51edo: Difference between revisions

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→Intervals: Add columns for additional ratios of 5 and 11 tending {flat/sharp}; move relevant ratios from main approximate ratios column to these; revert subgroup description to original specification.
Line 25: Line 25:
! [[Cent]]s
! [[Cent]]s
! Approximate ratios*
! Approximate ratios*
!Additional ratios of
5 and/or 11 tending
flat (51 patent val)
!Additional ratios of
5 and/or 11 tending
sharp (51ce val)
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
Line 30: Line 36:
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
|
|
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 36: Line 44:
| 1
| 1
| 23.5
| 23.5
| ''[[49/48]]'', [[64/63]]
| [[64/63]], ''[[49/48]]''
|
|[[81/80]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 44: Line 54:
| 47.1
| 47.1
| ''[[28/27]]''
| ''[[28/27]]''
|
|
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 51: Line 63:
| 70.6
| 70.6
| [[27/26]]
| [[27/26]]
|
|
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 58: Line 72:
| 94.1
| 94.1
|  
|  
|[[135/128]]
|
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 65: Line 81:
| 117.6
| 117.6
| [[14/13]]
| [[14/13]]
|[[16/15]]
|
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 72: Line 90:
| 141.2
| 141.2
| [[13/12]]
| [[13/12]]
|
|[[12/11]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 78: Line 98:
| 7
| 7
| 164.7
| 164.7
| [[11/10]]
|
|[[11/10]]
|11/10
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 86: Line 108:
| 188.2
| 188.2
|  
|  
|
|
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 93: Line 117:
| 211.8
| 211.8
| [[9/8]]
| [[9/8]]
|
|
| Major 2nd
| Major 2nd
| M2
| M2
Line 100: Line 126:
| 235.3
| 235.3
| [[8/7]]
| [[8/7]]
|
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 107: Line 135:
| 258.8
| 258.8
| [[7/6]]
| [[7/6]]
|
|
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 114: Line 144:
| 282.4
| 282.4
| ''[[32/27]]''
| ''[[32/27]]''
|
|[[13/11]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 121: Line 153:
| 305.9
| 305.9
|  
|  
|
|[[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 127: Line 161:
| 14
| 14
| 329.4
| 329.4
| [[40/33]]
|
|[[40/33]]
|40/33
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 135: Line 171:
| 352.9
| 352.9
| [[16/13]], [[39/32]]
| [[16/13]], [[39/32]]
|
|
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 142: Line 180:
| 376.5
| 376.5
| [[26/21]]
| [[26/21]]
|[[5/4]]
|
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 149: Line 189:
| 400.0
| 400.0
|  
|  
|
|''5/4''
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 156: Line 198:
| 423.5
| 423.5
| ''[[81/64]]''
| ''[[81/64]]''
|
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 163: Line 207:
| 447.1
| 447.1
| ''[[9/7]]''
| ''[[9/7]]''
|
|
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 170: Line 216:
| 470.6
| 470.6
| [[21/16]]
| [[21/16]]
|
|
| Down 4th
| Down 4th
| v4
| v4
Line 177: Line 225:
| 494.1
| 494.1
| [[4/3]]
| [[4/3]]
|
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 184: Line 234:
| 517.6
| 517.6
|  
|  
|
|
| Up 4th
| Up 4th
| ^4
| ^4
Line 190: Line 242:
| 23
| 23
| 541.2
| 541.2
| [[15/11]]
|
|[[11/8]], [[15/11]]
|15/11
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 198: Line 252:
| 564.7
| 564.7
| [[18/13]]
| [[18/13]]
|
