51edo: Difference between revisions

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Using the [[patent val]], 51et [[tempering out|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 [[patent val]], 51et [[tempering out|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 {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented (temperament)|augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. Alternatively, the 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]]. The 51ce val {{val| 51 81 '''119''' 143 '''177''' 189 }} supports a variant of rodan called [[aerodino]].  
Using the 51c val {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented (temperament)|augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. Alternatively, the 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]]. The 51ce val {{val| 51 81 '''119''' 143 '''177''' 189 }} supports a variant of rodan called [[Gamelismic_clan#Aerodino|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.
51edo's step is the closest direct approximation to the [[Pythagorean comma]] by edosteps, though that comma itself is mapped to a different interval.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|51}}
{{Harmonics in equal|51|intervals=odd|prec=2|columns=14}}
{{Harmonics in equal|51|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 51edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Line 20: Line 21:


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2 center-6 center-7 center-8"
|-
|-
! #
! rowspan="2" | #
! [[Cent]]s
! rowspan="2" | [[Cent]]s
! Approximate ratios*
! colspan="3" | Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
! rowspan="2" colspan="3" | [[Ups and downs notation]]
|-
! 2.3.7.11/5.13<br>subgroup
! Ratios of 5 and 11<br>tending flat (51 val)
! Ratios of 5 and 11<br>tending sharp (51ce val)
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| [[1/1]]
|
|
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 36: Line 43:
| 1
| 1
| 23.5
| 23.5
| ''49/48'', 64/63
| [[64/63]], ''[[49/48]]''
| ''40/39''
| [[81/80]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 43: Line 52:
| 2
| 2
| 47.1
| 47.1
| ''28/27''
| ''[[28/27]]''
| [[33/32]], ''25/24'', ''81/80''
| [[36/35]], [[40/39]]
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 50: Line 61:
| 3
| 3
| 70.6
| 70.6
| 27/26
| [[27/26]]
| ''36/35''
| ''21/20'', ''33/32''
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 58: Line 71:
| 94.1
| 94.1
|  
|  
| [[21/20]]
| ''16/15'', ''25/24''
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 64: Line 79:
| 5
| 5
| 117.6
| 117.6
| 14/13
| [[14/13]]
| [[15/14]], [[16/15]]
|
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 71: Line 88:
| 6
| 6
| 141.2
| 141.2
| 13/12
| [[13/12]]
|
| [[12/11]], ''15/14''
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 78: Line 97:
| 7
| 7
| 164.7
| 164.7
| [[11/10]]
| ''10/9'', ''12/11''
|  
|  
| Upmid 2nd
| Upmid 2nd
Line 86: Line 107:
| 188.2
| 188.2
|  
|  
|
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 92: Line 115:
| 9
| 9
| 211.8
| 211.8
| 9/8
| [[9/8]]
|
|
| Major 2nd
| Major 2nd
| M2
| M2
Line 99: Line 124:
| 10
| 10
| 235.3
| 235.3
| 8/7
| [[8/7]]
| ''15/13''
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 106: Line 133:
| 11
| 11
| 258.8
| 258.8
| 7/6
| [[7/6]]
|
| [[15/13]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 113: Line 142:
| 12
| 12
| 282.4
| 282.4
| ''32/27''
| ''[[32/27]]''
|
| [[13/11]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 121: Line 152:
| 305.9
| 305.9
|  
|  
| ''13/11''
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 127: Line 160:
| 14
| 14
| 329.4
| 329.4
| 63/52
| [[40/33]], [[63/52]]
| ''6/5'', ''11/9''
|
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 134: Line 169:
| 15
| 15
| 352.9
| 352.9
| 16/13, 39/32
| [[16/13]], [[39/32]]
|
| [[11/9]], [[27/22]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 141: Line 178:
| 16
| 16
| 376.5
| 376.5
| 26/21
| [[26/21]]
| [[5/4]], ''27/22''
|
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 149: Line 188:
| 400.0
| 400.0
|  
|  
|
| ''5/4'', ''14/11''
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 155: Line 196:
| 18
| 18
| 423.5
| 423.5
| ''81/64''
| ''[[81/64]]''
| [[14/11]]
