recursal/Fanatic-Fandom huggingface dataset:
208edo

The *208 equal division* divides the octave into 208 equal parts of size 5.769 cents each. It tempers out 15625/15552, the kleisma, and is the optimal patent val for the kleismic temperament metakleismic, and 7, 11 and 13 limit rank three tolerant temperament. It is also the optimal patent val for the rank four 11-limit temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.

208 = 16 \* 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.

# 13-limit transversal

[196/195, 100/99, 91/90, 64/63, 55/54, 49/48, 40/39, 77/75, 36/35, 28/27, 80/77, 25/24, 245/234, 22/21, 21/20, 81/77, 35/33, 52/49, 16/15, 77/72, 15/14, 14/13, 250/231, 13/12, 49/45, 12/11, 35/32, 100/91, 11/10, 54/49, 10/9, 49/44, 39/35, 28/25, 55/49, 9/8, 147/130, 25/22, 91/80, 8/7, 55/48, 147/128, 15/13, 196/169, 64/55, 7/6, 90/77, 75/64, 147/125, 13/11, 77/65, 25/21, 105/88, 117/98, 6/5, 77/64, 40/33, 63/52, 128/105, 11/9, 49/40, 16/13, 154/125, 26/21, 56/45, 96/77, 5/4, 49/39, 44/35, 63/50, 80/63, 14/11, 125/98, 32/25, 77/60, 9/7, 35/27, 100/77, 13/10, 64/49, 55/42, 21/16, 120/91, 33/25, 65/49, 4/3, 147/110, 75/56, 35/26, 66/49, 27/20, 49/36, 15/11, 175/128, 48/35, 11/8, 135/98, 18/13, 245/176, 39/28, 7/5, 108/77, 45/32, 147/104, 64/45, 77/54, 10/7, 56/39, 351/245, 13/9, 196/135, 16/11, 35/24, 143/98, 22/15, 72/49, 40/27, 49/33, 52/35, 112/75, 220/147, 3/2, 98/65, 50/33, 91/60, 32/21, 55/36, 49/32, 20/13, 77/50, 54/35, 14/9, 120/77, 25/16, 196/125, 11/7, 63/40, 100/63, 35/22, 78/49, 8/5, 77/48, 45/28, 21/13, 125/77, 13/8, 49/30, 18/11, 105/64, 104/63, 33/20, 81/49, 5/3, 147/88, 117/70, 42/25, 130/77, 22/13, 245/144, 75/44, 77/45, 12/7, 55/32, 169/98, 26/15, 256/147, 96/55, 7/4, 135/77, 44/25, 260/147, 16/9, 98/55, 25/14, 70/39, 88/49, 9/5, 49/27, 20/11, 91/50, 64/35, 11/6, 90/49, 24/13, 231/125, 13/7, 28/15, 144/77, 15/8, 49/26, 66/35, 91/48, 40/21, 21/11, 245/128, 25/13, 77/40, 27/14, 35/18, 150/77, 39/20, 49/25, 55/28, 63/32, 125/63, 99/50, 195/98, 2]
269edo

The *269 equal division* divides the octave into 269 equal parts of 4.461 cents each. The patent val tempers out 6144/6125 in the 7-limit, 540/539 and and 5632/5625 in the 11-limit and 364/363 and 676/675 in the 13-limit. The 269c val tempers out 225/224 and 4375/4374 in the 7-limit, and 269ce 385/384 in the 11-limit, so that it supports and provides an excellent tuning for catakleismic and marvel temperaments.
25-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 **25-odd-limit** intervals.

* 26/25, 25/13
* 25/24, 48/25
* 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
* 25/23, 46/25
* 12/11, 11/6
* 23/21, 42/23
* 11/10, 20/11
* 21/19, 38/21
* 10/9, 9/5
* 19/17, 34/19
* 28/25, 25/14
* 9/8, 16/9
* 26/23, 23/13
* 17/15, 30/17
* 25/22, 44/25
* 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
* 25/21, 42/25
* 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
* 32/25, 25/16
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 30/23, 23/15
* 17/13, 26/17
* 21/16, 32/21
* 25/19, 38/25
* 4/3, 3/2
* 23/17, 34/23
* 19/14, 28/19
* 34/25, 25/17
* 15/11, 22/15
* 26/19, 19/13
* 11/8, 16/11
* 18/13, 13/9
* 25/18, 36/25
* 32/23, 23/16
* 7/5, 10/7
* 24/17, 17/12

todo: also cents and descriptions
25/21

**25/21**

301.8465 cents

File:Jid 25 21 pluck adu dr220.mp3 sound sample

25/21, the quasi-tempered minor third, is the interval between 6/5 and 10/7, and also 7/5 and 5/3. It is 126/125 flat of 6/5, and tempering 126/125 out makes the 1-6/5-10/7 triad into a diminished triad where two minor thirds make up a tritone. It is extremely close to 3 steps of 12edo (300 cents).
25/24

**25/24**

70.6724 cents

File:Jid 25 24 pluck adu dr220.mp3 sound sample

The **5-limit chromatic semitone** (short: **chroma**), 25/24, 70.672 cents, is the superparticular ratio which marks the difference between the 5-limit thirds, 6/5 and 5/4, and sixths, 8/5 and 5/3. It is therefore the amount which sharpens or flattens a 5-limit third or sixth, and when notating 5-limit just intonation it can be associated with the sharp or flat symbol, and along with an additional symbol for the 81/80 comma, it can be used for a complete system of 5-limit notation.

