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=<span style="color: #ffa610; font-family: 'Times New Roman',Times,serif; font-size: 122%;">'''61 tone equal temperament'''</span>=
{{Infobox ET}}
''61-EDO'' refers to the equal division of [[2/1|2/1]] ratio into 61 equal parts, of 19.6721 [[cent|cent]]s each. It is the 18th [[prime_numbers|prime]] EDO, after of [[59edo|59edo]] and before of [[67edo|67edo]]. It provides the optimal patent val for the 24&amp;37 temperament in the 7-, 11- and 13-limit.
{{ED intro}}


=Poem=
== Theory ==
These 61 equal divisions of the octave,
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.


though rare are assuredly a ROCK-tave (har har),
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]].


while the 3rd and 5th harmonics are about six cents sharp,
=== Odd harmonics ===
{{Harmonics in equal|61}}


(and the flattish 15th poised differently on the harp),
=== 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.


the 7th and 11th err by less, around three,
== Intervals ==
{{Interval table}}


and thus mayhap, a good orgone tuning found to be;
== Notation ==
=== Ups and downs notation ===
61edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:


slightly sharp as well, is the 13th harmonic's place,
{{Sharpness-sharp8}}


but the 9th and 17th lack near so much grace,
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].


interestingly the 19th is good but a couple cents flat,
==== 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>


and the 21st and 23rd are but a cent or two sharp!
==== 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>


=='''61-EDO Intervals'''==
==== 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>


{| class="wikitable"
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.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| | '''Degrees'''
! rowspan="2" |[[Subgroup]]
| | '''Cent Value'''
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 0
![[TE error|Absolute]] (¢)
| | 0
![[TE simple badness|Relative]] (%)
|-
|-
| | 1
| 2.3
| | 19.6721
|{{Monzo| 97 -61 }}
|{{Mapping| 61 97 }}
| −1.97
| 1.97
| 10.0
|-
|-
| | 2
| 2.3.5
| | 39.3443
| 20000/19683, 262144/253125
|-
|{{Mapping| 61 97 142 }}
| | 3
| −2.33
| | 59.0164
| 1.69
|-
| 8.59
| | 4
|- style="border-top: double;"
| | 78.6885
| 2.3.5.7
|-
| 64/63, 2430/2401, 3125/3087
| | 5
|{{mapping| 61 97 142 172 }} (61d)
| | 98.3607
| −3.06
|-
| 1.93
| | 6
| 9.84
| | 118.0328
|- style="border-top: double;"
|-
| 2.3.5.7
| | 7
| 126/125, 1029/1024, 2240/2187
| | 137.7049
|{{Mapping| 61 97 142 171 }} (61)
|-
| −1.32
| | 8
| 2.29
| | 157.377
| 11.7
|-
|}
| | 9
 
| | 177.0492
=== Rank-2 temperaments ===
|-
{| class="wikitable center-all left-5"
| | 10
|+ style="font-size: 105%;" |Table of rank-2 temperaments by generator
| | 196.7213
|-
| | 11
| | 216.3934
|-
| | 12
| | 236.0656
|-
| | 13
| | 255.7377
|-
| | 14
| | 275.4098
|-
| | 15
| | 295.082
|-
|-
| | 16
! Periods<br>per 8ve
| | 314.7541
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| | 17
| 1
| | 334.4262
| 2\61
| 39.3
| 40/39
|[[Hemivalentine]] (61)
|-
|-
| | 18
| 1
| | 354.0984
| 3\61
| 59.0
| 28/27
|[[Dodecacot]] (61de…)
|-
|-
| | 19
| 1
| | 373.7705
| 4\61
| 78.7
| 22/21
|[[Valentine]] (61)
|-
|-
| | 20
| 1
| | 393.4426
| 5\61
| 98.4
| 16/15
|[[Passion]] (61de…) / [[passionate]] (61)
|-
|-
| | 21
| 1
| | 413.1148
| 7\61
| 137.7
| 13/12
|[[Quartemka]] (61)
|-
|-
| | 22
| 1
| | 432.7869
| 9\61
| 177.0
| 10/9
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
|-
| | 23
| 1
| | 452.459
| 11\61
| 236.1
| 8/7
|[[Slendric]] (61)
|-
|-
| | 24
| 1
| | 472.1311
| 16\61
| 314.8
| 6/5
|[[Parakleismic]] (61d)
|-
|-
| | 25
| 1
| | 491.8033
| 23\61
| 452.5
| 13/10
|[[Maja]] (61d)
|-
|-
| | 26
| 1
| | 511.4754
| 25\61
| 491.8
| 4/3
|[[Quasisuper]] (61d)
|-
|-
| | 27
| 1
| | 531.1475
| 28\61
|-
| 550.8
| | 28
| 11/8
| | 550.8197
|[[Freivald]] (61)
|-
| | 29
| | 570.4918
|-
| | 30
| | 590.1639
|-
| | 31
| | 609.8361
|-
| | 32
| | 629.5082
|-
| | 33
| | 649.1803
|-
| | 34
| | 668.8525
|-
| | 35
| | 688.5246
|-
| | 36
| | 708.1967
|-
| | 37
| | 727.8689
|-
| | 38
| | 747.541
|-
| | 39
| | 767.2131
|-
| | 40
| | 786.8852
|-
| | 41
| | 806.5574
|-
| | 42
| | 826.2295
|-
| | 43
| | 845.9016
|-
| | 44
| | 865.5738
|-
| | 45
| | 885.2459
|-
| | 46
| | 904.918
|-
| | 47
| | 924.5902
|-
| | 48
| | 944.2623
|-
| | 49
| | 963.9344
|-
| | 50
| | 983.6066
|-
| | 51
| | 1003.2787
|-
| | 52
| | 1022.9508
|-
| | 53
| | 1042.623
|-
| | 54
| | 1062.2951
|-
| | 55
| | 1081.9672
|-
| | 56
| | 1101.6393
|-
| | 57
| | 1121.3115
|-
| | 58
| | 1140.9836
|-
| | 59
| | 1160.6557
|-
| | 60
| | 1180.3279
|}
|}
<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 ==


[[Category:Edo]]
=== Introductory poem ===
[[Category:Prime EDO]]
[[Peter Kosmorsky]] wrote a poem on 61edo; see [[User:Spt3125/61edo poem|the 61edo poem]].


[[Category:todo:add sound examples]]
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
Retrieved from "https://en.xen.wiki/w/61edo"