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'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.
{{Infobox ET}}
{{ED intro}}


;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
== Theory ==
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.


== Some important MOS scales ==
=== Odd harmonics ===
{{Harmonics in equal|101}}


'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)
=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.


{| class="wikitable"
== Intervals ==
! Steps
{{Interval table}}
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}


'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)
== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} <!-- why is this significant? -->
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} <!-- which val? -->
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)


{| class="wikitable"
== Instruments ==
!Steps
* [[Lumatone mapping for 101edo]]
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}


'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)
== Music ==
; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]


{| class="wikitable"
== External links ==
!Steps
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}


'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)
{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}
'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)
{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}
'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)
{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament or allows one to use an MOS scale with 50 or more large steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|[[2L 9s]]
|10:9
|-
|12
|[[5L 7s]]
|9:8
|-
|13
|[[10L 3s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]
[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
{{todo|improve_layout|unify precision}}

Latest revision as of 11:03, 18 August 2025

← 100edo 101edo 102edo →
Prime factorization 101 (prime)
Step size 11.8812 ¢ 
Fifth 59\101 (700.99 ¢)
Semitones (A1:m2) 9:8 (106.9 ¢ : 95.05 ¢)
Consistency limit 3
Distinct consistency limit 3

101 equal divisions of the octave (abbreviated 101edo or 101ed2), also called 101-tone equal temperament (101tet) or 101 equal temperament (101et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 101 equal parts of about 11.9 ¢ each. Each step represents a frequency ratio of 21/101, or the 101st root of 2.

Theory

101edo is inconsistent in the 5-odd-limit, with harmonics 5 and 7 falling about halfway between its steps. As such, ⟨101 160 235 284] (patent val) and ⟨101 160 234 283] (101cd) are about as viable. Using the patent val, it tempers out 32805/32768 (schisma) and 51018336/48828125 in the 5-limit; 126/125 and 2430/2401 in the 7-limit. It can be used to tune the grackle temperament. The 101cd val provides an excellent tuning for witchcraft temperament, falling between the 13- and 15-odd-limit least squares tuning.

Odd harmonics

Approximation of prime harmonics in 101edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.96 +5.77 +5.43 -4.78 +3.04 +1.98 -0.48 +1.43 +4.09 -4.44
Relative (%) +0.0 -8.1 +48.5 +45.7 -40.3 +25.6 +16.6 -4.1 +12.0 +34.4 -37.4
Steps
(reduced) 		101
(0) 		160
(59) 		235
(33) 		284
(82) 		349
(46) 		374
(71) 		413
(9) 		429
(25) 		457
(53) 		491
(87) 		500
(96)

