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


61edo is the 18th [[prime edo]], after of [[59edo]] and before of [[67edo]].  
=== As a tuning of other temperaments ===
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]].


=== Odd harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|61|columns=11}}
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}}
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
 
[[Category:Prime EDO]]
== Notation ==
=== Ups and downs notation ===
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>
 
==== 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>
 
==== 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>
 
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"
|-
! 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
|}
 
=== 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
 
== 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]].
 
== Music ==
; [[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"