See also:

* Chroma -- a generalising concept for MOS scales
* Gallery of Just Intervals
* comma
26-3reduced-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

 **26-3reduced-limit**.

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

In 27-odd-limit we have octave equivalence, and the following odd harmonics: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27

In 26-3reduced-limit we have tritave equivalence, and the following 3reduced harmonics: 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26

* 27/26, 26/9
* 26/25, 75/26
* 25/24, 72/25
* 24/23, 23/8
* 23/22, 66/23
* 22/21, 63/22
* 21/20, 20/7
* 20/19, 57/20
* 19/18, 54/19
* 18/17, 17/6
* 17/16, 48/17
* 16/15, 45/16
* 15/14, 14/5
* 14/13, 39/14
* 27/25, 25/9
* 13/12, 36/13
* 25/23, 69/25
* 12/11, 11/4
* 23/21, 63/23
* 11/10, 30/11
* 21/19, 19/7
* 10/9, 27/10
* 19/17, 51/19
* 9/8, 8/3
* 26/23, 69/26
* 17/15, 45/17
* 25/22, 66/25
* 8/7, 21/8
* 23/20, 60/23
* 15/13, 13/5
* 22/19, 57/22
* 7/6, 18/7
* 27/23, 23/9
* 20/17, 51/20
* 13/11, 33/13
* 19/16, 48/19
* 25/21, 63/25
* 6/5, 5/2
* 23/19, 57/23
* 17/14, 42/17
* 11/9, 27/11
* 27/22, 22/9
* 16/13, 39/16
* 21/17, 17/7
* 26/21, 63/26
* 5/4, 12/5
* 24/19, 19/8
* 19/15, 45/19
* 33/26, 26/11
* 14/11, 33/14
* 23/18, 54/23
* 9/7, 7/3
* 22/17, 51/22
* 13/10, 30/13
* 30/23, 23/10
* 17/13, 39/17
* 21/16, 16/7
* 25/19, 57/25
* 33/25, 25/11
* 4/3, 9/4
* 27/20, 20/9
* 23/17, 51/23
* 19/14, 42/19
* 15/11, 11/5
* 26/19, 57/26
* 11/8, 24/11
* 18/13, 13/6
* 25/18, 54/25
* 7/5, 15/7
* 24/17, 17/8
* 17/12, 36/17
* 27/19, 19/9
* 10/7, 21/10
* 33/23, 23/11
* 23/16, 48/23
* 36/25, 25/12
* 13/9, 27/13
* 16/11, 33/16
* 19/13, 39/19
* 22/15, 45/22
* 25/17, 51/25
* 3/2, 2/1
* 26/17, 51/26
* 23/15, 45/23
* 20/13, 39/20
* 17/11, 33/17
* 14/9, 27/14
* 39/25, 25/13
* 25/16, 48/25
* 36/23, 23/12
* 11/7, 21/11
* 30/19, 19/10
* 19/12, 36/19
* 27/17, 17/9
* 8/5, 15/8
* 21/13, 13/7
* 13/8, 24/13
* 18/11, 11/6
* 23/14, 42/23
* 33/20, 20/11
* 5/3, 9/5
* 42/25, 25/14
* 27/16, 16/9
* 22/13, 39/22
* 39/23, 23/13
* 17/10, 30/17
* 12/7, 7/4
* 19/11, 33/19
* 45/26, 26/15
266edo

Welcome to 266edo

## 266edo = 14edo × 19edo

Distinctively interesting choice for an edo, due to its divisibility by both 14 and 19. This means it can be used to play 14edo music and 19edo music at the same time, using the 266/19 and 266/14 vals respectively.

## 4.511

266edo has a step size of 4.5112781954887 cents. It is a commatonal EDO, because the step size represents commas like 4374/4375, 176/175, 169/168. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 266edo (ordered by absolute error).
2190edo

The 2190 equal division divides the octave into 2190 equal parts of 0.5479 cents each. It is is a very strong 13-limit system; no smaller division has a smaller 13-limit relative error, and nothing beats it until 2684. A basis for the 13-limit commas is {9801/9800, 10648/10647, 105644/105625, 140625/140608, 196625/196608}; also tempered out are 123201/123200 and 151263/151250.

2160 factors as 2 \* 3 \* 5 \* 73; among its divisors is the Woolhouse unit system, 730.
225/128

**225/128**

976.5374 cents

File:Jid 225 128 pluck adu dr220.mp3 sound sample

The marvel five-limit harmonic seventh, 225/128, is the interval between the 5-limit minor second, 16/15, and the major second, 15/8. It is a marvel comma, 225/224, sharp of the harmonic seventh, 7/4, and so tempering out 225/224, which is marvel temperament, equates the two. Marvel tempering is a useful method of tempering the 5-limit to include septimal harmony, making two minor seconds a supermajor second, 8/7 and two major seconds into 7/6.
170edo

Welcome to 170edo

## 170edo = 10edo × 17edo

Distinctively interesting choice for an edo, due to its divisibility by both 10 and 17. This means it can be used to play 10edo music and 17edo music at the same time, using the 170/17 and 170/10 vals respectively.

## 7.059

170edo has a step size of 7.0588235294118 cents. It is a commatonal EDO, because the step size represents commas like 81/80.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 170edo (ordered by absolute error).
217edo

The *217 equal division* divides the octave into 217 equal parts of 5.530 cents each. It is a strong 19-limit system, the smallest uniquely consistent in the 19-limit and consistent to the 21-limit. In the 5-limit it tempers out the parakleisma, |8 14 -13>, and the escapade comma, |32 -7 -9>; in the 7-limit 10976/10935, 3136/3125 and 4375/4374; in the 11-limit 441/440, 4000/3993 and 5632/5625; in the 13-limit 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079; in 17-limit 595/594, 833/832, 936/935, 1156/1155, 1225/1224, 1701/1700; in the 19-limit 343/342, 476/475, 969/968, 1216/1215, 1445/1444, 1521/1520 and 1540/1539.
19ED3

