutral third, E-down, & get something more intuitive & useful for certain purposes.

But what if we want solfege syllables for these respellings?

### ru, mu, lu & tu

I offer the suffix "-u" to indicate neutral 2nds, 3rds, 6ths, & 7ths.

So if we encounter a "ru," we know to take it as a neutral second (not an augmented unison); if we encounter "lu," we know to take it as a neutral sixth (not an augmented fifth) & so on.

### fu & su

What about the two tritones? Perhaps you want to respell G-flat as F-up. For this, I offer "fu"; likewise, if you want G-down in place of F-up, I offer "su".

### other enharmonics

Some offers, to fill out the octave:

###

### side-by-side

For comparison, the full system:

See also 22edo Solfege
22/21

**22/21**

80.537 cents

File:Jid 22 21 pluck adu dr220.mp3 sound sample

22/21 is a small superparticular semitone of about 80.5¢ that appears in 11-limit Just Intonation. It makes the difference between the 21st and 22nd overtone. It is close in size to 21/20, a 7-limit superparticular interval, differing from it by 441/440, about 3.9¢.

The single degree of 88cET can function as both 21/20 and 22/21, thus tempering out 441/440.

See: Gallery of Just Intervals, List of Superparticular Intervals
23-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 **23-odd-limit** intervals. It contains all the wonders of 94edo. Also see 23-limit.

* 24/23, 23/12
* 23/22, 44/23
* 22/21, 21/11
* 21/20, 40/21
* 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
* 23/21, 42/23
* 11/10, 20/11
* 21/19, 38/21
* 10/9, 9/5
* 19/17, 34/19
* 9/8, 16/9
* 26/23, 23/13
* 17/15, 30/17
* 8/7, 7/4
* 23/20, 40/23
* 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
* 23/19, 38/23
* 17/14, 28/17
* 28/23, 23/14
* 11/9, 18/11
* 16/13, 13/8
* 21/17, 34/21
* 26/21, 21/13
* 5/4, 8/5
* 24/19, 19/12
* 19/15, 30/19
* 14/11, 11/7
* 23/18, 36/23
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 30/23, 23/15
* 17/13, 26/17
* 21/16, 32/21
* 4/3, 3/2
* 23/17, 34/23
* 19/14, 28/19
* 15/11, 22/15
* 26/19, 19/13
* 11/8, 16/11
* 18/13, 13/9
* 32/23, 23/16
* 7/5, 10/7
* 24/17, 17/12

todo: also cents and descriptions
1848edo

The 1848 equal division divides the octave into 1848 equal parts of 0.64935 cents each. It is a super strong 11-limit division, having the lowest 11-limit relative error than any division until 6079. It tempers out the 11-limit commas 9801/9800, 151263/151250, 1771561/1771470 and 3294225/3294172. It also tempers out the 7-limit landscape comma, 250047/250000. It is distinctly consistent through the 15-limit, and tempers out the 13-limit commas 4225/4224 and 6656/6655. In the 5-limit it is an atomic system, tempering out the atom, |161 -84 -12>; and also the minortone comma, |-16 35 -17>.

1848 is highly composite, factoring as 2^3 \* 3 \* 7 \* 11. Among its divisors are 12, 14, 22, 24, 28, 56, 77, 84, 88, 154, and 231.
11-limit comma temperaments

# Bosonic

Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [<26 0 -22 73|, <0 1 2 0|]

EDOs: 26, 130, 546, 676, 806c

Badness: 0.1558

## 11-limit

Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [<26 0 -22 73 90|, <0 1 2 0 0|]

EDOs: 26, 104c, 130

Badness: 0.0652

## 13-limit

Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [<26 0 -22 73 90 55|, <0 1 2 0 0 1|]

EDOs: 26, 104c, 130

Badness: 0.0329

# Fermionic

Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [<26 0 -22 73 -116|, <0 1 2 0 5|]

EDOs: 130, 286, 416, 546, 1508bcd

Badness: 0.0906

## 13-limit

Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [<26 0 -22 73 -116 55|, <0 1 2 0 5 1|]

EDOs: 130, 286, 416, 546, 962bf

Badness: 0.0436
18/17

**18/17**

98.954592230368 cents

File:Jid 18 17 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 18/17 is the "small septendecimal semitone" of about 99¢. It is very close to 12edo's "half step" of 100¢, and fairly close to the "large septendecimal semitone" of 17/16 (~105¢).

See: Gallery of Just Intervals
108edo

The *108 equal division* divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine temperament, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
27/11

Interval information

In 11-limit Just Intonation, **27/11** is a neutral tenth of about 1554.5¢, falling in between "major tenth" and "minor tenth" territory. It is one of the simplest neutral tenths in just intonation, but of course, only one of persony (others include 39/16, 22/9, 120/49 and 49/20). It is nearly halfway between two intervals of 12edo, implying that it is both very xenharmonic and well-represented in 24edo. It is the tritave inverse of 11/9.

In the 11-limit they xad, 27/11 appears between the harmonic 27th and the harmonic 11th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral tenth, 22/9, which together make a perfect fifth. Persony temperaments, including 17edo, 24edo, 31edo, 41edo, 58edo, 72edo, 130edo, 202edo, miracle, harry, and sesquart, conflate these two neutral tenths, allowing one neutral tenth interval to be stacked to generate a perfect fifth. 27/11 differs from 22/9 by 243/242, but also from 120/49 by 441/440 and 49/20 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament jove.

*See also Gallery of Just Intervals*
2-3reduced-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

 The 2-3reduced-limit embeds the perfect fifth and octave into the tritave. The perfect fourth is featured in 4-3reduced-limit.

