recursal/Fanatic-Fandom huggingface dataset:
12edo

Compact notation based on meantone notation:

<inline>

```
C  C# D  Eb E  F  F# G  G# A  Bb B
```
</inline>

**12edo** since it really is divisions per octave, perhaps better known as **12et** since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave (EDO) which can seriously claim to represent 5-limit harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents meantone.

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial (7/4) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the val <12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth zeta integral edo.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

## Rank two temperaments

* List of 12et rank two temperaments by badness
* List of 12et rank two temperaments by complexity
* List of edo-distinct 12f rank two temperaments

## Commas

12 EDO tempers out the following commas. (Note: This assumes val < 12 19 28 34 42 44 |.)

# Intervals

File:12ed2-11-001.svg

12ed2-11-001.svg

An expanded version of the above, including some higher-limit intervals:

File:12ed2-19-001e.svg

12ed2-19-001e.svg

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 12edo (ordered by absolute error).

## Split octave

* Tempered out: 26/25 50/49 56/55 99/98 100/99
1506edo

The 1506 division divides the octave into 1506 parts of 0.7968 cents each. It is a very strong 13 and 17 limit division, since it is the first past 494 with a lower 13-limit relative error, and likewise the first with a lower 17-limit relative error. Like 494 it is distinctly consistent through the 17 limit. It tends sharp, all of the odd primes to 17 being tuned sharply. A basis for the 13 limit commas is {4096/4095, 6656/6655, 9801/9800, 105644/105625, 371293/371250}, and for the 17-limit commas, {4096/4095, 4914/4913, 5832/5831, 6656/6655, 9801/9800, 28561/28560, 105644/105625}.
16-3reduced-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

 **16-3reduced-limit** is the tritave version of 15-odd-limit. It has proven helpful for showing JI properties of medium-sized edt systems (~24 .. 65 steps).

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

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

* 16/15, 45/16
* 15/14, 14/5
* 14/13, 39/14
* 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
* 16/13, 39/16
* 5/4, 12/5
* 14/11, 33/14
* 9/7, 7/3
* 13/10, 30/13
* 21/16, 16/7
* 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
* 16/11, 33/16
* 3/2, 2/1
* 14/9, 27/14
* 11/7, 21/11
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 5/3, 9/5
* 27/16, 16/9
* 12/7, 7/4
156edo

Welcome to 156edo

## 156edo = 12edo × 13edo

Distinctively interesting choice for an edo, due to its divisibility by both 12 and 13. This means it can be used to play 12edo music and 13edo music at the same time, using the 156/13 and 156/12 vals respectively.

## 7.692

156edo has a step size of 7.6923076923077 cents. Microtonal enthusiasts might notice it seems close to the 225/224 ratio, but it isn't in the patent val, because 156edo tempers out 225/224. The edo step size closest to 225/224, taking into account the primes that compose it, is in 171edo. You have to use the 2\*156 val, sacrificing 3 for a new 9, to make this approximation come true. It is up to the microtonalist to decide whether this tradeoff is worth it.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 156edo (ordered by absolute error).
15edf

# 15edf

0: 1/1 0.000 unison, perfect prime

1: 46.797 cents 46.797

2: 93.594 cents 93.594

3: 140.391 cents 140.391

4: 187.188 cents 187.188

5: 233.985 cents 233.985

6: 280.782 cents 280.782

7: 327.579 cents 327.579

8: 374.376 cents 374.376

9: 421.173 cents 421.173

10: 467.970 cents 467.970

11: 514.767 cents 514.767

12: 561.564 cents 561.564

13: 608.361 cents 608.361

14: 655.158 cents 655.158

15: 3/2 701.955 perfect fifth

Lookalikes: 26edo

* this is an interesting one
16/13

**16/13**

359.47234 cents

File:Jid 16 13 pluck adu dr220.mp3 sound sample

In 13-limit Just Intonation, 16/13 is a tridecimal (13-based) neutral third measuring about 359.5¢. It is the inversion of 13/8, the 13th harmonic. It differs from 5/4 by 65/64, about 26.8¢. It differs from the undecimal neutral third 11/9 by 144/143, about 12¢. A root-3rd-P5 triad featuring 16/13 is 26:32:39, which introduces another tridecimal neutral third, 39/32, which measures about 342.5¢. The interval between these two intervals is 512/507, about 17¢. While 16/13 is utonal, 39/32 is otonal, as it is the 39th overtone of the harmonic series.