# Division of 3/1 into 19 equal parts

Bernhard Stopper's OnlyPure tuning

Note: 19 equal divisions of the tritave is not a "real" xenharmonic tuning; it is a slightly stretched version (with an octave of 1201.2 cents) of the normal 12-tone scale. Although it is really just the normal "harmonic" tuning framed in a tritave equivalence, the "default" approach to it is as the tritave twin of godzilla temperament (with a generator of 400.4 cents and a 3:1 ratio superdiatonic scale, weird coincidence how 17edt and 19edt tonality have the same "default" scheme with two tones more or less), which has little connection to standard 12-tone practice in spite of using the 12-tone interval set. Beyond this, it is also the tritave twin of sensi or meantone temperament (with a generator of 700.7 or 1101.1 cents and a 2:1 ratio superdiatonic scale).
204edo

Welcome to 204edo

## 204edo = 12edo × 17edo

Distinctively interesting choice for an edo, due to its divisibility by both 12 and 17. This means it can be used to play 12edo music and 17edo music at the same time, using the 204/17 and 204/12 vals respectively.

## 5.882

204edo has a step size of 5.8823529411765 cents. It is a commatonal EDO, because the step size represents commas like 126/125, 225/224.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 204edo (ordered by absolute error).
22edo Modes

Some modes available in 22edo. Please add more, discovered or newly-composed! Note also that 22edo can play all 11edo Modes as well....

# Orwell

### 9 notes MOS, 4L+5s (fair bug)

 2 2 3 2 3 2 3 2 3

# Porcupine

See also Porcupine Temperament Modal Harmony for many more MODMOS modes.

### 8 notes MOS, 7L+1s (Grumpy octatonic)

 3 3 3 1 3 3 3 3
 3 3 1 3 3 3 3 3
 3 1 3 3 3 3 3 3
 1 3 3 3 3 3 3 3

### 15 notes MOS, 7L+8s

 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2

# Hedgehog

### 6 notes MOS, 2L+4s (Rice hexatonic)

 3
 5

### 8 notes MOS, 6L+2s (Biggie octatonic)

 2 3 3 3 2 3 3 3

### 14 notes MOS, 8L+6s

 2 2 1 2 1 2 1 2 2 1 2 1 2 1

# Superpyth

### 7 notes MOS, 5L+2s (diatonic)

 4 1 4 4 4 1 4
 1 4 4 4 1 4 4
 4 4 4 1 4 4 1
 4 4 1 4 4 1 4
 4 1 4 4 1 4 4
 1 4 4 1 4 4 4

## 5-or7-limitSuperpyth/11-limitSuprapyth[12]

### 12 notes MOS, 5L+7s

Scala files for all 12 modes

Audio comparison of the 12 modes

# Pajara

### 10 notes

"Symmetrical" modes are MOS, 2L+8s (Rice decatonic); "Pentachordal" modes are not MOS.

(The names of the modes are Paul Erlich's - see http://lumma.org/tuning/erlich/erlich-decatonic.pdf.)

# Doublewide/Fleetwood

### 6 notes MOS, 4L+2s (bicycle)

 5
 1

# Astrology

###

### 6 notes MOS, 4L+2s (bicycle)

 4
 3
253edo

# 253 tone equal temperament

***253-EDO*** or **253-tET** divides the octave into 253 equal steps of 4.743083 cents each. It approximates the fifth by **148\253**, which is 701.976285 cents, a mere **0.004487 cents sharp**. The primes from 5 to 17 are all slightly flat. It tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for higher-limit sesquiquartififths temperament.

**253 tone equal modes:**

63 32 63 63 32: Pentatonic

43 43 19 43 43 43 19: Pythagorean tuning

41 41 24 41 41 41 24: Meantonic tuning

24 19 19 24 19 24 19 24 19 19 24 19: Pythagorean chromatic

17 24 24 17 24 17 24 17 24 24 17 24: Meantone chromatic

19 22 21 20 23 19 23 20 21 22 19 24: Well temperament (fifth Pythagorean comma on C-G-D-A-E-B)

35 35 35 35 35 35 35 8: Porcupine tuning

33 33 33 11 33 33 33 33 11: "The Hendecapliqued superdiatonic of the Icositriphony"

31 31 31 18 31 31 31 31 18: Superdiatonic tuning in the way of Mavila

26 26 15 26 26 26 15 26 26 26 15: Sensi tuning

20 20 20 11 20 20 20 20 11 20 20 20 20 11: Ketradektriatoh tuning

**PRIME FACTORIZATION:**

253 = 11 \* 23
261edo

Welcome to 261edo

## 261edo = 9edo × 29edo

Distinctively interesting choice for an edo, due to its divisibility by both 9 and 29. This means it can be used to play 9edo music and 29edo music at the same time, using the 261/29 and 261/9 vals respectively.

## 4.598

261edo has a step size of 4.5977011494253 cents. It is a commatonal EDO, because the step size represents commas like 2401/2400, 385/384, 364/363. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 261edo (ordered by absolute error).
202edo

The 202 equal temperament divides the octave into 202 equal parts of 5.941 cents each. It tempers out 2401/2400, 19683/19600 and 65625/65536 in the 7-limit and 243/242, 441/440 and 4000/3993 in the 11-limit. It is the optimal patent val for the 11-limit harry and tertiaseptal rank two temperaments, the rank three temperament jove tempering out 243/242 and 441/440, and the rank four rastma temperament tempering out 243/242.
210edo

Welcome to 210edo

## 210edo = 14edo × 15edo

Distinctively interesting choice for an edo, due to its divisibility by both 14 and 15. This means it can be used to play 14edo music and 15edo music at the same time, using the 210/15 and 210/14 vals respectively.

