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61edo: Difference between revisions

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=61 tone equal temperament=
{{Infobox ET}}
'''61-EDO''' refers to the equal division of [[2/1|2/1]] ratio into 61 equal parts, of 19.6721 [[cent|cent]]s each. It is the 18th [[prime_numbers|prime]] EDO, after of [[59edo|59edo]] and before of [[67edo|67edo]]. It provides the optimal patent val for the 24&37 temperament in the 7-, 11- and 13-limit.
{{ED intro}}


=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),
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]].


while the 3rd and 5th harmonics are about six cents sharp,
=== Odd harmonics ===
{{Harmonics in equal|61}}


(and the flattish 15th poised differently on the harp),
=== 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.


the 7th and 11th err by less, around three,
== Intervals ==
{{Interval table}}


and thus mayhap, a good orgone tuning found to be;
== Notation ==
=== Ups and downs notation ===
61edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:


slightly sharp as well, is the 13th harmonic's place,
{{Sharpness-sharp8}}


but the 9th and 17th lack near so much grace,
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[54edo #Sagittal notation|54edo]].


interestingly the 19th is good but a couple cents flat,
==== 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>


and the 21st and 23rd are but a cent or two sharp!
==== 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>


==61-EDO Intervals==
==== 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>


{| class="wikitable"
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"
|-
|-
| | '''Degrees'''
! rowspan="2" |[[Subgroup]]
| | '''Cent Value'''
! rowspan="2" |[[Comma list]]
|Pions
! rowspan="2" |[[Mapping]]
|7mus
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 0
![[TE error|Absolute]] (¢)
| colspan="3"| 0
![[TE simple badness|Relative]] (%)
|-
|-
| | 1
| 2.3
| | 19.6721
|{{Monzo| 97 -61 }}
|20.8525
|{{Mapping| 61 97 }}
|25.1803 (19.2E2A<sub>16</sub>)
| −1.97
| 1.97
| 10.0
|-
|-
| | 2
| 2.3.5
| | 39.3443
| 20000/19683, 262144/253125
|41.7049
|{{Mapping| 61 97 142 }}
|50.3607 (32.5C54<sub>16</sub>)
| −2.33
|-
| 1.69
| | 3
| 8.59
| | 59.0164
|- style="border-top: double;"
|62.5574
| 2.3.5.7
|75.541 (4B.8A7E<sub>16</sub>)
| 64/63, 2430/2401, 3125/3087
|-
|{{mapping| 61 97 142 172 }} (61d)
| | 4
| −3.06
| | 78.6885
| 1.93
|83.4098
| 9.84
|100.7213 (64.B8A8<sub>16</sub>)
|- style="border-top: double;"
|-
| 2.3.5.7
| | 5
| 126/125, 1029/1024, 2240/2187
| | 98.3607
|{{Mapping| 61 97 142 171 }} (61)
|104.2623
| −1.32
|125.9016 (7D.E6D2<sub>16</sub>)
| 2.29
|-
| 11.7
| | 6
|}
| | 118.0328
 