See: Gallery of Just Intervals
14/11

**14/11**

417.50796 cents

File:Jid 14 11 pluck adu dr220.mp3 sound sample

In 11-limit Just Intonation, 14/11 is a supermajor third of about 417.5¢. It represents the difference between the 11th and 14th overtones of the harmonic series and appears in chords such as 8:11:14, the principal triad of Orgone temperament. 14/11 can also function as a Neo-Gothic major third, as it falls between 5/4 and 9/7. Indeed, it is the mediant ratio between those simpler intervals, as it is (5+9)/(4+7). Other relatively simple thirds in this region can be generated by taking the mediant between 5/4 and 14/11 (which is (5+14)/(4+11) = 19/15, about 409.2¢) and between 14/11 and 9/7 (which is (14+9)/(11+7) = 23/18, about 424.4¢. Also in this region is the Pythagorean (3-limit) major third of 81/64 (about 407.8¢), which can be generated by stacking four 3/2 perfect fifths and octave-reducing.

See: Gallery of Just Intonation Intervals List of root-3rd-P5 triads in JI, The Noble Mediant
15/13

Interval information

In 13-limit Just Intonation, **15/13** is an interval measuring about 247.7¢. In the language of Margo Schulter, 15/13 is an instance of an interseptimal interval, as it falls in an ambiguous zone between two septimal extremes -- namely the large major second 8/7 and the small minor third 7/6. (15/13)\*(13/10)=3/2, which implies that 15/13 and 13/10 make a 3/2 perfect fifth. Thus you can make a root-3rd-P5 triad that goes 26:30:39, with a 15/13 "inframinor third" up from the root. When being used as type of second, it is given the name "ultra second" as it is even sharper than 8/7 which is often called a "super major second".

## See also

* Gallery of Just Intervals,
* The Archipelago
* 26/15 its octave complement
105edo-a

*105edo, the wonderful meantone*
105edo belongs to a meantonic 13–limit system known as grosstone, can be considered an expanded version of 43edo just like how 94edo is an expanded version of 41edo.
## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 105edo (ordered by absolute error).
100/99

Deutsch

In Just Intonation,100/99 is a superparticular interval (the ptolemisma) of around 17.4¢ which makes the difference between 11/10, the greater undecimal neutral second of about 165¢, and 10/9, the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include 22edo, 41edo and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of 81/80, which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with 9/8. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an 11/8 and two 6/5s fall short of an octave, so that tempering out 100/99 allows for the ptolemismic triad which has intervals of 11/8, 6/5, 6/5.

See: Galley of Just Intervals
1-5 Syntonic Comma Meantone

In 1/5-comma meantone temperament, each perfect fifth is tempered by a fifth of a syntonic comma from its just value of 3/2. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of 43edo (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)
166edo

The 166 equal temperament (in short 166-EDO) divides the octave into 166 equal steps of size 7.229 cents each. Its principle interest lies in the usefulness of its approximations; it tempers out 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament marvel, in both the 11-limit and in the 13-limit extension hecate, and the rank two temperament wizard, which also tempers out 4000/3993, giving the optimal patent val for all of these. In the 13-limit it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives gizzard, the 72&94 temperament, for which 166 is an excellent tuning through the 19 limit.

Its prime factorization is 166 = 2 \* 83.

166edo (as 83edo) contains a very good approximation of the harmonic 7th. It's 0.15121 cent flat of the just interval 7:4.
15edt

# Properties

The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15> in the 5-limit, which is tempered out by 19edo but has an optimal patent val of 303edo. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&123, which has a mapping [<1 0 0 0|, <0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.

# Intervals of 15edt

15edt contains 4 intervals from 5edt and 2 intervals from 3edt, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.

15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.

# Z function

Below is a plot of the no-twos Z function in the vicinity of 15edt:

15edt.png

Music:

http://www.youtube.com/watch?v=bC\_Pc4jKm2k
152edo

The *152 equal division* divides the octave into 152 equally sized parts of 7.895 cents each. It is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It provides the optimal patent val for the 11-limit grendel and kwai linear temperaments, the 13-limit rank two temperament octopus, the planar temperament laka, and the rank five temperament tempering out 169/168. It tempers out 1600000/1594323, the amity comma, in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/15807 in the 7-limit; 540/539, 1375/1372, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit, and 169/168, 325/324, 351/350, 364/363, 1001/1000, and 4096/4095 in the 13-limit.

Paul Erlich has suggested that 152 could be considered a sort of universal tuning.

152 = 8 \* 19, with divisors 2, 4, 8, 19, 38, 76. It fuses 94edo and 58edo.
161edo

The *161 equal division* divides the octave into 161 equal parts of 7.453 cents each. It tempers out the Würschmidt comma, 393216/390625, in the 5-limit; 3136/3125, 6144/6125 and 2401/2400 in the 7-limit; 243/242, 441/440, 540/539 and 5632/5625 in the 11-limit; and 1188/1183, 351/350, 847/845, 1575/1573, 1001/1000 and 1716/1715 in the 13-limit. It serves as the optimal patent val for mintone temperament in the 5-, 7-, 11- and 13-limits.
147edo

**147edo** is the equal division of the octave into 147 parts of 8.1633 cents each. It tempers out 32805/32768 in the 5-limit; 225/224 and 3125/3087 in the 7-limit; 243/242 in the 11-limit; 364/363 in the 13-limit; 442/441 and 595/594 in the 17-limit. It is the optimal patent val for the 11-limit 41&106 temperament, and is the smallest division which is uniquely consistent through the 17-limit.