## 5.714

210edo has a step size of 5.7142857142857 cents. It is a commatonal EDO, because the step size represents commas like 78732/78125, 1728/1715, 121/120.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 210edo (ordered by absolute error).
227edo

The *227 equal temperament* divides the octave into 227 equal parts of 5.286 cents each. It tempers out the kleisma, 15625/15552, in the 5-limit and 5120/5103 in the 7-limit, so that it supports countercata temperament, and it also tempers out 117649/116640. In the 11-limit it tempers out 385/384 and it provides the optimal patent val for 11-limit countercata. 227 is the 49th prime EDO.
228edo

The *228 equal division* divides the octave into 228 equal parts of 5.263 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit, and 225/224 and 240047/240000 in the 7-limit, so that it supports 7-limit compton temperament and in fact provides the optimal patent val. In the 11-limit it tempers out 225/224, 441/440, 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the enneadeca or 19-tone-comma, and this is reflected in the fact that 228 = 12 \* 19.
Vamir Rolar

Vamir Rolar is a former member of The Lord's Army that was taken prisoner along with the rest of his group. He was the only one who survived and escaped, and now lives in hiding in Rugbug.

## Description/Personality

He is a high elf He is 6 foot 2 and has a slightly muscular build. He has a scraggly copper colored facial hair. He wears a worn beige tunic and black pants. He is slightly paranoid of being caught, but is otherwise a very chill and laid back person.

## Interactions

Mara and Alvie found him in his house in Rugbug. He sat them down and told them that Mara's parents, who were a part of his Lord's Army group that went missing, were dead.
1L 5s

There are two notable low-harmonic-entropy scales with this MOS pattern. The first is tetracot, in which four generators make a 3/2. The second is a scale in which two generators make a 5/4, which could be referred to by various names. The temperaments "luna", "hemithird", and "hemiwürschmidt" all have 6-note MOSes that are very similar. But perhaps it is best known as roulette6, the six note haplitonic scale for the roulette Chromatic pair.

Other scales include very improper (incomplete) versions of porcupine, negri, etc.
Malick Hrask

Malick Hrask is a goblin child living in Aldeburgh.

## Description/Personality

He is a young goblin. He is 2 and a half feet tall and averagely built for a goblin child. He is slightly reserved.

## Interactions

When Alvie, Mara, Rosalind, Heskan, Graynor, and Kuiekkel were in Aldeburgh looking for Zari Hrask, they found her house, where Xylac Branth and Malick were eating food.
231edo

The *231 equal temperament* divides the octave into 231 equal parts of 5.195 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports tritikleismic temperament, and in fact provides the optimal patent val. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.
18edf

# 18edf

0: 1/1 0.000 unison, perfect prime

1: 38.998 cents 38.998

2: 77.995 cents 77.995

3: 116.993 cents 116.993

4: 155.990 cents 155.990

5: 194.988 cents 194.988

6: 233.985 cents 233.985

7: 272.983 cents 272.983

8: 311.980 cents 311.980

9: 350.978 cents 350.978

10: 389.975 cents 389.975

11: 428.973 cents 428.973

12: 467.970 cents 467.970

13: 506.968 cents 506.968

14: 545.965 cents 545.965

15: 584.963 cents 584.963

16: 623.960 cents 623.960

17: 662.958 cents 662.958

18: 3/2 701.955 perfect fifth

Lookalikes: 31edo
Hype House

The "Hype House" is an estate in the wealthier parts of Drakontas.

## Interactions

It was given to Heskan, Graynor, and Rosalind by Belenog Dunztan, the steward for the Court of Elders after they safely returned Eldrin Tunalas to Drakontas. Alvie, Kuiekkel, and Mara were offered residence, but they declined. However, it was made clear that they are all welcome to visit anytime.

## People

* Graynor - Co-owner
* Heskan - Co-owner
* Rosalind - Co-owner
198edo

The *198 equal temperament* divides the octave into 198 parts of 6.061 cents each. It is contorted in the 7-limit, with the same the same tuning as 99edo, but makes for a good 11- and 13-limit system. Like 99, it tempers out 2401/2400, 4375/4374, 3136/3135, 5120/5103 and 6144/6125 in the 7-limit; in the 11-limit it tempers 3025/3024 and 9801/9800; and in the 13-limit 352/351, 847/845, 676/675, 1001/1000, 1716/1715 and 2080/2079.

It is the optimal patent val for the rank five temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as hemimist and namaka. It is distinctly consistent through the 15-limit, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
265edo

The *265 equal division* divides the octave into 265 equal parts of 4.528 cents each. It is contorted in the 5-limit, tempering out the same commas as 53edo, including 15625/15552 and 32805/32768. In the 7-limit it tempers out 16875/16807 and 420175/419904, so that it supports sqrtphi temperament, for which it provides the optimal patent val. In the 11-limit it tempers out 540/539, 1375/1372 and 4375/4356, and gives the optimal patent val for 11-limit sqrtphi temperament.
27-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 **27-odd-limit** intervals.