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 162: Line 205:
| 19
| 19
| 447.1
| 447.1
| ''9/7''
| ''[[9/7]]''
|
| [[13/10]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 169: Line 214:
| 20
| 20
| 470.6
| 470.6
| 21/16
| [[21/16]]
| ''13/10''
|
| Down 4th
| Down 4th
| v4
| v4
Line 176: Line 223:
| 21
| 21
| 494.1
| 494.1
| 4/3
| [[4/3]]
|
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 184: Line 233:
| 517.6
| 517.6
|  
|  
|
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 190: Line 241:
| 23
| 23
| 541.2
| 541.2
| [[15/11]]
| [[11/8]], ''27/20''
|  
|  
| Downdim 5th
| Downdim 5th
Line 197: Line 250:
| 24
| 24
| 564.7
| 564.7
| 18/13
| [[18/13]]
|
| ''7/5'', ''11/8''
| Dim 5th
| Dim 5th
| d5
| d5
Line 204: Line 259:
| 25
| 25
| 588.2
| 588.2
| 39/28
| [[39/28]]
| [[7/5]]
|
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 211: Line 268:
| 26
| 26
| 611.8
| 611.8
| 56/39
| [[56/39]]
| [[10/7]]
|
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 218: Line 277:
| 27
| 27
| 635.3
| 635.3
| 13/9
| [[13/9]]
|
| ''10/7'', ''16/11''
| Aug 4th
| Aug 4th
| A4
| A4
Line 225: Line 286:
| 28
| 28
| 658.8
| 658.8
| [[22/15]]
| [[16/11]], ''40/27''
|  
|  
| Upaug 4th
| Upaug 4th
Line 233: Line 296:
| 682.4
| 682.4
|  
|  
|
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
Line 239: Line 304:
| 30
| 30
| 705.9
| 705.9
| 3/2
| [[3/2]]
|
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 246: Line 313:
| 31
| 31
| 729.4
| 729.4
| 32/21
| [[32/21]]
| ''20/13''
|
| Up 5th
| Up 5th
| ^5
| ^5
Line 253: Line 322:
| 32
| 32
| 752.9
| 752.9
| ''14/9''
| ''[[14/9]]''
|
| [[20/13]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 260: Line 331:
| 33
| 33
| 776.5
| 776.5
| ''128/81''
| ''[[128/81]]''
| [[11/7]]
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 268: Line 341:
| 800.0
| 800.0
|  
|  
|
| ''8/5'', ''11/7''
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 274: Line 349:
| 35
| 35
| 823.5
| 823.5
| 21/13
| [[21/13]]
| [[8/5]], ''44/27''
|
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 281: Line 358:
| 36
| 36
| 847.1
| 847.1
| 13/8, 64/39
| [[13/8]], [[64/39]]
|
| [[18/11]], [[44/27]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 288: Line 367:
| 37
| 37
| 870.6
| 870.6
| 104/63
| [[33/20]], [[104/63]]
| ''5/3'', ''18/11''
|
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 296: Line 377:
| 894.1
| 894.1
|  
|  
| ''22/13''
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 302: Line 385:
| 39
| 39
| 917.6
| 917.6
| ''27/16''
| ''[[27/16]]''
|
| [[22/13]]
| Major 6th
| Major 6th
| M6
| M6
Line 309: Line 394:
| 40
| 40
| 941.2
| 941.2
| 12/7
| [[12/7]]
|
| [[26/15]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 316: Line 403:
| 41
| 41
| 964.7
| 964.7
| 7/4
| [[7/4]]
| ''26/15''
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 323: Line 412:
| 42
| 42
| 988.2
| 988.2
| 16/9
| [[16/9]]
|
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 331: Line 422:
| 1011.8
| 1011.8
|  
|  
|
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 337: Line 430:
| 44
| 44
| 1035.3
| 1035.3
| [[20/11]]
|  
|  
| ''9/5'', ''11/6''
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 344: Line 439:
| 45
| 45
| 1058.8
| 1058.8
| 24/13
| [[24/13]]
|
| [[11/6]], ''28/15''
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 351: Line 448:
| 46
| 46
| 1082.4
| 1082.4
| 13/7
| [[13/7]]
| [[15/8]], [[28/15]]
|
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 359: Line 458:
| 1105.9
| 1105.9
|  
|  
| [[40/21]]
| ''15/8'', ''48/25''
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 365: Line 466:
| 48
| 48
| 1129.4
| 1129.4
| 52/27
| [[52/27]]
| ''35/18''
| ''40/21'', ''64/33''
| Major 7th
| Major 7th
| M7
| M7
Line 372: Line 475:
| 49
| 49
| 1152.9
| 1152.9
| ''27/14''
| ''[[27/14]]''
| [[64/33]], ''48/25'', ''160/81''
| [[35/18]], [[39/20]]
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 379: Line 484:
| 50
| 50
| 1176.5
| 1176.5
| 63/32, ''96/49''
| [[63/32]], ''[[96/49]]''
| ''39/20''
| [[160/81]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 386: Line 493:
| 51
| 51
| 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, inconsistent intervals in italic
<nowiki>*</nowiki> inconsistent intervals in italic.