|125.11475
=== Rank-2 temperaments ===
|151.082 (97.14FC<sub>16</sub>)
{| class="wikitable center-all left-5"
|-
|+ style="font-size: 105%;" |Table of rank-2 temperaments by generator
| | 7
| | 137.7049
|145.9672
|176.2623 (B0.4326<sub>16</sub>)
|-
| | 8
| | 157.37705
|166.8197
|201.4426 (C9.715<sub>16</sub>)
|-
| | 9
| | 177.0492
|187.6721
|226.62295 (E2.9F7A<sub>16</sub>)
|-
| | 10
| | 196.7213
|208.5246
|251.8033 (FB.CDA4<sub>16</sub>)
|-
| | 11
| | 216.3934
|229.37705
|276.9836 (114.FBCE<sub>16</sub>)
|-
| | 12
| | 236.0656
|250.2295
|302.1639 (12E.29F8<sub>16</sub>)
|-
| | 13
| | 255.7377
|271.082
|327.3443 (147.5822<sub>16</sub>)
|-
| | 14
| | 275.4098
|291.9344
|352.5246 (160.864B8<sub>16</sub>)
|-
| | 15
| | 295.082
|312.7869
|377.7049 (179.B4758<sub>16</sub>)
|-
|-
| | 16
! Periods<br>per 8ve
| | 314.7541
! Generator*
|333.6393
! Cents*
|402.88525 (192.E29F8<sub>16</sub>)
! Associated<br>ratio*
! Temperament
|-
|-
| | 17
| 1
| | 334.4262
| 2\61
|354.4918
| 39.3
|428.0656 (1AC.10BC<sub>16</sub>)
| 40/39
|[[Hemivalentine]] (61)
|-
|-
| | 18
| 1
| | 354.0984
| 3\61
|375.3443
| 59.0
|453.2459 (1C5.3EF3<sub>16</sub>)
| 28/27
|[[Dodecacot]] (61de…)
|-
|-
| | 19
| 1
| | 373.7705
| 4\61
|396.1967
| 78.7
|478.4262 (1DE.6D1E<sub>16</sub>)
| 22/21
|[[Valentine]] (61)
|-
|-
| | 20
| 1
| | 393.4426
| 5\61
|417.0492
| 98.4
|503.6066 (1F7.9B47<sub>16</sub>)
| 16/15
|[[Passion]] (61de…) / [[passionate]] (61)
|-
|-
| | 21
| 1
| | 413.11475
| 7\61
|437.9016
| 137.7
|528.7869 (210.C971<sub>16</sub>)
| 13/12
|[[Quartemka]] (61)
|-
|-
| | 22
| 1
| | 432.7869
| 9\61
|458.7541
| 177.0
|553.9672 (229.F79B<sub>16</sub>)
| 10/9
|[[Modus]] (61de) / [[wollemia]] (61e)
|-
|-
| | 23
| 1
| | 452.459
| 11\61
|479.6066
| 236.1
|579.1475 (243.25C5<sub>16</sub>)
| 8/7
|[[Slendric]] (61)
|-
|-
| | 24
| 1
| | 472.13115
| 16\61
|500.459
| 314.8
|604.3279 (25C.53EF<sub>16</sub>)
| 6/5
|[[Parakleismic]] (61d)
|-
|-
| | 25
| 1
| | 491.8033
| 23\61
|521.3115
| 452.5
|629.5082 (275.8219<sub>16</sub>)
| 13/10
|[[Maja]] (61d)
|-
|-
| | 26
| 1
| | 511.4754
| 25\61
|542.1639
| 491.8
|654.6885 (28E.B043<sub>16)</sub>
| 4/3
|[[Quasisuper]] (61d)
|-
|-
| | 27
| 1
| | 531.1475
| 28\61
|563.0164
| 550.8
|679.86885 (2A7.DE6D<sub>16</sub>)
| 11/8
|-
|[[Freivald]] (61)
| | 28
| | 550.8197
|583.86885
|705.0492 (2C1.0C97<sub>16</sub>)
|-
| | 29
| | 570.4918
|604.7213
|730.2295 (2DA.3AC1<sub>16</sub>)
|-
| | 30
| | 590.1639
|625.5738
|755.4098 (2F3.68EB<sub>16</sub>)
|-
| | 31
| | 609.8361
|646.4262
|780.5902 (30C.9715<sub>16</sub>)
|-
| | 32
| | 629.5082
|667.2787
|805.7705 (325.C53F<sub>16</sub>)
|-
| | 33
| | 649.1803
|688.13115
|830.9508 (33E.F359<sub>16</sub>)
|-
| | 34
| | 668.8525
|708.9836
|856.13115 (358.2183<sub>16</sub>)
|-
| | 35
| | 688.5246
|729.8361
|881.3115 (371.4FAD<sub>16</sub>)
|-
| | 36
| | 708.1967
|750.6885
|906.4918 (38A.7DD7<sub>16</sub>)
|-
| | 37
| | 727.86885
|771.541
|931.6721 (3A3.AC11<sub>16</sub>)
|-
| | 38
| | 747.541
|792.3934
|956.8525 (3BC.DA3B<sub>16</sub>)
|-
| | 39
| | 767.2131
|813.2459
|982.0328 (3D6.0865<sub>16</sub>)
|-
| | 40
| | 786.88525
|834.0984
|1007.2169 (3EF.268F<sub>16</sub>
|-
| | 41
| | 806.5574
|854.9508
|1032.3934 (408.62B9<sub>16</sub>)
|-
| | 42
| | 826.2295
|875.8033
|1057.5738 (421.90E3<sub>16</sub>)
|-
| | 43
| | 845.9016
|896.6557
|1082.7541 (43A.BEFD<sub>16</sub>)
|-
| | 44
| | 865.5738
|917.5082
|1107.9344 (453.EF44<sub>16</sub>)
|-
| | 45
| | 885.2459
|938.3607
|1133.11475 (46D.1D608<sub>16</sub>)
|-
| | 46
| | 904.918
|959.2131
|1158.2951 (486.4B8A8<sub>16</sub>)
|-
| | 47
| | 924.5902
|980.0656
|1183.4754 (49F.79B48<sub>16</sub>)
|-
| | 48
| | 944.2623
|1000.918
|1208.6557 (4B8.A7DE<sub>16</sub>)
|-
| | 49
| | 963.9344
|1021.7705
|1233.8361 (4D1.D708<sub>16</sub>)
|-
| | 50
| | 983.6066
|1042.62295
|1259.0164 (4EA.0431<sub>16</sub>)
|-
| | 51
| | 1003.2787
|1063.4754
|1284.1967 (504.325C<sub>16</sub>)
|-
| | 52
| | 1022.9508
|1084.3279
|1309.37705 (51D.6085<sub>16</sub>)
|-
| | 53
| | 1042.62295
|1105.1803
|1334.5574 (536.8EB<sub>16</sub>)
|-
| | 54
| | 1062.2951
|1126.0328
|1359.7377 (54F.BCDA<sub>16</sub>)
|-
| | 55
| | 1081.9672
|1146.88525
|1384.918 (568.EB04<sub>16</sub>)
|-
| | 56
| | 1101.6393
|1167.7377
|1410.0984 (582.192E<sub>16</sub>)
|-
| | 57
| | 1121.3115
|1188.5902
|1435.2787 (59B.4748<sub>16</sub>)
|-
| | 58
| | 1140.9836
|1209.4426
|1460.459 (5B4.7581<sub>16</sub>)
|-
| | 59
| | 1160.6557
|1230.2951
|1485.6393 (5CD.A3AB<sub>16</sub>)
|-
| | 60
| | 1180.3279
|1251.1475
|1510.8197 (5D7.D1D5<sub>16</sub>)
|}
|}
<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 ==


[[Category:Edo]]
=== Introductory poem ===
[[Category:Prime EDO]]
[[Peter Kosmorsky]] wrote a poem on 61edo; see [[User:Spt3125/61edo poem|the 61edo poem]].


[[Category:todo:add sound examples]]
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
Retrieved from "https://en.xen.wiki/w/61edo"