* 28/27, 27/14
* 27/26, 52/27
* 26/25, 25/13
* 25/24, 48/25
* 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
* 27/25, 50/27
* 13/12, 24/13
* 25/23, 46/25
* 12/11, 11/6
* 23/21, 42/23
* 11/10, 20/11
* 21/19, 38/21
* 10/9, 9/5
* 19/17, 34/19
* 28/25, 25/14
* 9/8, 16/9
* 26/23, 23/13
* 17/15, 30/17
* 25/22, 44/25
* 8/7, 7/4
* 23/20, 40/23
* 15/13, 26/15
* 22/19, 19/11
* 7/6, 12/7
* 27/23, 46/27
* 20/17, 17/10
* 13/11, 22/13
* 32/27, 27/16
* 19/16, 32/19
* 25/21, 42/25
* 6/5, 5/3
* 23/19, 38/23
* 17/14, 28/17
* 28/23, 23/14
* 11/9, 18/11
* 27/22, 44/27
* 16/13, 13/8
* 21/17, 34/21
* 26/21, 21/13
* 5/4, 8/5
* 34/27, 27/17
* 24/19, 19/12
* 19/15, 30/19
* 14/11, 11/7
* 23/18, 36/23
* 32/25, 25/16
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 30/23, 23/15
* 17/13, 26/17
* 21/16, 32/21
* 25/19, 38/25
* 4/3, 3/2
* 27/20, 40/27
* 23/17, 34/23
* 19/14, 28/19
* 34/25, 25/17
* 15/11, 22/15
* 26/19, 19/13
* 11/8, 16/11
* 18/13, 13/9
* 25/18, 36/25
* 32/23, 23/16
* 7/5, 10/7
* 38/27, 27/19
* 24/17, 17/12

todo: also cents and descriptions
21-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 **21-odd-limit** intervals.

* 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
* 11/10, 20/11
* 21/19, 38/21
* 10/9, 9/5
* 19/17, 34/19
* 9/8, 16/9
* 17/15, 30/17
* 8/7, 7/4
* 15/13, 26/15
* 22/19, 19/11
* 7/6, 12/7
* 20/17, 17/10
* 13/11, 22/13
* 19/16, 32/19
* 6/5, 5/3
* 17/14, 28/17
* 11/9, 18/11
* 16/13, 13/8
* 21/17, 34/21
* 26/21, 21/13
* 5/4, 8/5
* 24/19, 19/12
* 19/15, 30/19
* 14/11, 11/7
* 9/7, 14/9
* 22/17, 17/11
* 13/10, 20/13
* 17/13, 26/17
* 21/16, 32/21
* 4/3, 3/2
* 19/14, 28/19
* 15/11, 22/15
* 26/19, 19/13
* 11/8, 16/11
* 18/13, 13/9
* 7/5, 10/7
* 24/17, 17/12

todo: also cents and descriptions
2460edo

The **2460** equal division divides the octave into 2460 equal parts of 0.4878 cents each. It has been used in Sagittal notation to define the "olympian level" of JI notation, and has been proposed as the basis for a unit, the mina, which could be used in place of the cent. It is uniquely consistent through to the 27-limit, which is not very remarkable in itself (388edo is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a zeta peak edo and has a lower 19-limit relative error than any edo until 3395, and a lower 23-limit relative error than any until 8269. Also it has a lower 23-limit  TE loglfat badness than any smaller edo and less than any until 16808.

As a micro (or nano) temperament, it is a landscape system in the 7-limit, tempering out 250047/250000, and in the 11-limit it tempers out 9801/9800. Beyond that, 10648/10647 in the 13-limit, 12376/12375 in the 17-limit, 5929/5928 and 6860/6859 in the 19-limit and 8281/8280 in the 23-limit.

Since its prime factorization is 2^2\*3\*5\*41, 2460 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230. Of these, 12edo is too well-known to need any introduction, 41edo is an important system, and 205edo has proponents such as Aaron Andrew Hunt, who uses it as the default tuning for Hi-pi Instruments (and as a unit: Mem). Aside from these, 15edo, 20edo, 30edo, 60edo, and 164edo all have drawn some attention. Moreover a cent is exactly 2.05 minas, and a mem, 1\205 octaves, is exactly 12 minas.
221edo

Welcome to 221edo

## 221edo = 13edo × 17edo

Distinctively interesting choice for an edo, due to its divisibility by both 13 and 17. This means it can be used to play 13edo music and 17edo music at the same time, using the 221/17 and 221/13 vals respectively.

## 5.430

221edo has a step size of 5.4298642533937 cents. It is a commatonal EDO, because the step size represents commas like 78732/78125, 225/224, 144/143. It is also a nanotonal EDO.
220edo

Welcome to 220edo

## 220edo = 11edo × 20edo

Distinctively interesting choice for an edo, due to its divisibility by both 11 and 20. This means it can be used to play 11edo music and 20edo music at the same time, using the 220/20 and 220/11 vals respectively.

## 5.455

220edo has a step size of 5.4545454545455 cents. It is a commatonal EDO, because the step size represents commas like 15625/15552, 2401/2400, 176/175, 169/168. It is also a nanotonal EDO.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 220edo (ordered by absolute error).
17edo tetrachords

Let a "17edo primary tetrachord" mean a set of four pitches in 17edo that span a perfect fourth (seven degrees) and include one of each of the following:

1. the unison - 0 (degrees of 17edo) - solfege name 'do'.
2. a second - includes 1 (ra, a minor second), 2 (ru, a neutral second), and 3 (re, a major second).
3. a third - includes 4 (me, a minor third), 5 (mu, a neutral third), and 6 (mi, a major third).
4. the perfect fourth - 7 (fa).

### tetrachord notation

Tetrachord notation will show three scalar steps (as degrees of 17edo) separated by hyphens.

For instance, tetrachord 3-3-1 consists of

0 (do), the unison;

3 (re), a major second, 3 degrees up from 0;

6 (mi), a major third, 3 degrees up from 3; and

7 (fa), the perfect fourth, 1 degree up from 6.

The numbers in a tetrachord name will always add to 7.

### 17edo primary tetrachords

We have 9 primary tetrachords in 17edo.