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould 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.  
[[Stein–Zimmermann–Gould notation]] for 51edo uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
51edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.  
{{Sharpness-sharp6a}}
{{Sharpness-sharp6a}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
{{Sharpness-sharp6b}}
In 51edo, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
{{Sharpness-sharp6}}
If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}


=== Ivan Wyschnegradsky's notation ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
{{Sharpness-sharp6-iw}}


=== Sagittal notation ===
=== Sagittal notation ===
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|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.
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|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 ====
==== Evo flavor ====
Line 451: Line 558:
default [[File:51-EDO_Evo-SZ_Sagittal.svg]]
default [[File:51-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|51}}
{{Q-odd-limit intervals|51.1|apx=val|header=none|tag=none|title=15-odd-limit intervals in 51edo (51ce val mapping)}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 472: Line 584:
|-
|-
| 2.3.7.13
| 2.3.7.13
| 512/507, 1029/1024, 2197/2187
| 343/338, 512/507, 2197/2187
| {{Mapping| 51 81 143 }}
| {{Mapping| 51 81 143 }}
| −0.695
| −0.695
Line 521: Line 633:
| 117.6
| 117.6
| 15/14
| 15/14
| [[Miracle]] (51e, out of tune) / oracle (51)
| [[Miracle]] (51e, out of tune)
|-
|-
| 1
| 1
Line 533: Line 645:
| 235.3
| 235.3
| 8/7
| 8/7
| [[Rodan]] (51cf…, out of tune) / aerodino (51ce)
| [[Rodan]] (51cf, out of tune) / aerodino (51ce)
|-
|-
| 1
| 1
| 5\51
| 19\51
| 447.1
| 13/10
| [[Supersensi]] (51cde)
|-
| 1
| 22\51
| 517.6
| 27/20
| [[Gravity]] (51ce) / [[abergravity]] (51ce)
|-
| 1
| 23\51
| 541.2
| 541.2
| 15/11
| 15/11
| [[Necromanteion]] (51ce)
| [[Necromanteion]] (51ce)<br>[[Oracle]] (51)<br>[[Cypress]] (51cde…)
|-
|-
| 3
| 3
Line 547: Line 671:
| [[Hemiaug]] (51ce)
| [[Hemiaug]] (51ce)
|-
|-
| 3
| rowspan="2" | 3
| 21\51<br>(4\51)
| rowspan="2" | 21\51<br>(4\51)
| 494.1<br>(94.1)
| rowspan="2" | 494.1<br>(94.1)
| 4/3<br>(16/15)
| 4/3<br>(16/15)
| [[Augmented (temperament)|Augmented]] (51c)
| [[Augmented (temperament)|Augmented]] (7-limit, 51cd)
|-
| style="text-align: center;" | 4/3<br>(21/20)
| style="text-align: left;" | [[Fog]] (51)
|}
|}
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 566: Line 693:
== Music ==
== Music ==
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=sySLQUXnQ70 ''Preludio Sentimentale (microtonal improvisation in 28edo)''] (2023)
* [https://www.youtube.com/watch?v=sCE0MjUyRUk ''28edo blues''] (2023)
* [https://www.youtube.com/shorts/sTPJtuHUwkg ''51edo improv''] (2025-02-03)
* [https://www.youtube.com/shorts/sTPJtuHUwkg ''51edo improv''] (2025-02-03)
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025-05-02)
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025-05-02)
* [https://www.youtube.com/shorts/Fymg9vYO6iQ ''Northernlight - Deltarune (microtonal cover in 51edo)''] (2025)
* [https://www.youtube.com/shorts/Fymg9vYO6iQ ''Northernlight - Deltarune (microtonal cover in 51edo)''] (2025)
* [https://www.youtube.com/shorts/SJW-JTHyeIA ''51edo prelude''] (2026)
* [https://www.youtube.com/watch?v=k3NOBYbiqpo ''51edo improv''] (2026-04-22)


