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{{Infobox ET}}
{{Infobox ET}}
'''61edo''' refers to the equal division of [[2/1]] ratio into 61 equal parts, of 19.6721 [[cent]]s each.
{{ED intro}}


61edo provides the [[optimal patent val]] for the [[freivald]] (24&37) temperament in the 7-, 11- and 13-limit.
== Theory ==
61edo is only [[consistent]] to the [[5-odd-limit]]. Its [[3/1|3rd]] and [[5/1|5th]] [[harmonic]]s are sharp of just by more than 6 cents, and the [[7/1|7th]] and [[11/1|11th]], though they err by less, are on the flat side. This limits its harmonic inventory. However, it does possess reasonably good approximations of [[21/16]] and [[23/16]], only a bit more than one cent off in each case.  


61edo is the 18th [[prime edo]], after of [[59edo]] and before of [[67edo]].  
As an equal temperament, 61et is characterized by [[tempering out]] 20000/19683 ([[tetracot comma]]) and 262144/253125 ([[passion comma]]) in the 5-limit. In the 7-limit, the [[patent val]] {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] ({{nowrap| 15 & 46 }}), and is the [[optimal patent val]] for [[freivald]] ({{nowrap| 24 & 37 }}) in the 7-, 11- and 13-limit. The 61d [[val]] {{val| 61 97 142 '''172''' }} is a great tuning for [[modus]] and [[quasisuper]], and is a simple but out-of-tune edo tuning for [[parakleismic]].  


== Poem ==
=== Odd harmonics ===
These 61 equal divisions of the octave,
{{Harmonics in equal|61}}


though rare are assuredly a ROCK-tave (har har),
=== Subsets and supersets ===
61edo is the 18th [[prime edo]], after [[59edo]] and before [[67edo]]. [[183edo]], which triples it, corrects its approximation to many of the lower harmonics.


while the 3rd and 5th harmonics are about six cents sharp,
== Intervals ==
{{Interval table}}


(and the flattish 15th poised differently on the harp),
== Notation ==
=== Ups and downs notation ===
61edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:


the 7th and 11th err by less, around three,
{{Sharpness-sharp8}}


and thus mayhap, a good orgone tuning found to be;
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].


slightly sharp as well, is the 13th harmonic's place,
==== Evo flavor ====
<imagemap>
File:61-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 704 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:61-EDO_Evo_Sagittal.svg]]
</imagemap>


but the 9th and 17th lack near so much grace,
==== Revo flavor ====
<imagemap>
File:61-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:61-EDO_Revo_Sagittal.svg]]
</imagemap>


interestingly the 19th is good but a couple cents flat,
==== Evo-SZ flavor ====
<imagemap>
File:61-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 696 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:61-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


and the 21st and 23rd are but a cent or two sharp!
In the diagrams above, 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.


{{Harmonics in equal|61|columns=11}}
== Regular temperament properties ==
 
{| class="wikitable center-4 center-5 center-6"
== Intervals ==
 
{| class="wikitable center-1 right-2"
|-
|-
! #
! rowspan="2" |[[Subgroup]]
! Cents
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| 0
![[TE error|Absolute]] (¢)
| 0.000
![[TE simple badness|Relative]] (%)
|-
|-
| 1
| 2.3
| 19.672
|{{Monzo| 97 -61 }}
|{{Mapping| 61 97 }}
| −1.97
| 1.97
| 10.0
|-
|-
| 2
| 2.3.5
| 39.344
| 20000/19683, 262144/253125
|{{Mapping| 61 97 142 }}
| −2.33
| 1.69
| 8.59
|- style="border-top: double;"
| 2.3.5.7
| 64/63, 2430/2401, 3125/3087
|{{mapping| 61 97 142 172 }} (61d)
| −3.06
| 1.93
| 9.84
|- style="border-top: double;"
| 2.3.5.7
| 126/125, 1029/1024, 2240/2187
|{{Mapping| 61 97 142 171 }} (61)
| −1.32
| 2.29
| 11.7
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" |Table of rank-2 temperaments by generator
|-
|-
| 3
! Periods<br>per 8ve
| 59.016
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 4
| 1
| 78.689
| 2\61
| 39.3
| 40/39
|[[Hemivalentine]] (61)
|-
|-
| 5
| 1
| 98.361
| 3\61
| 59.0
| 28/27
|[[Dodecacot]] (61de…)
|-
|-
| 6
| 1
| 118.033
| 4\61
| 78.7
| 22/21
|[[Valentine]] (61)
|-
|-
| 7
| 1
| 137.705
| 5\61
| 98.4
| 16/15
|[[Passion]] (61de…) / [[passionate]] (61)
|-
|-
| 8
| 1
| 157.377
| 7\61
| 137.7
| 13/12
|[[Quartemka]] (61)
|-
|-
| 9
| 1
| 177.049
| 9\61
| 177.0
| 10/9
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
|-
| 10
| 1
| 196.721
| 11\61
| 236.1
| 8/7
|[[Slendric]] (61)
|-
|-
| 11
| 1
| 216.393
| 16\61
| 314.8
| 6/5
|[[Parakleismic]] (61d)
|-
|-
| 12
| 1
| 236.066
| 23\61
| 452.5
| 13/10
|[[Maja]] (61d)
|-
|-
| 13
| 1
| 255.738
| 25\61
| 491.8
| 4/3
|[[Quasisuper]] (61d)
|-
|-
| 14
| 1
| 275.410
| 28\61
|-
| 550.8
| 15
| 11/8
| 295.082
|[[Freivald]] (61)
|-
| 16
| 314.754
|-
| 17
| 334.426
|-
| 18
| 354.098
|-
| 19
| 373.770
|-
| 20
| 393.443
|-
| 21
| 413.115
|-
| 22
| 432.787
|-
| 23
| 452.459
|-
| 24
| 472.131
|-
| 25
| 491.803
|-
| 26
| 511.475
|-
| 27
| 531.148
|-
| 28
| 550.820
|-
| 29
| 570.492
|-
| 30
| 590.164
|-
| 31
| 609.836
|-
| 32
| 629.508
|-
| 33
| 649.180
|-
| 34
| 668.852
|-
| 35
| 688.525
|-
| 36
| 708.197
|-
| 37
| 727.869
|-
| 38
| 747.541
|-
| 39
| 767.213
|-
| 40
| 786.885
|-
| 41
| 806.557
|-
| 42
| 826.230
|-
| 43
| 845.902
|-
| 44
| 865.574
|-
| 45
| 885.246
|-
| 46
| 904.918
|-
| 47
| 924.590
|-
| 48
| 944.262
|-
| 49
| 963.934
|-
| 50
| 983.607
|-
| 51
| 1003.279
|-
| 52
| 1022.951
|-
| 53
| 1042.623
|-
| 54
| 1062.295
|-
| 55
| 1081.967
|-
| 56
| 1101.639
|-
| 57
| 1121.311
|-
| 58
| 1140.984
|-
| 59
| 1160.656
|-
| 60
| 1180.328
|-
| 61
| 1200.000
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
== Instruments ==
A [[Lumatone mapping for 61edo]] has now been demonstrated (see the Valentine mapping for full gamut coverage).
== See also ==
=== Introductory poem ===
[[Peter Kosmorsky]] wrote a poem on 61edo; see [[User:Spt3125/61edo poem|the 61edo poem]].


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Music ==
[[Category:Prime EDO]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
Retrieved from "https://en.xen.wiki/w/61edo"