Another way of showing them:

 mu
 mi

### 17edo tetrachords complete

A more generalized tetrachord system would allow multiple seconds or multiple thirds: for instance, 1-1-5 or 5-1-1. Thus, a complete chart of 17edo tetrachords looks like this (with primary tetrachords in bold):

Thus, by allowing multiples seconds or multiple thirds, we add 6 new tetrachords to our 9 primary tetrachords, for a total of 15. Our new ones:

See also: tetrachord, 22edo tetrachords, Tricesimoprimal Tetrachordal Tesseract.
1920edo

The 1920 division divides the octave into 1920 equal parts of exactly 0.625 cents each. It is distinctly consistent through the 25 limit, and in terms of 23-limit relative error, only 1578 and 1889 are both smaller and with a lower relative error. In the 29-limit, only 1578 beats it, and in the 31, 37, 41, 43 and 47 limits, nothing beats it. Because of this and because it is a highly composite number divisible by 12, it is another candidate for interval size measure.

1920 = 2^7 \* 3 \* 5; some of its divisors are 10, 12, 15, 16, 24, 60, 80, 96, 128, 240, 320 and 640.
173ed49

Division of the just 49:1 interval into 173 equal parts yields a remarkable non-octave-equivalent equally tempered scale with step width very close to 31edo, yet it is much more suited to melodies and harmonies spanning more than one octave. The octaves in this scale are stretched by 7.3 cents, and the step width is about 0.23 cents wider than in 31edo (which has *174* equal steps in its tempered version of 49:1).

This scale and 18edf perform almost identically except over very large distances.

Out of all the harmonics between 1 and 49 and other low-harmonic entropy intervals within this range, this tuning matches the overwhelming majority with tolerable accuracy. The only harmonics that aren't matched well are 27, 33, and 37. This scale also has very good perfect fifths (within a cent of just intonation), although its fourths are not as good.

The 49:1 interval, 6737.6518 cents, could be called a "wide fortieth" since it consists of five octaves plus a 49:32 "wide fifth". This is more than half the average hearing range of a human, and thus actual instruments would find it seldom necessary to cover much more than this range. Extending too far beyond this range offers diminishing returns anyway, since harmonics above the 51st become very poorly matched compared to those below. Thus, this temperament is best used over a finite range (173 or perhaps 175 steps at the maximum).

The intervals within this scale are:

* An eka-tritone (named by analogy with the periodic table) is an octave plus a tritone. A dvi-tritone is two octaves plus a tritone.

* + The 5:1, 7:1, 8:1, 12:1, and 18:1 intervals are split, yet all have a relatively high tolerance for mistuning, so in each case, both approximations are reasonable. When designing instruments to play in this tuning, it might be a good idea to dampen the 5th, 7th, 8th, and 12th harmonics while detuning the others slightly toward their corresponding scale degrees.
256/243

**256/243**

90.225 cents

File:Jid 256 243 pluck adu dr220.mp3 sound sample

The Pythagorean limma, or Pythagorean semitone, is the interval of size 256/243 = 2^8/3^5 (about 90.2¢), which is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five 4/3 just perfect fourths and octave-reducing the resulting interval.

See: Gallery of Just Intervals, comma

5edo, 10edo, 15edo, 20edo, 25edo and 30edo temper it out.
24/11

Interval information

In 11-limit Just Intonation, 24/11 is an undecimal (11-based) subninth of about 1350.6¢. Falling about halfway between 12edo's perfect ninth and diminished ninth, it is very xenharmonic. It is one of the simplest subninths in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

*See also Gallery of Just Intervals*
17/10

Interval information

In 17-limit Just Intonation, **17/10** is the "septendecimal major sixth," measuring about 918.6¢. It is the mediant between 5/3 and 12/7. Its octave complement is 20/17, the "septendecimal minor third".

## See also

* 3\4 or 9\12 (900 cents)
* 10\13 (923.08 cents)
* 13\17 (917.65 cents)
* Gallery of Just Intervals
171edo

*171edo* is a remarkable division of the octave which serves as a microtemperament for the 7-limit, approximating the 9-limit tonality diamond within about 2/5 of a cent. It divides the octave into 171 parts of 7.016 cents each, nearly the size of 225/224 as well as the third-comma, making it suitable for approximating tunings that use the third comma tempered fifth of some sort (third comma meantone is approximated as the built-in 19edo, the alternative is to link three such fifths with the patent fifth to produce a 12-tone well temperament). The excellence of its 7-limit approximations is good enough to make it the eleventh zeta integral edo but not enough to make it a gap edo.

171 supports a number of 7-limit rank-two temperaments: pontiac, with commas 4375/4374 and 32805/32768; sesquiquartififths, with 2401/2400 and 32805/32768; term, with 32805/32768 and 250047/250000; ennealimmal, with 2401/2400 and 4375/4374; tertiaseptal with 2401/2400 and 65635/65536; supermajor, with 4375/4374 and 52734275/52706752; enneadecal with 4375/4374 and 703125/702464; neptune, with 2401/2400 and 48828125/488771072; mitonic, with 4375/4374 and 2100875/2097152; and mutt, with 65635/65536 and 250047/250000. It is also an excellent tuning for the 5-limit schismatic microtemperament, tempering out 32805/32768, and the no-fives temperament tempering out <59 -39 0 1|.

171 factors into primes as 3^2 \* 19, and it shares the nearly pure 7/6 of 9edo and the nearly pure 6/5 of 19edo, with every 7-limit interval expressible in terms of 2, 6/5 and 7/6. 171 is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-three temperament jove, which tempers out 243/242 and 441/440, not to mention 540/539 and 2401/2400. Jove can be extended by adding 364/363 for the 13 limit and 595/594 for the 17 limit, which 171 also supports.

Also, see the Table of 171edo intervals.