; [[Frédéric Gagné]]
; [[Frédéric Gagné]]

Latest revision as of 03:35, 28 May 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 edosteps, though that comma itself is mapped to a different interval.

Odd harmonics

Approximation of odd harmonics in 51edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Error Absolute (¢) +3.93 -9.84 -4.12 +7.85 -10.14 +6.53 -5.92 -10.84 +8.37 -0.19 +7.02 +3.84 -11.75 +5.72
Relative (%) +16.7 -41.8 -17.5 +33.4 -43.1 +27.8 -25.1 -46.1 +35.6 -0.8 +29.8 +16.3 -49.9 +24.3
Steps
(reduced) 		81
(30) 		118
(16) 		143
(41) 		162
(9) 		176
(23) 		189
(36) 		199
(46) 		208
(4) 		217
(13) 		224
(20) 		231
(27) 		237
(33) 		242
(38) 		248
(44)
Approximation of odd harmonics in 51edo (continued)
Harmonic 31 33 35 37 39 41 43 45 47 49 51 53 55 57
Error Absolute (¢) +7.91 -6.21 +9.57 +7.48 +10.46 -5.53 +6.13 -1.99 -6.68 -8.24 -6.91 -2.92 +3.54 -11.23
Relative (%) +33.6 -26.4 +40.7 +31.8 +44.4 -23.5 +26.0 -8.5 -28.4 -35.0 -29.4 -12.4 +15.1 -47.7
Steps
(reduced) 		253
(49) 		257
(2) 		262
(7) 		266
(11) 		270
(15) 		273
(18) 		277
(22) 		280
(25) 		283
(28) 		286
(31) 		289
(34) 		292
(37) 		295
(40) 		297
(42)