|''11/8''
| Dim 5th
| Dim 5th
| d5
| d5
Line 205: Line 261:
| 588.2
| 588.2
| [[39/28]]
| [[39/28]]
|
|
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 212: Line 270:
| 611.8
| 611.8
| [[56/39]]
| [[56/39]]
|
|
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 219: Line 279:
| 635.3
| 635.3
| [[13/9]]
| [[13/9]]
|
|''16/11''
| Aug 4th
| Aug 4th
| A4
| A4
Line 225: Line 287:
| 28
| 28
| 658.8
| 658.8
| [[22/15]]
|
|[[16/11]], [[22/15]]
|22/15
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 233: Line 297:
| 682.4
| 682.4
|  
|  
|
|
| Down 5th
| Down 5th
| v5
| v5
Line 240: Line 306:
| 705.9
| 705.9
| [[3/2]]
| [[3/2]]
|
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 247: Line 315:
| 729.4
| 729.4
| [[32/21]]
| [[32/21]]
|
|
| Up 5th
| Up 5th
| ^5
| ^5
Line 254: Line 324:
| 752.9
| 752.9
| ''[[14/9]]''
| ''[[14/9]]''
|
|
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 261: Line 333:
| 776.5
| 776.5
| ''[[128/81]]''
| ''[[128/81]]''
|
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 268: Line 342:
| 800.0
| 800.0
|  
|  
|
|''8/5''
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 275: Line 351:
| 823.5
| 823.5
| [[21/13]]
| [[21/13]]
|[[8/5]]
|
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 282: Line 360:
| 847.1
| 847.1
| [[13/8]], [[64/39]]
| [[13/8]], [[64/39]]
|
|
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 288: Line 368:
| 37
| 37
| 870.6
| 870.6
| [[33/20]]
|
|[[33/20]]
|33/20
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 296: Line 378:
| 894.1
| 894.1
|  
|  
|
|[[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 303: Line 387:
| 917.6
| 917.6
| ''[[27/16]]''
| ''[[27/16]]''
|
|[[22/13]]
| Major 6th
| Major 6th
| M6
| M6
Line 310: Line 396:
| 941.2
| 941.2
| [[12/7]]
| [[12/7]]
|
|
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 317: Line 405:
| 964.7
| 964.7
| [[7/4]]
| [[7/4]]
|
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 324: Line 414:
| 988.2
| 988.2
| [[16/9]]
| [[16/9]]
|
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 331: Line 423:
| 1011.8
| 1011.8
|  
|  
|
|
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 337: Line 431:
| 44
| 44
| 1035.3
| 1035.3
| [[20/11]]
|
|[[20/11]]
|20/11
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 345: Line 441:
| 1058.8
| 1058.8
| [[24/13]]
| [[24/13]]
|
|[[11/6]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 352: Line 450:
| 1082.4
| 1082.4
| [[13/7]]
| [[13/7]]
|[[15/8]]
|
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 359: Line 459:
| 1105.9
| 1105.9
|  
|  
|[[256/135]]
|
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 366: Line 468:
| 1129.4
| 1129.4
| [[52/27]]
| [[52/27]]
|
|
| Major 7th
| Major 7th
| M7
| M7
Line 373: Line 477:
| 1152.9
| 1152.9
| ''[[27/14]]''
| ''[[27/14]]''
|
|
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 380: Line 486:
| 1176.5
| 1176.5
| [[63/32]], ''[[96/49]]''
| [[63/32]], ''[[96/49]]''
|
|[[160/81]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 387: Line 495:
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
|
|
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki>*</nowiki> As a 2.3.7.13-subgroup temperament, with the addition of selected intervals of 5 and 11; inconsistent intervals in italic.
<nowiki>*</nowiki> As a 2.3.7.13-subgroup temperament; inconsistent intervals in italic.