Subsets and supersets

101edo is the 26th prime edo, following 97edo and before 103edo. 202edo, which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 11.9 ^D, ^^E♭♭
2 23.8 ^^D, ^3E♭♭
3 35.6 ^3D, ^4E♭♭
4 47.5 37/36, 38/37 ^4D, v4E♭
5 59.4 29/28, 30/29 v4D♯, v3E♭
6 71.3 24/23 v3D♯, vvE♭
7 83.2 21/20, 43/41 vvD♯, vE♭
8 95 19/18 vD♯, E♭
9 106.9 17/16, 33/31 D♯, ^E♭
10 118.8 15/14 ^D♯, ^^E♭
11 130.7 14/13, 41/38 ^^D♯, ^3E♭
12 142.6 ^3D♯, ^4E♭
13 154.5 ^4D♯, v4E
14 166.3 v4D𝄪, v3E
15 178.2 41/37 v3D𝄪, vvE
16 190.1 29/26 vvD𝄪, vE
17 202 9/8 E
18 213.9 26/23, 43/38 ^E, ^^F♭
19 225.7 41/36 ^^E, ^3F♭
20 237.6 31/27, 39/34 ^3E, ^4F♭
21 249.5 15/13, 37/32 ^4E, v4F
22 261.4 43/37 v4E♯, v3F
23 273.3 34/29 v3E♯, vvF
24 285.1 vvE♯, vF
25 297 19/16 F
26 308.9 43/36 ^F, ^^G♭♭
27 320.8 ^^F, ^3G♭♭
28 332.7 23/19 ^3F, ^4G♭♭
29 344.6 39/32 ^4F, v4G♭
30 356.4 27/22 v4F♯, v3G♭
31 368.3 26/21 v3F♯, vvG♭
32 380.2 vvF♯, vG♭
33 392.1 vF♯, G♭
34 404 24/19 F♯, ^G♭
35 415.8 ^F♯, ^^G♭
36 427.7 41/32 ^^F♯, ^3G♭
37 439.6 ^3F♯, ^4G♭
38 451.5 ^4F♯, v4G
39 463.4 17/13 v4F𝄪, v3G
40 475.2 v3F𝄪, vvG
41 487.1 45/34 vvF𝄪, vG
42 499 4/3 G
43 510.9 39/29, 43/32 ^G, ^^A♭♭
44 522.8 23/17 ^^G, ^3A♭♭
45 534.7 ^3G, ^4A♭♭
46 546.5 37/27 ^4G, v4A♭
47 558.4 29/21, 40/29 v4G♯, v3A♭
48 570.3 32/23 v3G♯, vvA♭
49 582.2 7/5 vvG♯, vA♭
50 594.1 31/22, 38/27 vG♯, A♭
51 605.9 27/19, 44/31 G♯, ^A♭
52 617.8 10/7 ^G♯, ^^A♭
53 629.7 23/16 ^^G♯, ^3A♭
54 641.6 29/20, 42/29 ^3G♯, ^4A♭
55 653.5 ^4G♯, v4A
56 665.3 v4G𝄪, v3A
57 677.2 34/23 v3G𝄪, vvA
58 689.1 vvG𝄪, vA
59 701 3/2 A
60 712.9 ^A, ^^B♭♭
61 724.8 41/27 ^^A, ^3B♭♭
62 736.6 26/17 ^3A, ^4B♭♭
63 748.5 37/24 ^4A, v4B♭
64 760.4 45/29 v4A♯, v3B♭
65 772.3 v3A♯, vvB♭
66 784.2 vvA♯, vB♭
67 796 19/12 vA♯, B♭
68 807.9 43/27 A♯, ^B♭
69 819.8 45/28 ^A♯, ^^B♭
70 831.7 21/13 ^^A♯, ^3B♭
71 843.6 44/27 ^3A♯, ^4B♭
72 855.4 ^4A♯, v4B
73 867.3 38/23 v4A𝄪, v3B
74 879.2 v3A𝄪, vvB
75 891.1 vvA𝄪, vB
76 903 32/19 B
77 914.9 39/23 ^B, ^^C♭
78 926.7 29/17, 41/24 ^^B, ^3C♭
79 938.6 ^3B, ^4C♭
80 950.5 26/15, 45/26 ^4B, v4C
81 962.4 v4B♯, v3C
82 974.3 v3B♯, vvC
83 986.1 23/13 vvB♯, vC
84 998 16/9 C
85 1009.9 43/24 ^C, ^^D♭♭
86 1021.8 ^^C, ^3D♭♭
87 1033.7 ^3C, ^4D♭♭
88 1045.5 ^4C, v4D♭
89 1057.4 v4C♯, v3D♭
90 1069.3 13/7 v3C♯, vvD♭
91 1081.2 28/15, 43/23 vvC♯, vD♭
92 1093.1 32/17 vC♯, D♭
93 1105 36/19 C♯, ^D♭
94 1116.8 40/21 ^C♯, ^^D♭
95 1128.7 23/12 ^^C♯, ^3D♭
96 1140.6 29/15 ^3C♯, ^4D♭
97 1152.5 37/19 ^4C♯, v4D
98 1164.4 45/23 v4C𝄪, v3D
99 1176.2 v3C𝄪, vvD
100 1188.1 vvC𝄪, vD
101 1200 2/1 D

Scales

Mos scales

  • 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101)[clarification needed]
  • Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
  • Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
  • Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
  • Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101)[clarification needed]
  • Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

Instruments

Music

Francium

External links

Retrieved from "https://en.xen.wiki/index.php?title=101edo&oldid=207462"