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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|61}}
{{ED intro}}


== Theory ==
== Theory ==
61edo provides the [[optimal patent val]] for the [[freivald]] (24 & 37) temperament in the 7-, 11- and 13-limit.
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]].
=== Odd harmonics ===
{{Harmonics in equal|61}}


=== Odd harmonics ===
=== As a tuning of other temperaments ===
{{Harmonics in equal|61|columns=11}}
There are three reasonable [[val]]s in the [[13-limit]]: the [[patent val]] ({{val| 61 97 142 171 211 226 }}), 61d ({{val| 61 97 142 '''172''' 211 226 }}), and 61de ({{val| 61 97 142 '''172''' '''212''' 226 }}). In any case, it 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]]. The 61d val supports 11-limit [[quasisupra]], and the 61de val supports 11- and 13-limit [[modus]].
 
=== 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.


== Table of intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}


== Miscellany ==
== Notation ==
=== Mnemonic descriptive poem ===
=== Ups and downs notation ===
These 61 equal divisions of the octave,
61edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
 
{{Sharpness-sharp8}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].
 
==== 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>


though rare are assuredly a ROCK-tave (har har),
==== 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>


while the 3rd and 5th harmonics are about six cents sharp,
==== 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 flattish 15th poised differently on the harp),
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.


the 7th and 11th err by less, around three,
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
| 2.3
|{{Monzo| 97 -61 }}
|{{Mapping| 61 97 }}
| −1.97
| 1.97
| 10.0
|-
| 2.3.5
| 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
|}


and thus mayhap, a good orgone tuning found to be;
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" |Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 2\61
| 39.3
| 40/39
|[[Hemivalentine]] (61)
|-
| 1
| 3\61
| 59.0
| 28/27
|[[Dodecacot]] (61de…)
|-
| 1
| 4\61
| 78.7
| 22/21
|[[Valentine]] (61)
|-
| 1
| 5\61
| 98.4
| 16/15
|[[Passion]] (61de…) / [[passionate]] (61)
|-
| 1
| 7\61
| 137.7
| 13/12
|[[Quartemka]] (61)
|-
| 1
| 9\61
| 177.0
| 10/9
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
| 1
| 11\61
| 236.1
| 8/7
|[[Slendric]] (61)
|-
| 1
| 16\61
| 314.8
| 6/5
|[[Parakleismic]] (61d)
|-
| 1
| 23\61
| 452.5
| 13/10
|[[Maja]] (61d)
|-
| 1
| 25\61
| 491.8
| 4/3
|[[Quasisuper]] (61d)
|-
| 1
| 28\61
| 550.8
| 11/8
|[[Freivald]] (61)
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


slightly sharp as well, is the 13th harmonic's place,
== Instruments ==
A [[Lumatone mapping for 61edo]] has now been demonstrated (see the Valentine mapping for full gamut coverage).


but the 9th and 17th lack near so much grace,
== See also ==


interestingly the 19th is good but a couple cents flat,
=== Introductory poem ===
[[Peter Kosmorsky]] wrote a poem on 61edo; see [[User:Spt3125/61edo poem|the 61edo poem]].


and the 21st and 23rd are but a cent or two sharp!
== Music ==
=== Modern renderings ===
; [https://genius.com/artists/Sam-haft Sam Haft] and [https://genius.com/artists/Andrew-underberg Andrew Underberg]
* [https://www.youtube.com/watch?v=LqCwEc6tXGY&list=WL&index=262 ''Wilting''] via {{W|List_of_Roblox_games#Dandy's_World|Dandy's World}} (2024) — microtonal cover by [[Bryan Deister]] (2026)


—by [[Peter Kosmorsky]]
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
* [https://www.youtube.com/shorts/S9bJnllI7CI ''61edo prelude''] (2025)
Retrieved from "https://en.xen.wiki/w/61edo"