== Notation ==
== Notation ==

Revision as of 10:23, 4 January 2026

← 50edo 51edo 52edo →
Prime factorization 3 × 17
Step size 23.5294 ¢ 
Fifth 30\51 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 6:3 (141.2 ¢ : 70.59 ¢)
Consistency limit 3
Distinct consistency limit 3

51 equal divisions of the octave (abbreviated 51edo or 51ed2), also called 51-tone equal temperament (51tet) or 51 equal temperament (51et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 51 equal parts of about 23.5 ¢ each. Each step represents a frequency ratio of 21/51, or the 51st root of 2.

Theory

Since 51 = 3 × 17, 51edo shares its fifth with 17edo. Compared to other multiples of 17edo, notably 34edo and 68edo, 51edo's harmonic inventory seems lacking, getting few harmonics very well considering its step size. However, it does possess excellent approximations of 11/10 and 21/16, only about 0.3 cents off in each case.

Using the patent val, 51et tempers out 250/243 in the 5-limit, 225/224 and 2401/2400 in the 7-limit, and 55/54 and 100/99 in the 11-limit. It is the optimal patent val for sonic, the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to 22edo for porcupine, with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit porky temperament, which is sonic plus 225/224. It contains an archeotonic (6L 1s) scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top.

Using the 51c val ⟨51 81 119 143], the 5/4 is mapped to 1\3 (400 cents), supporting augmented. In the 7-limit it tempers out 245/243 and supports hemiaug and rodan. Alternatively, the 51cd val ⟨51 81 119 144] takes the same 7/4 from 17edo, and supports augene. The 51ce val ⟨51 81 119 143 177 189] supports a variant of rodan called aerodino.

51edo's step is the closest direct approximation to the Pythagorean comma by edo steps, though that comma itself is mapped to a different interval.

Odd harmonics

Approximation of prime harmonics in 51edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +3.9 -9.8 -4.1 -10.1 +6.5 -10.8 +8.4 +7.0 +5.7 +7.9
Relative (%) +0.0 +16.7 -41.8 -17.5 -43.1 +27.8 -46.1 +35.6 +29.8 +24.3 +33.6
Steps
(reduced) 		51
(0) 		81
(30) 		118
(16) 		143
(41) 		176
(23) 		189
(36) 		208
(4) 		217
(13) 		231
(27) 		248
(44) 		253
(49)

Subsets and supersets

51edo contains 3edo and 17edo as subsets.

One of the very powerful (but very complex) supersets of 51edo is 612edo, which divides each step of 51edo into 12 equal parts, for which the name "skisma" has been proposed.

Intervals

# Cents Approximate ratios* Additional ratios of

5 and/or 11 tending flat (51 patent val)

Additional ratios of

5 and/or 11 tending sharp (51ce val)

Ups and downs notation
0 0.0 1/1 Perfect 1sn P1 D
1 23.5 64/63, 49/48 81/80 Up 1sn ^1 ^D
2 47.1 28/27 Downminor 2nd vm2 vEb
3 70.6 27/26 Minor 2nd m2 Eb
4 94.1 135/128 Upminor 2nd ^m2 ^Eb
5 117.6 14/13 16/15 Downmid 2nd v~2 ^^Eb
6 141.2 13/12 12/11 Mid 2nd ~2 vvvE, ^^^Eb
7 164.7 11/10 11/10 Upmid 2nd ^~2 vvE
8 188.2 Downmajor 2nd vM2 vE
9 211.8 9/8 Major 2nd M2 E
10 235.3 8/7 Upmajor 2nd ^M2 ^E
11 258.8 7/6 Downminor 3rd vm3 vF
12 282.4 32/27 13/11 Minor 3rd m3 F
13 305.9 6/5 Upminor 3rd ^m3 ^F
14 329.4 40/33 40/33 Downmid 3rd v~3 ^^F
15 352.9 16/13, 39/32 Mid 3rd ~3 ^^^F, vvvF#
16 376.5 26/21 5/4 Upmid 3rd ^~3 vvF#
17 400.0 5/4 Downmajor 3rd vM3 vF#
18 423.5 81/64 Major 3rd M3 F#
19 447.1 9/7 Upmajor 3rd ^M3 ^F#
20 470.6 21/16 Down 4th v4 vG
21 494.1 4/3 Perfect 4th P4 G
22 517.6 Up 4th ^4 ^G
23 541.2 11/8, 15/11 15/11 Downdim 5th vd5 vAb
24 564.7 18/13 11/8 Dim 5th d5 Ab
25 588.2 39/28 Updim 5th ^d5 ^Ab
26 611.8 56/39 Downaug 4th vA4 vG#
27 635.3 13/9 16/11 Aug 4th A4 G#
28 658.8 16/11, 22/15 22/15 Upaug 4th ^A4 ^G#
29 682.4 Down 5th v5 vA
30 705.9 3/2 Perfect 5th P5 A
31 729.4 32/21 Up 5th ^5 ^A
32 752.9 14/9 Downminor 6th vm6 vBb
33 776.5 128/81 Minor 6th m6 Bb
34 800.0 8/5 Upminor 6th ^m6 ^Bb
35 823.5 21/13 8/5 Downmid 6th v~6 ^^Bb
36 847.1 13/8, 64/39 Mid 6th ~6 vvvB, ^^^Bb
37 870.6 33/20 33/20 Upmid 6th ^~6 vvB
38 894.1 5/3 Downmajor 6th vM6 vB
39 917.6 27/16 22/13 Major 6th M6 B
40 941.2 12/7 Upmajor 6th ^M6 ^B
41 964.7 7/4 Downminor 7th vm7 vC
42 988.2 16/9 Minor 7th m7 C
43 1011.8 Upminor 7th ^m7 ^C
44 1035.3 20/11 20/11 Downmid 7th v~7 ^^C
45 1058.8 24/13 11/6 Mid 7th ~7 ^^^C, vvvC#
46 1082.4 13/7 15/8 Upmid 7th ^~7 vvC#
47 1105.9 256/135 Downmajor 7th vM7 vC#
48 1129.4 52/27 Major 7th M7 C#
49 1152.9 27/14 Upmajor 7th ^M7 ^C#
50 1176.5 63/32, 96/49 160/81 Down 8ve v8 vD
51 1200.0 2/1 Perfect 8ve P8 D

* As a 2.3.7.13-subgroup temperament; inconsistent intervals in italic.

