101edo: Difference between revisions

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+todo: Please don't add trailing 5 digits! (rounding doesn't really harm and it's not really hard to calculate the "correct" values for everyone)
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Link to lumatone mapping
 
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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
|-
|13
|154.455
|-
| '''25'''
| '''297.030'''
|-
| '''38'''
| '''451.485'''
|-
|51
|605.9405
|-
| '''63'''
| '''748.515'''
|-
|75
|891.089
|-
| '''88'''
| '''1045.5445'''
|}


'''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
|-
|9
|106.059
|-
| '''17'''
| '''201.980'''
|-
|26
|308.911
|-
| '''34'''
| '''403.960'''
|-
| '''42'''
| '''499.010'''
|-
|51
|605.940
|-
| '''59'''
| '''700.990'''
|-
|68
|807.921
|-
| '''76'''
| '''902.970'''
|-
|85
|1009.901
|-
| '''93'''
| '''1104.950'''
|}


'''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'''
|}
'''7 3 7 3 7 7 3 7 3 7 3 7 7 3 7 3 7 3 7:''' ''8L3s MOS'' (Improper Sensi-11)
{| class="wikitable"
! Steps
! Cents
|-
|7
|83.168
|-
| '''10'''
| '''118.812'''
|-
|17
|201.980
|-
| '''20'''
| '''237.624'''
|-
| '''27'''
| '''320.792'''
|-
|34
|403.960
|-
| '''37'''
| '''439.604'''
|-
|44
|527.772
|-
| '''47'''
| '''558.416'''
|-
|54
|641.584
|-
| '''57'''
| '''677.228'''
|-
| '''64'''
| '''760.396'''
|-
|71
|843.564
|-
| '''74'''
| '''879.218'''
|-
|81
|962.376
|-
| '''84'''
| '''998.020'''
|-
|91
|1081.188
|-
| '''94'''
| '''1116.842'''
|}
'''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
|}
== 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"