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the poem was not only redundant and unserious but also slightly misleading (in the sense that it doesn't actually support orgone)
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== Theory ==
== Theory ==
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]]. [[Peter Kosmorsky]] has an interesting poem about its tuning profile, as follows.
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.  
 
=== Introductory poem ===
[[Peter Kosmorsky]] wrote a poem on 61edo; see [[User:Spt3125/61edo poem|the 61edo poem]].


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|61}}
{{Harmonics in equal|61}}
=== 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]].


=== Subsets and supersets ===
=== Subsets and supersets ===
Line 70: Line 70:
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |[[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
![[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
![[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{Monzo| 97 -61 }}
|{{Monzo| 97 -61 }}
| {{Mapping| 61 97 }}
|{{Mapping| 61 97 }}
| −1.97
| −1.97
| 1.97
| 1.97
Line 88: Line 88:
| 2.3.5
| 2.3.5
| 20000/19683, 262144/253125
| 20000/19683, 262144/253125
| {{Mapping| 61 97 142 }}
|{{Mapping| 61 97 142 }}
| −2.33
| −2.33
| 1.69
| 1.69
Line 95: Line 95:
| 2.3.5.7
| 2.3.5.7
| 64/63, 2430/2401, 3125/3087
| 64/63, 2430/2401, 3125/3087
| {{mapping| 61 97 142 172 }} (61d)
|{{mapping| 61 97 142 172 }} (61d)
| −3.06
| −3.06
| 1.93
| 1.93
Line 102: Line 102:
| 2.3.5.7
| 2.3.5.7
| 126/125, 1029/1024, 2240/2187
| 126/125, 1029/1024, 2240/2187
| {{Mapping| 61 97 142 171 }} (61)
|{{Mapping| 61 97 142 171 }} (61)
| −1.32
| −1.32
| 2.29
| 2.29
Line 110: Line 110:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" |Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
Line 122: Line 122:
| 39.3
| 39.3
| 40/39
| 40/39
| [[Hemivalentine]] (61)
|[[Hemivalentine]] (61)
|-
|-
| 1
| 1
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| 59.0
| 59.0
| 28/27
| 28/27
| [[Dodecacot]] (61de…)
|[[Dodecacot]] (61de…)
|-
|-
| 1
| 1
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| 78.7
| 78.7
| 22/21
| 22/21
| [[Valentine]] (61)
|[[Valentine]] (61)
|-
|-
| 1
| 1
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| 98.4
| 98.4
| 16/15
| 16/15
| [[Passion]] (61de…) / [[passionate]] (61)
|[[Passion]] (61de…) / [[passionate]] (61)
|-
|-
| 1
| 1
Line 146: Line 146:
| 137.7
| 137.7
| 13/12
| 13/12
| [[Quartemka]] (61)
|[[Quartemka]] (61)
|-
|-
| 1
| 1
Line 152: Line 152:
| 177.0
| 177.0
| 10/9
| 10/9
| [[Modus]] (61de) / [[wollemia]] (61e)
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
|-
| 1
| 1
Line 158: Line 158:
| 236.1
| 236.1
| 8/7
| 8/7
| [[Slendric]] (61)
|[[Slendric]] (61)
|-
|-
| 1
| 1
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| 314.8
| 314.8
| 6/5
| 6/5
| [[Parakleismic]] (61d)
|[[Parakleismic]] (61d)
|-
|-
| 1
| 1
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| 452.5
| 452.5
| 13/10
| 13/10
| [[Maja]] (61d)
|[[Maja]] (61d)
|-
|-
| 1
| 1
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| 491.8
| 491.8
| 4/3
| 4/3
| [[Quasisuper]] (61d)
|[[Quasisuper]] (61d)
|-
|-
| 1
| 1
Line 182: Line 182:
| 550.8
| 550.8
| 11/8
| 11/8
| [[Freivald]] (61)
|[[Freivald]] (61)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


== Instruments ==
== Instruments ==
A [[Lumatone mapping for 61edo]] has now been demonstrated (see the Valentine mapping for full gamut coverage).
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 ==
== Music ==
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
* [https://www.youtube.com/shorts/S9bJnllI7CI ''61edo prelude''] (2025)

Latest revision as of 09:45, 30 May 2026

← 60edo 61edo 62edo →
Prime factorization 61 (prime)
Step size 19.6721 ¢ 
Fifth 36\61 (708.197 ¢)
Semitones (A1:m2) 8:3 (157.4 ¢ : 59.02 ¢)
Consistency limit 5
Distinct consistency limit 5

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.

Theory

61edo is only consistent to the 5-odd-limit. Its 3rd and 5th harmonics 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.

