//61-EDO// refers to the equal division of [[xenharmonic/2_1|2/1]] ratio into 61 equal parts, of 19.6721 [[xenharmonic/cent|cent]]s each. It is the 18th [[prime numbers|prime]] EDO, after of [[59edo]] and before of [[67edo]].
=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),
while the 3rd and 5th harmonics are about six cents sharp,
(and the flattish 15th poised differently on the harp),
the 7th and 11th err by less, around three,
and thus mayhap, a good orgone tuning found to be;
slightly sharp as well, is the 13th harmonic's place,
but the 9th and 17th lack near so much grace,
interestingly the 19th is good but a couple cents flat,
and the 21st and 23rd are but a cent or two sharp!
==**61-EDO Intervals**==
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]].
<em>61-EDO</em> refers to the equal division of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2_1">2/1</a> ratio into 61 equal parts, of 19.6721 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s each. It is the 18th <a class="wiki_link" href="/prime%20numbers">prime</a> EDO, after of <a class="wiki_link" href="/59edo">59edo</a> and before of <a class="wiki_link" href="/67edo">67edo</a>.<br />
61edo is the 18th [[prime edo]], after [[59edo]] and before [[67edo]]. [[183edo]], which triples it, corrects its approximation to many of the lower harmonics.
== Intervals ==
{{Interval table}}
== 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 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]].
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
61 equal divisions of the octave (abbreviated 61edo or 61ed2), also called 61-tone equal temperament (61tet) or 61 equal temperament (61et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 61 equal parts of about 19.7 ¢ each. Each step represents a frequency ratio of 21/61, or the 61st root of 2.
61edo is only consistent to the 5-odd-limit. Its 3rd and 5thharmonics are sharp of just by more than 6 cents, and the 7th and 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.
This notation uses the same sagittal sequence as 54edo.
Evo flavor
Revo flavor
Evo-SZ flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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.
