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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*
!Additional ratios of<br>5 and/or 11 tending<br>flat (51 patent val)
! rowspan="2" colspan="3" | [[Ups and downs notation]]
!Additional ratios of<br>5 and/or 11 tending<br>sharp (51ce val)
|-
! 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
Line 41: Line 44:
| 23.5
| 23.5
| [[64/63]], ''[[49/48]]''
| [[64/63]], ''[[49/48]]''
|
| ''40/39''
|[[81/80]]
| [[81/80]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 50: Line 53:
| 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 59: Line 62:
| 70.6
| 70.6
| [[27/26]]
| [[27/26]]
|
| ''36/35''
|
| ''21/20'', ''33/32''
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 68: Line 71:
| 94.1
| 94.1
|  
|  
|[[135/128]]
| [[21/20]]
|
| ''16/15'', ''25/24''
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 77: Line 80:
| 117.6
| 117.6
| [[14/13]]
| [[14/13]]
|[[16/15]]
| [[15/14]], [[16/15]]
|
|  
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 87: Line 90:
| [[13/12]]
| [[13/12]]
|
|
|[[12/11]]
| [[12/11]], ''15/14''
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 94: Line 97:
| 7
| 7
| 164.7
| 164.7
|
| [[11/10]]
|[[11/10]]
| ''10/9'', ''12/11''
|11/10
|
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 105: Line 108:
|  
|  
|
|
|
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 122: Line 125:
| 235.3
| 235.3
| [[8/7]]
| [[8/7]]
|
| ''15/13''
|
|  
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 131: Line 134:
| 258.8
| 258.8
| [[7/6]]
| [[7/6]]
|
|  
|
| [[15/13]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 141: Line 144:
| ''[[32/27]]''
| ''[[32/27]]''
|
|
|[[13/11]]
| [[13/11]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 149: Line 152:
| 305.9
| 305.9
|  
|  
|
| ''13/11''
|[[6/5]]
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 157: Line 160:
| 14
| 14
| 329.4
| 329.4
|
| [[40/33]], [[63/52]]
|[[40/33]]
| ''6/5'', ''11/9''
|40/33
|
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 167: Line 170:
| 352.9
| 352.9
| [[16/13]], [[39/32]]
| [[16/13]], [[39/32]]
|
|  
|
| [[11/9]], [[27/22]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 176: Line 179:
| 376.5
| 376.5
| [[26/21]]
| [[26/21]]
|[[5/4]]
| [[5/4]], ''27/22''
|
|  
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 185: Line 188:
| 400.0
| 400.0
|  
|  
|
|  
|''5/4''
| ''5/4'', ''14/11''
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 194: Line 197:
| 423.5
| 423.5
| ''[[81/64]]''
| ''[[81/64]]''
|
| [[14/11]]
|
|
| Major 3rd
| Major 3rd
Line 204: Line 207:
| ''[[9/7]]''
| ''[[9/7]]''
|
|
|
| [[13/10]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 212: Line 215:
| 470.6
| 470.6
| [[21/16]]
| [[21/16]]
|
| ''13/10''
|
|
| Down 4th
| Down 4th
Line 231: Line 234:
|  
|  
|
|
|
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 238: Line 241:
| 23
| 23
| 541.2
| 541.2
|
| [[15/11]]
|[[11/8]], [[15/11]]
| [[11/8]], ''27/20''
|15/11
|  
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 249: Line 252:
| [[18/13]]
| [[18/13]]
|
|
|''11/8''
| ''7/5'', ''11/8''
| Dim 5th
| Dim 5th
| d5
| d5
Line 257: Line 260:
| 588.2
| 588.2
| [[39/28]]
| [[39/28]]
|
| [[7/5]]
|
|
| Updim 5th
| Updim 5th
Line 266: Line 269:
| 611.8
| 611.8
| [[56/39]]
| [[56/39]]
|
| [[10/7]]
|
|
| Downaug 4th
| Downaug 4th
Line 276: Line 279:
| [[13/9]]
| [[13/9]]
|
|
|''16/11''
| ''10/7'', ''16/11''
| Aug 4th
| Aug 4th
| A4
| A4
Line 283: Line 286:
| 28
| 28
| 658.8
| 658.8
|
| [[22/15]]
|[[16/11]], [[22/15]]
| [[16/11]], ''40/27''
|22/15
|  
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 294: Line 297:
|  
|  
|
|
|
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
Line 311: Line 314:
| 729.4
| 729.4
| [[32/21]]
| [[32/21]]
|
| ''20/13''
|
|
| Up 5th
| Up 5th
Line 321: Line 324:
| ''[[14/9]]''
| ''[[14/9]]''
|
|
|
| [[20/13]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 329: Line 332:
| 776.5
| 776.5
| ''[[128/81]]''
| ''[[128/81]]''
|
| [[11/7]]
|
|
| Minor 6th
| Minor 6th
Line 339: Line 342:
|  
|  
|
|
|''8/5''
| ''8/5'', ''11/7''
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 347: Line 350:
| 823.5
| 823.5
| [[21/13]]
| [[21/13]]
|[[8/5]]
| [[8/5]], ''44/27''
|
|  
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 356: Line 359:
| 847.1
| 847.1
| [[13/8]], [[64/39]]
| [[13/8]], [[64/39]]
|
|  
|
| [[18/11]], [[44/27]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 364: Line 367:
| 37
| 37
| 870.6
| 870.6
|
| [[33/20]], [[104/63]]
|[[33/20]]
| ''5/3'', ''18/11''
|33/20
|
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 374: Line 377:
| 894.1
| 894.1
|  
|  
|
| ''22/13''
|[[5/3]]
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 384: Line 387:
| ''[[27/16]]''
| ''[[27/16]]''
|
|
|[[22/13]]
| [[22/13]]
| Major 6th
| Major 6th
| M6
| M6
Line 392: Line 395:
| 941.2
| 941.2
| [[12/7]]
| [[12/7]]
|
|  
|
| [[26/15]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 401: Line 404:
| 964.7
| 964.7
| [[7/4]]
| [[7/4]]
|
| ''26/15''
|
|  
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 420: Line 423:
|  
|  
|
|
|
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 427: Line 430:
| 44
| 44
| 1035.3
| 1035.3
|
| [[20/11]]
|[[20/11]]
|  
|20/11
| ''9/5'', ''11/6''
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 438: Line 441:
| [[24/13]]
| [[24/13]]
|
|
|[[11/6]]
| [[11/6]], ''28/15''
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 446: Line 449:
| 1082.4
| 1082.4
| [[13/7]]
| [[13/7]]
|[[15/8]]
| [[15/8]], [[28/15]]
|
|
| Upmid 7th
| Upmid 7th
Line 455: Line 458:
| 1105.9
| 1105.9
|  
|  
|[[256/135]]
| [[40/21]]
|
| ''15/8'', ''48/25''
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 464: Line 467:
| 1129.4
| 1129.4
| [[52/27]]
| [[52/27]]
|
| ''35/18''
|
| ''40/21'', ''64/33''
| Major 7th
| Major 7th
| M7
| M7
Line 473: Line 476:
| 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 482: Line 485:
| 1176.5
| 1176.5
| [[63/32]], ''[[96/49]]''
| [[63/32]], ''[[96/49]]''
|
| ''39/20''
|[[160/81]]
| [[160/81]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 497: Line 500:
| 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 632: 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 644: 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 662: Line 675:
| rowspan="2" | 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: center;" | 4/3<br>(21/20)
| style="text-align: left;" | [[Fog]] (51)
| 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 680: 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]]
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