147 = 3 \* 72, with divisors 3, 7, 21 and 49.

# Scales

baldy6

baldy11

baldy17
14/13

**14/13**

128.29824 cents

File:Jid 14 13 pluck adu dr220.mp3 sound sample

14/13, the 2/3-tone or trienthird (one-third third) interval is a somewhat exotic 13-limit interval which, measuring about 128.3¢, is almost exactly 1/3 of a major third of 5/4. The small comma (5/4)/(14/13)^3 = 10985/10976 is tempered out in such notable EDOs as 46, 84, 94, 103, 111, 121, 130, 224, 270, 494 and 764, leading to a variety of temperaments in which three trienthirds make up a major third.

The trienthird was a favorite interval of Avicenna (Ibn Sina) for his scale constructions, and may be considered a smaller size of neutral second (a second between major and minor.)

In 13-limit Just Intonation, 14/13 represents the difference in size between the tridecimal minor third of 13/11 and tridecimal supermajor third of 14/11. It is also the difference between 13/10 and 7/5; 13/12 and 7/6; 13/9 and 14/9, and of course 13/8 and 7/4 and the inversions of the above. As it combines the primes 7 and 13, it appears in JI subgroup tunings involving those primes.

See: Gallery of Just Intervals
11/6

Interval information

**11/6** is the undecimal neutral seventh of about 1049.4 cents. It can be treated as a consonance in 11-limit harmony, forming a part of such chords as 6:7:9:11 (1-7/6-3/2-11/6) and the neutral tetrad 18:22:27:33 (1-11/9-3/2-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.
13edt

The 13 equal division of 3, the tritave, divides it into 13 equal parts of 146.304 cents each, corresponding to 8.202 edo. An alternative name for it is the Bohlen-Pierce scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as bohpier temperament. It is less impressive in higher p-limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 26edt, 39edt and 52edt come to the fore.

Below is a plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak edt.

13edt.png

## See also

* Catalog of 3.5.7 subgroup rank two temperaments
12/11

Interval information

The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the superparticular ratio 12/11, and is about 150.6 cents large. One step of 8edo is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8>. It follows that EDOs which are multiples of 8, such as 16edo and 24edo, will also represent this interval well.

**12/11** differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include 15edo, 22edo, 31edo, orwell, porcupine, mohajira and valentine.
11edo Zine

In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, Andrew Heathwaite assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to 11edo on this here wiki and taking a listen to the files there under "Compositions".

And if you're interested in some audio traces from the semester, see the blog post for the Oddmusic Radio Shower on February 16, 2011.

More to come!

~Andrew
12edt

# Division of the tritave (3/1) into 12 equal parts

12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for hemikleismic temperament. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.

# A scala formatted description of the tuning

! C:\Cakewalk\scales\tritave-in-12.scl

!

3/1 in 12

12

!

158.49625

316.99250

475.48875

633.98500

792.48125

950.97750

1109.47375

1267.97000

1426.46625

1584.96250

1743.45875

3/1

# Exactly analogous to meantone

In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is hyperpyth which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

# Compositions

Instant Gamelan play by Carlo Serafini

Tritave in 12 by Chris Vaisvil
12edf

# 12 ED 3/2

0: 1/1 0.000 unison, perfect prime

1: 58.496 cents 58.496

2: 116.993 cents 116.993

3: 175.489 cents 175.489

4: 233.985 cents 233.985

5: 292.481 cents 292.481

6: 350.978 cents 350.978

7: 409.474 cents 409.474

8: 467.970 cents 467.970

9: 526.466 cents 526.466

10: 584.963 cents 584.963

11: 643.459 cents 643.459

12: 3/2 701.955 perfect fifth

Lookalikes:
16/11

**16/11**

648.68206 cents

File:Jid 16 11 pluck adu dr220.mp3 sound sample

In 11-limit Just Intonation, 16/11 is an undecimal subfifth measuring about 648.7¢. It is the inversion of 11/8, the undecimal superfourth. While the name suggests some variation of a perfect fifth, the subfifth is generally considered an interval in it's own right being like neither a perfect fifth nor the tritone. The character is something very unique in that it produces a sound of overtones that resembles that of a large bell.