Notation

Ups and downs notation

51edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

In 51edo, a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 51edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 51edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
15/14, 28/15 1.796 7.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
11/9, 18/11 5.533 23.5
7/5, 10/7 5.723 24.3
9/5, 10/9 5.832 24.8
15/8, 16/15 5.916 25.1
11/7, 14/11 6.021 25.6
13/8, 16/13 6.531 27.8
13/11, 22/13 6.857 29.1
13/10, 20/13 7.155 30.4
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
11/6, 12/11 9.461 40.2
5/3, 6/5 9.759 41.5
5/4, 8/5 9.843 41.8
11/8, 16/11 10.141 43.1
13/7, 14/13 10.651 45.3
15/13, 26/15 11.082 47.1
9/7, 14/9 11.555 49.1
15-odd-limit intervals in 51edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
15/14, 28/15 1.796 7.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
7/5, 10/7 5.723 24.3
15/8, 16/15 5.916 25.1
11/7, 14/11 6.021 25.6
13/8, 16/13 6.531 27.8
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
5/4, 8/5 9.843 41.8
11/8, 16/11 10.141 43.1
13/7, 14/13 10.651 45.3
9/7, 14/9 11.975 50.9
15/13, 26/15 12.447 52.9
5/3, 6/5 13.770 58.5
11/6, 12/11 14.069 59.8
13/10, 20/13 16.374 69.6
13/11, 22/13 16.673 70.9
9/5, 10/9 17.698 75.2
11/9, 18/11 17.996 76.5
15-odd-limit intervals in 51edo (51ce val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
11/9, 18/11 5.533 23.5
9/5, 10/9 5.832 24.8
13/8, 16/13 6.531 27.8
13/11, 22/13 6.857 29.1
13/10, 20/13 7.155 30.4
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
11/6, 12/11 9.461 40.2
5/3, 6/5 9.759 41.5
13/7, 14/13 10.651 45.3
15/13, 26/15 11.082 47.1
9/7, 14/9 11.975 50.9
11/8, 16/11 13.388 56.9
5/4, 8/5 13.686 58.2
11/7, 14/11 17.508 74.4
15/8, 16/15 17.614 74.9
7/5, 10/7 17.806 75.7
15/14, 28/15 21.734 92.4

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.7 1029/1024, [17 -16 3⟩ [⟨51 81 143]] −0.339 1.63 6.92
2.3.7.13 343/338, 512/507, 2197/2187 [⟨51 81 143]] −0.695 1.54 6.54
2.3.5 128/125, [-13 17 -6⟩ [⟨51 81 119]] (51c) −2.789 2.41 10.3
2.3.5.7 128/125, 245/243, 1029/1000 [⟨51 81 119 143]] (51c) −1.730 2.79 11.9
2.3.5 250/243, 34171875/33554432 [⟨51 81 118]] (51) +0.581 2.77 11.8
2.3.5.7 225/224, 250/243, 1029/1024 [⟨51 81 118 143]] (51) +0.803 2.43 10.3

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 5\51 117.6 15/14 Miracle (51e, out of tune) / oracle (51)
1 7\51 164.7 11/10 Porky (51)
1 10\51 235.3 8/7 Rodan (51cf…, out of tune) / aerodino (51ce)
1 5\51 541.2 15/11 Necromanteion (51ce)
3 19\51
(2\51) 		447.1
(47.1) 		9/7
(36/35) 		Hemiaug (51ce)
3 21\51
(4\51) 		494.1
(94.1) 		4/3
(16/15) 		Augmented (51c)
4/3
(21/20) 		Fog (51)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Instruments

Lumatone
See Lumatone mapping for 51edo.

Music

Bryan Deister
Frédéric Gagné
James Mulvale (FASTFAST)
Ray Perlner
  • Fugue (2023) – for organ in 51edo Porcupine[7] ssssssL "Pandian"
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