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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]] (15 & 46), and is the [[optimal patent val]] for [[freivald]] (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.  
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]] (15 & 46), and is the [[optimal patent val]] for [[freivald]] (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.  


=== Introductory poem ===
=== Introductory poem ===
Line 36: Line 36:


== Notation ==
== Notation ==
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].


===Sagittal notation===
==== Evo flavor ====
This notation uses the same sagittal sequence as [[54edo#Sagittal notation|54-EDO]].
====Evo flavor====
 
<imagemap>
<imagemap>
File:61-EDO_Evo_Sagittal.svg
File:61-EDO_Evo_Sagittal.svg
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</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:61-EDO_Revo_Sagittal.svg
File:61-EDO_Revo_Sagittal.svg
Line 67: Line 65:
</imagemap>
</imagemap>


====Evo-SZ flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:61-EDO_Evo-SZ_Sagittal.svg
File:61-EDO_Evo-SZ_Sagittal.svg
Line 81: Line 78:
</imagemap>
</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.
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
|}


[[Category:Todo:add rank 2 temperaments table]]
=== 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
| 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
| [[Rodan]] (61, 7-limit, out of tune)
|-
| 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

Revision as of 13:00, 23 January 2025

← 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

Template:EDO intro

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 ⟨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. Peter Kosmorsky has an interesting poem about its tuning profile, as follows.

Introductory poem

These 61 equal divisions of the octave,

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!

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)

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

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 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 Rodan (61, 7-limit, out of tune)
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

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