# Scales

nestoria7

nestoria12

Third comma meantone inherited from 19edo: 27 27 18 27 27 27 18

Simulate Pythagorean: 29 29 13 29 29 29 13

Simulate Well temperament with third comma in G-D-A-E: 28 27 16 29 27 28 16 (can be linked to 12-note scale: 13 15 14 13 16 13 16 13 14 15 13 16)
20平均律

20平均律（20tet、20edo）は、オクターブをそれぞれ正確に60セントで20パートに分割したものである。それはより小さい、2、4、5、10、5n平均律ファミリーを持つ。20平均律は5平均律から見ると7倍音に非常に近く、10平均律からみると11、13、15倍音に近く、4平均律から見ると19、27倍音にちかい。29、31、そしてそのほかの倍音も、また漠然とだが近似している。したがって、20平均律は合理的な、説得力ある4:7:11:13:15の響きに近似する。

# 楽曲

* Etude in 20-tone equal tuning play by Herman Miller
* 20ET — Prelude & Fugue a3 by Aaron Hunt
* We Call It Sharing Breath play by Andrew Heathwaite
* The Walrus and the Carpenter by Chris Vaisvil and Bethan Mathis
* 20 edo Electric Guitar Improvisation by Chris Vaisvil
* Göbekli Tepe (20 edo guitar solo) by Chris Vaisvil
* A Paradigm Shift (20 edo) by Chris Vaisvil
* Composition for unconventional tempered clavier by Pyotr Chernobrivets
* 5 compositions for violin and clavier in 20-EDO by Pyotr Chernobrivets

# 20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=20, interval of equivalence=2, integer limit=16, threshold of JI pitch inclusion=0.3]というパラメータで生成したものである。「The “neighborhood” of JI」の一覧はこちら（huygens-fokker）を参照のこと。

# コンマをなだらかにする

20平均律を< 20 32 46 56 69 74|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。
187edo

The 187 equal temperament, which divides the octave into 187 parts of 6.417 cents each, is a bit of a chimera. It has a fifth which is 2.9 cents flat and a major third 1.3 cents flat, but the 7, 11 and 13 are more accurate. It tempers out 78732/78125, the sensipent comma, in the 5-limit and 225/224 in the 7-limit, providing the optimal patent val for sensei temperament which tempers out both. In the 11-limit it tempers out 441/440, 1375/1372 and 4000/3993 and in the 13-limit 1188/1183, 351/350, 625/624, 1716/1715 and 2200/2197.
24/17

**24/17**

596.9996 cents

File:Jid 24 17 pluck adu dr220.mp3 sound sample

In 17-limit Just Intonation, 24/17 is the "first septendecimal tritone," measuring very nearly 597¢. It is the mediant between 7/5 and 17/12, the "second septendecimal tritone." The two septendecimal tritones are each 3¢ away from the 600¢ half-octave, and so they are well-represented in all even-numbered EDO systems, including 12edo. Indeed, the latter system, containing good approximations of the 3rd and 17th harmonics, can use the half-octave as 24/17 and 17/12 in close approximations to chords such as 8:12:17 and 16:17:24. 22edo is another good EDO system for using the half-octave in this way.

See: Gallery of Just Intervals
241edo

The 241 equal temperament divides the octave in 241 parts of 4.979 cents each. It tempers out 78732/78125 in the 5-limit, 19683/19600 and 3136/3125 in the 7-limit, 65536/65219, 540/539, 43923/43904, and 151263/151250 in the 11-limit, and 351/350, 676/675, 729/728, 1001/1000 and 2080/2079 in the 13-limit. It provides the optimal patent val to subpental temperament. 241 is the 53rd prime EDO.
1L 11s

This MOS, properly called the Happy dodecatonic, is not a complete scale for any "interesting" temperament―the simplest absolute harmonic entropy minimum for it is where +7 generators equals a meantone fifth, and even there it implies a val with warts for 3 and (for the most part) 5. Its generators will not be listed here as the only interesting use for it is to extend it to 13 tones, i. e. to make it into a different MOS.
245edo

**245edo** uses a step size of 4.8979591836735 cents, and is a commatonal and nanotonal EDO. It can be thought of as seven sets of 35edo. Divisors are {{#rreplace: 245 49 35 7 5 1|/(\d+)(\D+)?/|\1\2}}. In 7-limit, it has the same 1/1—225/224—126/125—81/80 split as 171edo does, but not as accurate. It is consistent in the 5-odd-limit.
255edo

The *255 equal division* divides the octave into 255 equal parts of 4.706 cents each. It tempers out the parakleisma, |8 14 -13>, and the septendecima, |-52 -17 34>, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports mirkat temperament, and in fact provides the optimal patent val. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.
2100edo

The 2100 equal division divides the octave into 2100 parts of precisely 4/7 cents (0.571429 cents) each. The patent val tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, and thereby provides an excellent tuning for 5-limit schismatic aka helmholtz temperament, and the 7-limit sesquiquartififths temperament. As with any equal division of this size, it supports a number of possible meantone tunings, but <2100 3319 4876 5890| is notable for being nearly identical to quarter-comma meantone. Also the relative cent of 21edo.
17L 12s

This MOS, belonging to the Leapday and Grackle temperaments, is generated by taking a chain of 29 fifths tempered 1.49 to 3.93 cents sharp. That is to say, its region includes all linear combinations of 17edo and 29edo and divides at 80edo into Margo Schulter's gentle region extending down to 29edo and a yet-unnamed region extending up to 17edo. Only the latter region will be catalouged here, as cataloguing the former in a second place is inefficient.
21/16

**21/16**

470.7809 cents

File:Jid 21 16 pluck adu dr220.mp3 sound sample

21/16, the septimal sub-fourth, is an interval of the 7 prime-limit measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of 4/3 by 64/63, a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3\*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of 3/2 between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths (7/4) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

21/16 is 21/20 away from 5/4. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by 8/7, the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an 11-limit system, 11/8 is also nearby, so that 21/16 can step up by the small semitone of 22/21 (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.

