61edo: Difference between revisions
the poem was not only redundant and unserious but also slightly misleading (in the sense that it doesn't actually support orgone) |
+intro to the tuning profile, as a compensation for the removal of the poem |
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== 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. | |||
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]]. | |||
[[ | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 70: | Line 69: | ||
{| 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 87: | ||
| 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 94: | ||
| 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 101: | ||
| 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 109: | ||
=== 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 121: | ||
| 39.3 | | 39.3 | ||
| 40/39 | | 40/39 | ||
| [[Hemivalentine]] (61) | |[[Hemivalentine]] (61) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 128: | Line 127: | ||
| 59.0 | | 59.0 | ||
| 28/27 | | 28/27 | ||
| [[Dodecacot]] (61de…) | |[[Dodecacot]] (61de…) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 134: | Line 133: | ||
| 78.7 | | 78.7 | ||
| 22/21 | | 22/21 | ||
| [[Valentine]] (61) | |[[Valentine]] (61) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 140: | Line 139: | ||
| 98.4 | | 98.4 | ||
| 16/15 | | 16/15 | ||
| [[Passion]] (61de…) / [[passionate]] (61) | |[[Passion]] (61de…) / [[passionate]] (61) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 146: | Line 145: | ||
| 137.7 | | 137.7 | ||
| 13/12 | | 13/12 | ||
| [[Quartemka]] (61) | |[[Quartemka]] (61) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 152: | Line 151: | ||
| 177.0 | | 177.0 | ||
| 10/9 | | 10/9 | ||
| [[Modus]] (61de) / [[wollemia]] (61e) | |[[Modus]] (61de) / [[wollemia]] (61e) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 158: | Line 157: | ||
| 236.1 | | 236.1 | ||
| 8/7 | | 8/7 | ||
| [[Slendric]] (61) | |[[Slendric]] (61) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 164: | Line 163: | ||
| 314.8 | | 314.8 | ||
| 6/5 | | 6/5 | ||
| [[Parakleismic]] (61d) | |[[Parakleismic]] (61d) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 170: | Line 169: | ||
| 452.5 | | 452.5 | ||
| 13/10 | | 13/10 | ||
| [[Maja]] (61d) | |[[Maja]] (61d) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 176: | Line 175: | ||
| 491.8 | | 491.8 | ||
| 4/3 | | 4/3 | ||
| [[Quasisuper]] (61d) | |[[Quasisuper]] (61d) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 182: | Line 181: | ||
| 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 lists|Octave-reduced form]], reduced to the first half-octave | ||
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== 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) | ||