See: Gallery of Just Intervals
11-notwos-3reduced-limit

**Notwos-3reduced-limits:**

* 1-notwos-3reduced-limit
* 5-notwos-3reduced-limit
* 7-notwos-3reduced-limit
* **11-notwos-3reduced-limit**
* 13-notwos-3reduced-limit
* 17-notwos-3reduced-limit
* 19-notwos-3reduced-limit
* 23-notwos-3reduced-limit
* 25-notwos-3reduced-limit
* 29-notwos-3reduced-limit
* 31-notwos-3reduced-limit
* 35-notwos-3reduced-limit

**11-notwos-3reduced-limit** is used in notwos tunings. It is naturally based on 11-notwos-limit, which utilizes the following primes: 3, 5, 7, 11. 11-notwos-3reduced-limit however assumes tritave equivalence and restricts the remaining factors to a maximum of 11. Here is the interval list:

* 11/9, 27/11
* 9/7, 7/3
* 15/11, 11/5
* 7/5, 15/7
* 11/7, 21/11
* 5/3, 9/5
1106edo

The 1106 division is a zeta peak edo which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit relative error being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.
110edo

110edo has a step size of 10.909090909091 cents. It reinforces 55edo with the same 3, but a flat 5 instead of sharp 5 (in patent val), it also gains a new 7, 11 and especially 13, now quite accurate. But if you want the meantone system back you can still use every other note, experiencing the wonders of generic meantone with non-standard 7, or the new 7, or whatever you choose. It's wonderful either way.
15/14

**15/14**

119.44281 cents

File:Jid 15 14 pluck adu dr220.mp3 sound sample

The septimal diatonic semitone, 15/14, is a superparticular ratio with a numerator which is the fifth triangular number.

It may be found as the interval between many 7-limit ratios, including:

* 16/15 and 8/7
* 14/13 and 15/13
* 7/6 and 5/4
* 6/5 and 9/7
* 14/11 and 15/11
* 4/3 and 10/7
* 7/5 and 3/2
* 22/15 and 11/7
* 14/9 and 5/3
* 8/5 and 12/7
* 26/15 and 13/7
* 7/4 and 15/8

http://en.wikipedia.org/wiki/Septimal\_diatonic\_semitone
1-2 syntonic comma meantone

In 1/2-comma meantone temperament, each perfect fifth is tempered by a half of a syntonic comma from its just value of 3/2. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the "major thirds" are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of 26edo, which is the most likely candidate for a closed system approximating this meantime.
142edo

*142edo* divides the octave into 142 equal parts, each of size 8.45 cents. It tempers out 1728/1715, 3136/3125, 16875/16807, 32805/32768, 176/175, 540/539 and 1375/1372. It is an excellent tuning for the 7-limit rank three temperament tempering out 1728/1715 and a good one for the 11-limit temperament also tempering out 176/175. It is also excellent for semisept, the 31&111 temperament tempering out 1728/1715 and 3136/3125 and the 53&89 temperament tempering out both 1728/1715 and 32805/32768.
13/11

**13/11**

289.20972 cents

File:Jid 13 11 pluck adu dr220.mp3 sound sample

In 13-limit Just Intonation, 13/11 is the tridecimal minor third (or Neo-Gothic minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (11/8, about 551.3¢) and 13th harmonic (13/8, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a 3/2 perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant 16/11 as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.

13/11 is the classic mediant between the simpler and more familiar ratios 6/5 and 7/6, as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = 20/17, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = 19/16, the overtone minor third of 19-limit JI, about 297.5¢). (See the diagram below.)

13/11 is also 352/351 (about 4.9¢) narrower than 32/27, the minor third in Pythagorean (3-limit) tuning.

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

The Noble Mediant (earliest description of 13:11 as the "Neo-Gothic" minor third)
10平均律

10平均律、または10tetは、オクターブを正確に120セントで10個に分割したものだ。それは120セントで分割した5平均律が2つ、もしくは2平均律の5つの円であると考えることができる。10平均律は5平均律に120セントの小さい中立的な2度「small neutral second」（または大きな長2度「large minor 2nd」）と、その補完音程である1080セントの大きい中立的な7度「large neutral seventh」、または小さい長7度「small major 7th」を加える。そしてとても13/8（840セント）に近い音とその補完音程の16/13（360セント）に近似的である音程を加える。最後に、番号づけられた各偶数のEDOにあらわれる、おどけた600セントの3全音「tritone」を加える。360セントの大きい中立的な3度「large neutral third」をジェネレーターとして取得するとき、1 2 1 2 1 2 1（3L 4s - mosh）という形のヘプタトニックMOSを生成する。整数のEDOにしろ、その間のEDOにしろ、それはゼータピークEDO「zeta peak edo」である。JIのテンペラメントの用語として解釈する一つの方法は、105/104, 225/224, 16808/16384をテンパーアウトしたような、2.7.13.15サブグループとすることだ。完全な13リミットテンペラメントとして扱うものの、前記で述べたサブグループと密接にマッチする。