Odd harmonics

Approximation of odd harmonics in 61edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +6.24 +7.13 -4.89 -7.19 -0.50 +5.37 -6.30 -6.59 -2.43 +1.35 +1.23
Relative (%) +31.7 +36.2 -24.9 -36.5 -2.5 +27.3 -32.0 -33.5 -12.4 +6.9 +6.3
Steps
(reduced) 		97
(36) 		142
(20) 		171
(49) 		193
(10) 		211
(28) 		226
(43) 		238
(55) 		249
(5) 		259
(15) 		268
(24) 		276
(32)

As a tuning of other temperaments

There are three reasonable vals in the 13-limit: the patent val (⟨61 97 142 171 211 226]), 61d (⟨61 97 142 172 211 226]), and 61de (⟨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 ⟨61 97 142 171] supports valentine (15 & 46), and is the optimal patent val for freivald (24 & 37) in the 7-, 11- and 13-limit. The 61d 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.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 19.7 ^D, vvE♭
2 39.3 ^^D, vE♭
3 59 29/28, 32/31 ^3D, E♭
4 78.7 22/21, 23/22 ^4D, ^E♭
5 98.4 35/33 v3D♯, ^^E♭
6 118 31/29 vvD♯, ^3E♭
7 137.7 13/12 vD♯, v4E
8 157.4 23/21, 34/31, 35/32 D♯, v3E
9 177 21/19, 31/28 ^D♯, vvE
10 196.7 19/17 ^^D♯, vE
11 216.4 26/23 E
12 236.1 ^E, vvF
13 255.7 22/19 ^^E, vF
14 275.4 34/29 F
15 295.1 19/16 ^F, vvG♭
16 314.8 6/5 ^^F, vG♭
17 334.4 17/14, 23/19 ^3F, G♭
18 354.1 ^4F, ^G♭
19 373.8 26/21 v3F♯, ^^G♭
20 393.4 vvF♯, ^3G♭
21 413.1 14/11, 33/26 vF♯, v4G
22 432.8 F♯, v3G
23 452.5 13/10 ^F♯, vvG
24 472.1 21/16 ^^F♯, vG
25 491.8 G
26 511.5 35/26 ^G, vvA♭
27 531.1 19/14 ^^G, vA♭
28 550.8 11/8 ^3G, A♭
29 570.5 25/18, 32/23 ^4G, ^A♭
30 590.2 31/22 v3G♯, ^^A♭
31 609.8 vvG♯, ^3A♭
32 629.5 23/16 vG♯, v4A
33 649.2 16/11, 35/24 G♯, v3A
34 668.9 28/19 ^G♯, vvA
35 688.5 ^^G♯, vA
36 708.2 A
37 727.9 29/19, 32/21, 35/23 ^A, vvB♭
38 747.5 20/13 ^^A, vB♭
39 767.2 ^3A, B♭
40 786.9 11/7 ^4A, ^B♭
41 806.6 35/22 v3A♯, ^^B♭
42 826.2 21/13 vvA♯, ^3B♭
43 845.9 31/19 vA♯, v4B
44 865.6 28/17, 33/20 A♯, v3B
45 885.2 5/3 ^A♯, vvB
46 904.9 32/19 ^^A♯, vB
47 924.6 29/17 B
48 944.3 19/11 ^B, vvC
49 963.9 ^^B, vC
50 983.6 23/13 C
51 1003.3 34/19 ^C, vvD♭
52 1023 ^^C, vD♭
53 1042.6 31/17 ^3C, D♭
54 1062.3 24/13 ^4C, ^D♭
55 1082 v3C♯, ^^D♭
56 1101.6 vvC♯, ^3D♭
57 1121.3 21/11 vC♯, v4D
58 1141 31/16 C♯, v3D
59 1160.7 ^C♯, vvD
60 1180.3 ^^C♯, vD
61 1200 2/1 D

Notation

Ups and downs notation

61edo can be notated using ups and downs notation using Helmholtz–Ellis accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sharp symbol  
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as 54edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8033/3227/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8033/3227/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8033/3227/26

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.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [97 -61⟩ [⟨61 97]] −1.97 1.97 10.0
2.3.5 20000/19683, 262144/253125 [⟨61 97 142]] −2.33 1.69 8.59
2.3.5.7 64/63, 2430/2401, 3125/3087 [⟨61 97 142 172]] (61d) −3.06 1.93 9.84
2.3.5.7 126/125, 1029/1024, 2240/2187 [⟨61 97 142 171]] (61) −1.32 2.29 11.7

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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)

* 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 the 61edo poem.

Music

Bryan Deister
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