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61edo: Difference between revisions

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


== 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.
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 ===
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 ===
=== 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 ===
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== Notation ==
== Notation ==
=== Ups and downs notation ===
61edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
{{Sharpness-sharp8}}
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].
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{| 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
| 10.0
| 10.0
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| 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
| 8.59
| 8.59
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| 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
| 9.84
| 9.84
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| 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
| 11.7
| 11.7
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=== 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
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! Associated<br>ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
| 1
| 2\61
| 39.3
| 40/39
|[[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
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| 137.7
| 137.7
| 13/12
| 13/12
| [[Quartemka]] (61)
|[[Quartemka]] (61)
|-
|-
| 1
| 1
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| 177.0
| 177.0
| 10/9
| 10/9
| [[Modus]] (61de) / [[wollemia]] (61e)
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
|-
| 1
| 1
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| 236.1
| 236.1
| 8/7
| 8/7
| [[Rodan]] (61, 7-limit, out of tune)
|[[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
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| 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 ==
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"