# 10平均律の音程と近似値

このリストは原文で紹介されている音程をまとめたものである。紹介されているものは主に、特徴的な音程と近似純正音程である。近似純正音程は各パラメータの数を上げればほぼ無限に生成される。その点原文の近似純正音程は適度に各パラメータが下げられているため、まとめることには大きな意義があると考えられる。

　各周波数比の大きさが16以内で表現される純正音程は、edjirulerのパラメータを、[number of equal divisions=7, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.16]にして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。

10平均律を2.7.13.15サブグループテンペラメントとして扱うことにもとづいた音程である。周波数3、5､9を加えることは、いくつかのより10平均律らしい、独自の素晴らしいエラーを生み出す。

# リニアーテンペラメント

per octave
 1
 2

# コンマをなだらかにする

10平均律を<10 16 23 28 35 37|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。

# 楽器

10平均律はギターやフレットのある弦楽器と非常に相性が良い。480セントの4度が5つ、2オクターブの間隔（480\*5=2400）に位置しているためである。これは開放弦が4度で均等に調律されていることを意味する。そのことは12平均律よりもコードや音階の指使いをより素晴らしく統一し、学びやすくする。たとえば、「E」コードの指使いは低いほうから高い方にかけて0 - 2 - 2 - 1 - 0 - 0のように、「A」コードは0 - 0 - 2 - 2 - 1 - 0のように、「D」コードは0 - 0 - 2 - 2 - 1 - 0のように表されるだろう。5の倍数にあるすべての平均律でこのことは適用できるが、10平均律は特にシンプルである。

慣習的なキーボードを10平均律にリチューニングする方法はおそらくたくさんある。しかし白鍵にsLsLsLs（smallとLargeの音程）を保持するEbやAbキーをおざなりにし、不要なものを作ってしまう。しかし余剰性により転調は簡単に行える。もう一つの選択は無関係なキーを、多くの特徴を持つ20平均律から選択するチューニングをすることである。20平均律は正確な11倍音と関連性がある。
12276edo

The 12276 division divides the octave into equal steps of size 0.097752 cents. This creates a unit known as the **prima,** useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three **tuning units**.

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from 6691. It factors as 12276 = 2^2 \* 3^2 \* 11 \* 31, and among its divisors are 12, 22, 31, 99 and 198. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.
11-limit

The *11-limit* consists of all justly tuned intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are 14/11, 11/8, 27/22 and 99/98. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the 7-limit introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of 12edo, the 11-limit introduces neutral intervals, superfourths and subfifths, which fall in between major, minor and perfect interval categories and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, edos which do (relatively) well in supporting 11-limit intervals are: 1edo, 2edo, 3edo, 4edo, 5edo, 6edo, 7edo, 9edo, 10edo, 12edo, 15edo, 22edo, 26edo, 31edo, 41edo, 63edo, 72edo, 87edo, 109edo, 161edo.

11-limit\_compare.png

## Intervals

Some of the simplest intervals of 11 include:

See: Gallery of Just Intervals

# Music

Study #3 play by Dave Hill

Brief 11-ratio composition play by Dave Hill

11 Limit Piano by Chris Vaisvil

11-limit singtervals by Andrew Heathwaite
131edo

# 131 tone equal temperament

**131-EDO**, or **131-tET**, divides the octave into 131 equal steps of approx. 9.1603 Cents, each one. 131edo is the next EDO, after 81edo, on the "Golden Tone System" (Das Goldene Tonsystem) of Thorvald Kornerup, using the 131b val. The patent val has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out 81/80 it tempers out the immunity comma, 1638400/1594323. In the 7-limit it tempers out 3125/3087 and 245/243, so that it supports bophier temperament.

131edo is the 32nd prime EDO.

**Some MOS Scales in 131-EDO:**
10-3reduced-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

 10-3reduced-limit focuses on the harmonics, 2, 4, 5, 7, 8, 10. The tritave equivalent to the 9-odd-limit.

* 10/9, 27/10
* 9/8, 8/3
* 8/7, 21/8
* 7/6, 18/7
* 6/5, 5/2
* 5/4, 12/5
* 9/7, 7/3
* 4/3, 9/4
* 7/5, 15/7
* 10/7, 21/10
* 3/2, 2/1
* 8/5, 15/8
* 5/3, 9/5
* 12/7, 7/4
17/16

**17/16**

104.9554 cents

File:Jid 17 16 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 17/16 is the 17th overtone, octave reduced, and may be called the "large septendecimal semitone". Measuring about 105¢, it is close to the 12edo semitone of 100¢, and thus 12edo can be said to approximate it closely. In a chord, it can function similarly to a jazz "minor ninth" -- for instance, 8:10:12:14:17 (although here the interval is 17/8, which is a little less harsh sounding than 17/16). In 17-limit JI, it is treated as the next basic consonance after 13 and 15.