* 3/2, 2/1
132edo

Welcome to 132edo

## 132edo = 11edo × 12edo

Distinctively interesting choice for an edo, due to its divisibility by both 11 and 12. This means it can be used to play 11edo music and 12edo music at the same time, using the 132/12 and 132/11 vals respectively.

## 9.091

132edo has a step size of 9.0909090909091 cents. It is a commatonal EDO, because the step size represents commas like 190/189, 3125/3072, 78/77, 153/152.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 132edo (ordered by absolute error).
26/25

**26/25**

67.9002 cents

File:Jid 26 25 pluck adu dr220.mp3 sound sample

In 13-limit Just Intonation, 26/25 appears as the difference between the 26th harmonic and the 25th. Thus it makes the difference between 13/8 and 25/16 (a stack of two 5/4's). It measures about 67.9¢.

See: Gallery of Just Intervals, List of Superparticular Intervals
26edt

The 26 equal division of 3 (the tritave), divides it into 26 equal parts of 73.152 cents each, corresponding to 16.404 edo. It is contorted in the 7-limit, tempering out the same commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh zeta peak tritave division. A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

## Intervals

It is a weird coincidence how 26edt intones any 26edo intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:

…and so on

It's not actually a coincidence. It is simply a demonstration of rational approximations to log₂(3). In this table are shown 8/5, 11/7, 19/12, 27/17, 30/19, 38/24, 46/29, 57/36. So all this table does is demonstrate that {{#rreplace: 5edo, 7edo, 12edo, 17edo, 19edo, 24edo, 29edo, 36edo|/(\d+)(\D+)?/|\1\2}} are good for 3-limit.

The Eel And Loach To Attack In Lasciviousness Are Insane play by Omega9
23edf

Twenty-three equal divisions of the perfect fifth (23ed3/2)

Rank 1 scale with step size of 30.52 cents.

Close to 39ed2 and/or 62ed3, however, the respective

octave and twelfth would need to be nearly 10 cents flat.

A proponent of this scale is Petr Pařízek.

Some intervals in table below, selected on the basis of

single-use of primes (for most cases):

–Todd Harrop (June 2015)
1200edo

The *1200 division* divides the octave in 1200 equal parts of exactly 1 cent each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely consistent through the 11-limit, which means the intervals of the 11-limit tonality diamond, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val <1200 1902 2786 3369 4141|. It is contorted in the 5-limit, having the same mapping as 600edo. In the 7-limit, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by 171edo. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by 494edo. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of quadritikleismic temperament: <1200 1902 2785 3368|. It also provides the optimal patent val for the 224&752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
27/10

Interval information

In 5-limit Just Intonation, **27/10** is a large minor eleventh of about 1719.6¢. It is a tritave inverse of a superparticular interval, as you can find it in the harmonic series between the 27th and the 10th overtones. It is one of two essential minor elevenths in the 5-limit; the other one is 8/3 (about 1698.0¢), which is 81/80 (about 21.5¢) lower than 27/10. 27/10 is a tritave-reduced undertone, and it is further from 12edo's single minor eleventh of 1700¢. Thus, 8/3 is more familiar and less difficult to tune by ear than 27/10.

The last three notes of a tritave JI major scale -- 12/5, 8/3, 3/1 -- move by a step of 10/9 followed by a step of 9/8 (or alternatively 12/5, 27/10, 3/1 -- move by a step of 9/8 followed by a step of 10/9). In systems where 81/80 is tempered out (in 12edo, 19edo, 31edo and other meantone systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

*See also Gallery of Just Intervals*
1L 10s

This MOS, generated by any interval up to a diatonic semitone of 1/11edo (109.091 cents), achieves its simplest harmonic entropy minimum where two generators equal 9/8. The temperament which occupies this harmonic entropy minimum is called Ripple, but there are several lower (and more complex) harmonic entropy minima of note including (in descending order of generator height): Passion (6/5=+3 generators), Octacot (3/2=+8 generators), Nautilus (3/2=-6 generators) and Valentine (7/4=-3 generators).
17-comma

134217728/129140163 = |27 -17>, called the 17-comma, is an interval of 66.765 cents, is the amount by which seventeen fifths fall short of ten octaves, or 2^10/(3/2)^17. For EDOs up to 150 the 17-comma is tempered out if and only if 17 divides it, for example 17edo, 34edo or 68edo.

See: comma, Gallery of Just Intervals
243edo

The *243 equal division* divides the octave into 243 equal parts of 4.938 cents each. It tempers out the semicomma (5-limit orwell comma) 2109375/2097152 in the 5-limit, and 2401/2400 and 4375/4374 in the 7-limit. In the 11-limit it tempers out 243/242 and 441/440, and provides the optimal patent val for ennealimnic temperament. In the 13-limit it tempers out 364/363 and 625/624, and provides the optimal temperament for 13-limit ennealimnic and the rank three jovial temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.
185edo

**185edo** divides the octave into 185 equal steps of size 6.4865 cents each. It tempers out the commas 126/125, 1029/1024, 6144/6125 and 243/242, making it useful for various purposes. It is an excellent tuning for starling temperament, the 7-limit planar temperament tempering out 126/125; valentine temperament, which also tempers out 1029/1024; and the rank three 89&154&185 temperament tempering out 126/125 and 243/242. With 185dff, the val <185 293 429 519 640 686|, it makes for a first-class tuning of the 13-limit miraculous temperament.

## Scales

* nova
* novadene