The 7-limit is known for its subminor and supermajor 2nds, 3rds, 6ths and 7ths. 21/16 is also an essential interval of the 7-limit and worth distinguishing.

See: Gallery of Just Intervals
176edo

Welcome to 176edo

## 176edo = 11edo × 16edo

Distinctively interesting choice for an edo, due to its divisibility by both 11 and 16. This means it can be used to play 11edo music and 16edo music at the same time, using the 176/16 and 176/11 vals respectively.

## 6.818

176edo has a step size of 6.8181818181818 cents. It is a commatonal EDO, because the step size represents commas like 32805/32768, 126/125.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 176edo (ordered by absolute error).
214edo

**214edo** uses a step size of 5.607476635514 cents, and is a commatonal EDO. It can be thought of as two sets of 107edo. Divisors are {{#rreplace: 214 107 2 1|/(\d+)(\D+)?/|\1\2}}. In 7-limit, it has the same 1/1—225/224—126/125—81/80 split as 171edo does, but not as accurate. It is consistent and uniquely consistent in the 7-odd-limit.
Quincy Fernborn

Quincy Fernborn is a doctor working at the free hospital in Estanrena.

## Interactions

Morthos, Zamena, and Yulseph met him while at a dinner party fundraiser event in the castle.

## Relations

* Queen Wilma Fernborn - mother
* Claude Fernborn - father
* Queen Norma Fernborn - sister
* Bishop Fernborn - brother
* Murphy Fernborn - brother
* Davis Percy - half-brother
22-3reduced-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

 **22-3reduced-limit** is the tritave version of 21-odd-limit.

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

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

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

Rocan is a baker at Rocan's Baked Goods in Taur Ylvanor.

## Description/Personality

He is a wood elf. He is of an average height and build. He wears dark green and brown clothes with a white apron and gloves.

He grew up on the streets of Taur Ylvanor, so he is familiar with every nook and cranny of the lower and middle districts. He, Alvie, and several other orphans founded the Monadika Orphanage and now regularly brings bread to the establishment free of charge. He is humble, caring, and stands up for the little guy. He is also willing to go to extreme measures to protect his fellow commoners.

## Interactions

He and Alvie grew up on the streets together, and he helped found the orphanage.

When Mara, Alvie, Heskan, Graynor, Rosalind, and Kuiekkel arrived in Taur Ylvanor, he offered them food at a discounted price. He asked Alvie to take care of a shady magic shop owner (Sigis Morralei) after he sold him a rug that ended up killing his assistant.
Bishop Fernborn

Murphy Fernborn is a trained mage teaching magic to kids at the Drakontas School of Magic in Drakontas.

## Interactions

Morthos, Zamena, and Yulseph met him while at a dinner party fundraiser event in the castle.

## Relations

* Queen Wilma Fernborn - mother
* Claude Fernborn - father
* Queen Norma Fernborn - sister
* Murphy Fernborn - brother
* Quincy Fernborn - brother
* Davis Percy - half-brother
17音理論

English

17平均律は12平均律や純正調の5度ととても近い5度を持つ。その結果として、5線譜上の通常の5度圏を使う。しかしそれは、5度の異なったサイズの積み重ねによって、新しいエンハーモニックと同様のねじれを持つ。閉じた5度圏のエンハーモニックネームを理解することで、次のように書き出すことができる。

Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E#

そしてその時望ましいサイズの円に包まれる。もし17を望むならば、Cb＝A#となる。フラット・シャープの値がとても大きい。その近い2つの音は円の中にあり、なじみある和声の機能をもつ。

17fifths

線は5つの4度を横切ることを意味し、それは1/17オクターブ、「短2度」である。これは17平均律の完全4度はオクターブの7/17であり、7\*5（完全4度が5つ）/17=2 …1を示す。

それゆえ、もしふたたびこれらの線をもとに円を包むならば、順に音階の17音を上げ下げする。

17versus12names

C#はDbより高い音である。12音と同様正確に、オクターブ/ユニゾンの音程を保持する音である。17平均律のハーフステップまたは短2度は、増1度、シャープ、フラットの2倍に当たる。2つのシャープまたはフラットは、合計で短3度になる。2つの中立3度は、合計で完全5度になる。17はこの最後の特性を24平均律と共有する。

17versus12cents

次にピッチ/音程クラスを示す

17versus12pcs
19/18

Interval information

In 19-limit Just Intonation, **19/18** is the "greater undevicesimal semitone," measuring about 93.6¢. Its octave complement is 36/19, the "undevicesimal major seventh".

The small comma (39/35)/(19/18)^2 = 12636/12635 is tempered out in {{#rreplace: 53, 130, 140, 243, 270, 422, 513, 665, 742, 795, 935, 1178, 1065, 1205, 1308, 1395, 1448, 1578, 2000, 2190, 2243, 2460, 3125, 3178, 3395, and 4703 EDOs|/(\d+)(\D+)?/|\1\2}}.

The password comma, (256/243)/(20/19) = (19/18)/(135/128) = 1216/1215 is tempered out in {{#rreplace: 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692 EDOs|/(\d+)(\D+)?/|\1\2}}, leading to a variety of temperaments which identifies small undevicesimal semitone with Pythagorean limma and greater undevicesimal semitone with major limma respectively.
21/8

Interval information

In Just Intonation, 21/8 is the "septimal sub minor eleventh" of approximately 1670.8¢. Although it falls between the familiar minor eleventh and major tenth of 12edo, it generally sounds more like a narrow eleventh than a wide tenth. It can be found between the 21st and 8th overtones in the harmonic series. It is the tritave inverse of 8/7. It differs from the Pythagorean minor eleventh of 8/3 by 64/63, a microtone of about 27.3¢.

## See also

* Gallery of Just Intervals
* Septimal whole tone - Wikipedia.