17/16 is one of two superparticular semitones in the 17-limit; the other is 18/17, which measures about 99¢. The difference between them is 289/288, about 6¢. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.

See: Gallery of Just Intervals
16edt

# Properties

As the double of 8edt, this division of the tritave is harmonically fraternal to 10edo. Its unit step is ~1.128 cents flat of 1\10edo. Unlike 10edo, it does not really have a 7 or 13 because it is not using its approximation of 2 as equivalent though the accumulated flatness of a stack of its unit step leads to an excellent 13:21 and a decent 7:13. When twos are admitted, it turns into a tritave-repeating version of Blackwood temperament.

# Music

A Short Tune in 16EDT by Peter 'Rush' Kosmorsky
11/7

**11/7**

782.49204 cents

File:Jid 11 7 pluck adu dr220.mp3 sound sample

In 11-limit Just Intonation, 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of 14/11, the undecimal supermajor third. Another subminor sixth is 14/9, at about 764.9¢; the interval between these is 99/98, about 17.6¢.

11/7 is 22/21 (about 80.5¢) above the 3/2 perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

See: Gallery of Just Intervals
137edo

The *137 equal division* divides the octave into 137 equal parts of 8.759 cents each. It is the optimal patent val for 7-limit orwell temperament and for the planar temperament tempering out 2430/2401. It tempers out 2109375/2097152 (the semicomma) in the 5-limit; 225/224 and 1728/1715 in the 7-limit; 243/242 in the 11-limit; 351/350 in the 13-limit; 375/374 and 442/441 in the 17-limit; and 324/323 and 495/494 in the 19-limit. Since it is the 33rd prime number, 137edo has no proper divisors aside from 1.

A diagram of 7-limit Orwell based on the 31\137edo generator:

137edo\_MOS\_031\_demo\_correction.png

137edo\_MOS\_031.svg
10ed5

# 10 equal divisions of the 5th harmonic

Half of 20ed5 (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest hyperpyth tuning, analogous to 5edo and 4edt in their own spheres, this, its double, can be compared, structurally, to, 10edo. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to 53edo.

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

Music:

Weird Blues -- Kosmorsky
10edt

**10 Equal Divisions of the Tritave**

10edt, like 5edt, has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
1236edo

The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a zeta peak edo, though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit comma basis of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12 (12 \* 103 = 1236).
11664edo

The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit relative error than any division until 18355. It is a zeta peak edo unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until 16808. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 \* 3^6. Among its divisiors are 12, 16, 24, 27, 72, 81 and 243.
13-Hendrix diamond

The **13-'***Hendrix diamond'* is the tonality diamond generated from the ratios 1/1, 13/12, 5/4, 3/2 and 13/8.

All notes in the Hendrix diamond are also found in the 15-limit tonality diamond, because 1/1:13/12:5/4:3/2:13/8 is a 15-limit anomalous saturated suspension.

## Symmetry group

The symmetry group of the 13-Hendrix diamond has 16 elements. Its generators are:

* 3:2, 5:4, 13:8 → 13:12, 5:3, 13:9
* 3:2, 5:4, 13:8 → 3:2, 15:13, 18:13
* 3:2, 5:4, 13:8 → 4:3, 8:5, 16:13
130edo

*130edo* divides the octave into 130 parts of size 9.231 cents each. It is the tenth zeta integral edo but not a gap edo. It can be used to tune a variety of temperaments, including hemiwürschmidt, sesquiquartififths, harry and hemischismic. It also can be used to tune the rank-three temperament jove, tempering out 243/242 and 441/440, plus 364/363 for the 13-limit and 595/594 for the 17-limit. It gives the optimal patent val for 11-limit [[Würschmidt\_family#Hemiw�rschmidt|hemiwürschmidt]] and sesquart and 13-limit harry temperaments.

7-limit commas: 2401/2400, 3136/3125, 19683/19600

11-limit commas: 441/440, 540/539, 3136/3125, 4000/3993

13-limit commas: 3136/3125, 243/242, 441/440, 351/350, 364/363

17-limit commas: 221/220, 364/363, 442/441, 595/594, 1275/1274, 4913/4875

## Music

The Paradise of Cantor play by Gene Ward Smith
148edo

The *148 equal division* divides the octave into 148 equal parts of 8.108 cents each, near a kleisma. It provides the optimal patent val for 11-limit echidnic temperament, the 10&46 temperament. It has a fifth on the sharp side, 3.45 cents sharp. It tempers out 2048/2025 in the 5-limit, making it a diaschismic system. In the 7-limit, the patent val tempers out 686/675 and 1029/1024, but an alternative mapping <148 235 344 416| with a sharp rather than a flat 7 tempers out 3136/3125 instead, and provides a better tuning than the patent val tuning of 80edo for 7- and 13- limit bidia temperament, the 12&68 temperament. In the 11-limit, the patent val tempers out 385/384 and 441/440, and the alternative mapping with the sharp 7 tempers out 176/175, 896/891 and 1375/1372 instead. In the 13-limit, the patent val tempers out 325/324 and 364/363, and the alternative val 325/324 again, as well as 640/637 and 847/845.