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* Ups and downs notation
2.3.7.11/5.13
subgroup 		Ratios of 5 and 11
tending flat (51 val) 		Ratios of 5 and 11
tending sharp (51ce val)
0 0.0 1/1 Perfect 1sn P1 D
1 23.5 64/63, 49/48 40/39 81/80 Up 1sn ^1 ^D
2 47.1 28/27 33/32, 25/24, 81/80 36/35, 40/39 Downminor 2nd vm2 vEb
3 70.6 27/26 36/35 21/20, 33/32 Minor 2nd m2 Eb
4 94.1 21/20 16/15, 25/24 Upminor 2nd ^m2 ^Eb
5 117.6 14/13 15/14, 16/15 Downmid 2nd v~2 ^^Eb
6 141.2 13/12 12/11, 15/14 Mid 2nd ~2 vvvE, ^^^Eb
7 164.7 11/10 10/9, 12/11 Upmid 2nd ^~2 vvE
8 188.2 10/9 Downmajor 2nd vM2 vE
9 211.8 9/8 Major 2nd M2 E
10 235.3 8/7 15/13 Upmajor 2nd ^M2 ^E
11 258.8 7/6 15/13 Downminor 3rd vm3 vF
12 282.4 32/27 13/11 Minor 3rd m3 F
13 305.9 13/11 6/5 Upminor 3rd ^m3 ^F
14 329.4 40/33, 63/52 6/5, 11/9 Downmid 3rd v~3 ^^F
15 352.9 16/13, 39/32 11/9, 27/22 Mid 3rd ~3 ^^^F, vvvF#
16 376.5 26/21 5/4, 27/22 Upmid 3rd ^~3 vvF#
17 400.0 5/4, 14/11 Downmajor 3rd vM3 vF#
18 423.5 81/64 14/11 Major 3rd M3 F#
19 447.1 9/7 13/10 Upmajor 3rd ^M3 ^F#
20 470.6 21/16 13/10 Down 4th v4 vG
21 494.1 4/3 Perfect 4th P4 G
22 517.6 27/20 Up 4th ^4 ^G
23 541.2 15/11 11/8, 27/20 Downdim 5th vd5 vAb
24 564.7 18/13 7/5, 11/8 Dim 5th d5 Ab
25 588.2 39/28 7/5 Updim 5th ^d5 ^Ab
26 611.8 56/39 10/7 Downaug 4th vA4 vG#
27 635.3 13/9 10/7, 16/11 Aug 4th A4 G#
28 658.8 22/15 16/11, 40/27 Upaug 4th ^A4 ^G#
29 682.4 40/27 Down 5th v5 vA
30 705.9 3/2 Perfect 5th P5 A
31 729.4 32/21 20/13 Up 5th ^5 ^A
32 752.9 14/9 20/13 Downminor 6th vm6 vBb
33 776.5 128/81 11/7 Minor 6th m6 Bb
34 800.0 8/5, 11/7 Upminor 6th ^m6 ^Bb
35 823.5 21/13 8/5, 44/27 Downmid 6th v~6 ^^Bb
36 847.1 13/8, 64/39 18/11, 44/27 Mid 6th ~6 vvvB, ^^^Bb
37 870.6 33/20, 104/63 5/3, 18/11 Upmid 6th ^~6 vvB
38 894.1 22/13 5/3 Downmajor 6th vM6 vB
39 917.6 27/16 22/13 Major 6th M6 B
40 941.2 12/7 26/15 Upmajor 6th ^M6 ^B
41 964.7 7/4 26/15 Downminor 7th vm7 vC
42 988.2 16/9 Minor 7th m7 C
43 1011.8 9/5 Upminor 7th ^m7 ^C
44 1035.3 20/11 9/5, 11/6 Downmid 7th v~7 ^^C
45 1058.8 24/13 11/6, 28/15 Mid 7th ~7 ^^^C, vvvC#
46 1082.4 13/7 15/8, 28/15 Upmid 7th ^~7 vvC#
47 1105.9 40/21 15/8, 48/25 Downmajor 7th vM7 vC#
48 1129.4 52/27 35/18 40/21, 64/33 Major 7th M7 C#
49 1152.9 27/14 64/33, 48/25, 160/81 35/18, 39/20 Upmajor 7th ^M7 ^C#
50 1176.5 63/32, 96/49 39/20 160/81 Down 8ve v8 vD
51 1200.0 2/1 Perfect 8ve P8 D

* inconsistent intervals in italic.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation for 51edo uses sharps and flats combined with quartertone accidentals and arrows:

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 quartertone accidentals:

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

Kite's ups and downs notation

51edo can also be notated with Kite's 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  
    
    
    

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)
1 7\51 164.7 11/10 Porky (51)
1 10\51 235.3 8/7 Rodan (51cf, out of tune) / aerodino (51ce)
1 19\51 447.1 13/10 Supersensi (51cde)
1 22\51 517.6 27/20 Gravity (51ce) / abergravity (51ce)
1 23\51 541.2 15/11 Necromanteion (51ce)
Oracle (51)
Cypress (51cde…)
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 (7-limit, 51cd)
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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