148 = 4 \* 37, with divisors 2, 4, 37, 74.
128/125

The 41.059 cent interval of **128/125** is called the **diesis** or **augmented comma**; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in 28, 31 or 34 EDO, and by two steps of 53, 59 or 65 EDO. In any tuning with just major thirds, such as quarter-comma meantone, it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called **enharmonic comma** for this reason. Tempering it out leads to augmented temperament.

de:128/125
122edo

**122edo** is the equal division of the octave into 122 parts of 9.836 cents each. It is the optimal patent val for 7-limit tritonic temperament and 11-limit tritoni temperament, and the planar squalentine temperament. It tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 \* 61.
111edo

**111edo** is the equal division of the octave into 111 parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent optimal patent val tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&111 temperament, and buzzard, the 58&111 temperament. The Trio piece below is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

# 111
3
⋅
37

{\displaystyle 111 = 3\cdot 37}

## Music

Trio for SoftSaturn, NebulaSing and TromBonehead play by Gene Ward Smith
1578edo

The **1578** equal division divides the octave into 1578 equal parts of 0.7605 cents each. It is a very strong higher limit system, and is a zeta peak, integral and gap edo. It is distinctly consistent through the 29 limit, and is the first edo past 311 with a lower 29-limit relative error. It is also the lowest past 311 in the 31 limit, the lowest past 581 in the 23 limit, and the lowest past 1178 in the 19 limit. It is also quite strong taken just as an 11-limit system; the only smaller edo with a lower 11-limit relative error is 342.

Some 31 limit or lower superpaticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999... .
140edo

The 140 equal division divides the octave into 140 parts of 8.571 cents each. In the 5-limit, it tempers out 15625/15552, making it a kleismic system, and the kwazy comma, |-53 10 16>. It is most notable, however, in the 7-limit, where it tempers out 2401/2400, 5120/5103, 10976/10935 and 65625/65536. It supports the 7-limit rank-two temperaments tertiaseptal, hemififths, countercata and bisupermajor, and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the optimal patent val for 13-limit countercata. In the 11-limit it tempers out 1331/1323, 385/384, 1375/1372, 6250/6237, 5632/5625 and 9801/9800, and in the 13-limit 325/324, 352/351, 847/845, 625/624, 676/675, 1001/1000, 1716/1715 and 2080/2079.

If we use the val <140 223 325 394| we obtain a tuning for porcupine temperament; the generator 19\140 is 0.023 cents flat of the POTE generator.
200edo

# 200 tone equal temperament

200 EDO divides the octave into 200 parts of exactly **6 cents** each, and contains a perfect fifth of exactly **702 cents** and a perfect fourth of exactly **498** cents, which is quite accurate, with an error of about 1/22 cent. It tempers out the schisma, 32805/32768, in the 5-limit and the gamelisma, 1029/1024, in the 7-limit, so that it supports guiron temperament.

**200 tone equal modes:**

34 34 15 34 34 34 15 = Pythagorean tuning

32 32 20 32 32 32 20 = Meantone tuning in the same way of 50edo

27 27 27 27 27 27 27 11 = Porcupine tuning

26 26 26 9 26 26 26 26 9 = Superdiatonic tuning

24 24 24 16 24 24 24 24 16 = Superdiatonic tuning in the same way of 25edo

22 22 8 22 22 22 8 22 22 22 8 = Sensi

16 16 16 8 16 16 16 16 8 16 16 16 16 8 = Ketradektriatoh tuning

The prime factorization

200 = 23 \* 52

leads to these further divisors

4, 8, 10, 20, 25, 40, 50, 100

# Music

Fugue on Elgar’s Enigma Theme play by Claudi Meneghin
129edo

**129edo** is the equal division of the octave into 129 parts of 9.302 cents each. It provides the optimal patent val for the 11-limit rank three clio temperament. It tempers out 81/80 in the 5-limit; 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit.

The factorization of 129 is 3 and 43
1-notwos-3reduced-limit

**Notwos-3reduced-limits:**

* **1-notwos-3reduced-limit**
* 5-notwos-3reduced-limit
* 7-notwos-3reduced-limit
* 11-notwos-3reduced-limit
* 13-notwos-3reduced-limit
* 17-notwos-3reduced-limit
* 19-notwos-3reduced-limit
* 23-notwos-3reduced-limit
* 25-notwos-3reduced-limit
* 29-notwos-3reduced-limit
* 31-notwos-3reduced-limit
* 35-notwos-3reduced-limit

 The only tritave–reduced 1-notwos-3reduced-limit interval is 1/1.
10L 3s

This is the chromatic scale of the Sephiroth temperament explicated on the page for 3L 7s. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
11平均律

11平均律は1200セントのオクターブを11個に均等分割したものであり、最小の音程は約109.09セントである。2、3、5、7平均律に次ぐ5番目の素数平均律である。

12よりも少ないけれど、11平均律のマップはスタンダードキーボードへ簡単にマッピングできる。Ab/G#を無視してマッピングすることを提案する。そしてまた、白鍵にOrgan[7]（Orgone）と呼ばれる音階を配置することから始めることを提案する。（無関係なA♭は22平均律によって生成され、このチューニングはelevenplusとして知られている）

# チューニング

12平均律と比較して、11平均律の音程は拡大解釈される。

・1ステップ→109.09セントの「短2度」は、12平均律の100セントで表される短2度と、非常によく似たメロディーやハーモニーの機能を持つ。

・2ステップ→218.18セントの「長2度」は、12平均律の200セントで表される長2度と似た様相を示す。しかし「長9度」とする場合、協和音にはあまりならない。その反行系である981.82セントは、ブルース調の7度として機能する。7/4から13セントはなれているけれども。

・3ステップ→327.27セントの「短3度」は、だいぶシャープされ中立3度に近づく。

・4ステップ→436.36セントの「長3度」は、すっかりシャープされ、5/4（386.31cent）の単純な3度よりも9/7（435.08cent）のスーパーメジャーサードに近くなる。

・5ステップ→545.45セントの「完全4度」は、全く完全5度のようには聞こえない。そして4/3（498.04cent）の単純な完全4度よりも11/8（551.32cent）のスーパーフォースとしてより簡単に近づく。

# サブグループ

11平均律は22平均律とする2\*11サブグループサブグループ2.9.15.7.11と同じチューニングを提供する。そしてこのサブグループは22と同じコンマをテンパーアウトする。また、このサブグループは8:9:11:14:15:16コードとそのサブコードの近似値である。エラーは比較的大きいけれども、JIコードの様々な近似コードとともに11を提供する。

# 11平均律の音程と近似値

11平均律はまた、2.7.9.11.15.17サブグループテンペラメントとして考えられる。その場合は以下のダイアグラムになる。

11edo\_approx\_2-7-9-11-15-17\_2ndsave.png

一方、各周波数比の大きさが15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=11, interval of equivalence=2, integer limit=14, threshold of JI pitch inclusion=0.2]というパラメータで生成したものである。

# 11平均律のソルフェージュ

11平均律のソルフェージュシステムは、22平均律のソルフェージュシステムから適用するのが簡単な方法である。クロマティックスケールはしたがって、「do, ra, re, me, mo, fu, su, lo, la, ta, ti, do」と歌う。

* 2.7.9.11.15.17サブグループの中で表されるソルフェージュ

# MOS音階

11平均律は12平均律よりも音程が少ないけれども、MOS音階の観点から見ると、多くのさまざまなものを提供する。これは11が素数だからである。12は素数ではない。2*11（11平均律の2ステップ・音程）、3*11、4*11、5*11は11音程すべてが発生するまで、オクターブを繰り返さない音階を生成する。

・211は2 2 2 2 3を生成する。これはMachine[5]と呼ばれる1L 4sの音階を生成する。そしてMachine[6]（Machine）と呼ばれる2 2 2 2 2 1の5L 1sの音階を生成する。

・311は3 3 3 2とOrgone[7]と呼ばれる1 2 1 2 1 2 2の4L3s（4L 3s）音階を生成する。

・411は4 4 3、1 3 1 3 3という3L 2sという音階と、1 1 2 1 1 2 1 2という3L 5sの音階を作る

・511は5 5 1、1 4 1 4 1という2L 3sという音階と、1 1 3 1 1 3 1という2L 5sの音階を作る。また、1 1 1 2 1 1 1 2 1という2L 7sの音階も作る。

11平均律のモードもまた参照。

# コンマをなだらかにする

11平均律を<11 17 26 31 38 41|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。

# 11平均律のインスタントアンサンブル

2011年2月、微分音デザインセミナーの一部のOddmusic U-Cで、7作品のアンサンブルが11平均律で演奏された。楽器はautotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbotである。

# 11平均律のジン

11平均律のジンがある。私たちが知る限りでは、11平均律は自身のジンを持った初めてのゼンハーモニックチューニングシステムである。11平均律のジン（11edo Zine）を参照のこと。