106edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
m →Theory: (''See regular temperament for more about what all this means and how to use it.'')
Tag: Reverted
Overthink (talk | contribs)
autopatrolled
2,697 edits
→Theory: Removed confusing MMTM sentence from years ago
 
(3 intermediate revisions by 3 users not shown)
Line 3: Line 3:


== Theory ==
== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. (''See [[regular temperament]] for more about what all this means and how to use it.'')
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. Unfortunately, it is now only consistent to the [[5-odd-limit]] due to 7/5 being closer to 51 steps rather than its patent val mapping of 52 steps. A superset of 53edo with much a higher consistency limit is [[159edo]].


The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π<sup>¢</sup>, π<sup>r¢</sup>), which most closely approximates equally dividing an exact [[3/2]].
The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.


=== Prime harmonics ===
=== Prime harmonics ===
Line 15: Line 15:
== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}
== Instruments ==
* [[Lumatone mapping for 106edo]]


== See also ==
== See also ==

Latest revision as of 22:41, 18 November 2025

← 105edo 106edo 107edo →
Prime factorization 2 × 53
Step size 11.3208 ¢ 
Fifth 62\106 (701.887 ¢) (→ 31\53)
Semitones (A1:m2) 10:8 (113.2 ¢ : 90.57 ¢)
Consistency limit 5
Distinct consistency limit 5

106 equal divisions of the octave (abbreviated 106edo or 106ed2), also called 106-tone equal temperament (106tet) or 106 equal temperament (106et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 106 equal parts of about 11.3 ¢ each. Each step represents a frequency ratio of 21/106, or the 106th root of 2.

Theory

Since 106 = 2 × 53, 106edo is closely related to 53edo, and is contorted through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3087, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports spectacle temperament and borwell temperament. Unfortunately, it is now only consistent to the 5-odd-limit due to 7/5 being closer to 51 steps rather than its patent val mapping of 52 steps. A superset of 53edo with much a higher consistency limit is 159edo.

The division is notable for the fact that it is related to the turkish cent, or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the relative cent division for 106edo.

Prime harmonics

Approximation of prime harmonics in 106edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 +3.40 -2.79 -3.07 -3.17 -5.63 +0.61 -1.64 -2.29 +1.13 -2.08 +2.42 -1.81
Relative (%) +0.0 -0.6 -12.4 +42.0 +30.0 -24.7 -27.1 -28.0 -49.8 +5.4 -14.5 -20.2 +9.9 -18.4 +21.4 -16.0
Steps
(reduced) 		106
(0) 		168
(62) 		246
(34) 		298
(86) 		367
(49) 		392
(74) 		433
(9) 		450
(26) 		479
(55) 		515
(91) 		525
(101) 		552
(22) 		568
(38) 		575
(45) 		589
(59) 		607
(77)

53edo for comparison:

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68 -2.29 +1.13 +9.24 -8.90 +9.51
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1 -47.3 +42.8 -10.1 +5.0 +40.8 -39.3 +42.0
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28) 		257
(45) 		263
(51) 		276
(11) 		284
(19) 		288
(23) 		294
(29) 		304
(39)

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 11.3 ^D, ^3E♭♭
2 22.6 ^^D, ^4E♭♭
3 34 ^3D, v5E♭
4 45.3 37/36, 38/37, 39/38, 40/39 ^4D, v4E♭
5 56.6 30/29, 31/30, 32/31 ^5D, v3E♭
6 67.9 26/25 v4D♯, vvE♭
7 79.2 22/21, 45/43 v3D♯, vE♭
8 90.6 20/19, 39/37 vvD♯, E♭
9 101.9 35/33 vD♯, ^E♭
10 113.2 16/15, 31/29 D♯, ^^E♭
11 124.5 29/27, 43/40, 44/41 ^D♯, ^3E♭
12 135.8 40/37 ^^D♯, ^4E♭
13 147.2 37/34 ^3D♯, v5E
14 158.5 34/31 ^4D♯, v4E
15 169.8 32/29, 43/39 ^5D♯, v3E
16 181.1 10/9 v4D𝄪, vvE
17 192.5 19/17 v3D𝄪, vE
18 203.8 9/8 E
19 215.1 17/15, 43/38 ^E, ^3F♭
20 226.4 41/36 ^^E, ^4F♭
21 237.7 31/27, 39/34 ^3E, v5F
22 249.1 15/13, 37/32 ^4E, v4F
23 260.4 36/31, 43/37 ^5E, v3F
24 271.7 41/35 v4E♯, vvF
25 283 20/17, 33/28 v3E♯, vF
26 294.3 32/27, 45/38 F
27 305.7 31/26, 37/31, 43/36 ^F, ^3G♭♭
28 317 6/5 ^^F, ^4G♭♭
29 328.3 29/24 ^3F, v5G♭
30 339.6 45/37 ^4F, v4G♭
31 350.9 38/31 ^5F, v3G♭
32 362.3 37/30 v4F♯, vvG♭
33 373.6 31/25, 36/29, 41/33 v3F♯, vG♭
34 384.9 5/4 vvF♯, G♭
35 396.2 39/31, 44/35 vF♯, ^G♭
36 407.5 19/15, 43/34 F♯, ^^G♭
37 418.9 14/11 ^F♯, ^3G♭
38 430.2 41/32 ^^F♯, ^4G♭
39 441.5 31/24, 40/31 ^3F♯, v5G
40 452.8 13/10 ^4F♯, v4G
41 464.2 17/13 ^5F♯, v3G
42 475.5 25/19 v4F𝄪, vvG
43 486.8 45/34 v3F𝄪, vG
44 498.1 4/3 G
45 509.4 43/32 ^G, ^3A♭♭
46 520.8 27/20 ^^G, ^4A♭♭
47 532.1 34/25 ^3G, v5A♭
48 543.4 26/19, 37/27 ^4G, v4A♭
49 554.7 40/29 ^5G, v3A♭
50 566 43/31 v4G♯, vvA♭
51 577.4 v3G♯, vA♭
52 588.7 45/32 vvG♯, A♭
53 600 41/29 vG♯, ^A♭
54 611.3 37/26 G♯, ^^A♭
55 622.6 43/30 ^G♯, ^3A♭
56 634 ^^G♯, ^4A♭
57 645.3 29/20, 45/31 ^3G♯, v5A
58 656.6 19/13 ^4G♯, v4A
59 667.9 25/17 ^5G♯, v3A
60 679.2 37/25, 40/27 v4G𝄪, vvA
61 690.6 v3G𝄪, vA
62 701.9 3/2 A
63 713.2 ^A, ^3B♭♭
64 724.5 38/25, 41/27 ^^A, ^4B♭♭
65 735.8 26/17 ^3A, v5B♭
66 747.2 20/13, 37/24 ^4A, v4B♭
67 758.5 31/20, 45/29 ^5A, v3B♭
68 769.8 39/25 v4A♯, vvB♭
69 781.1 11/7 v3A♯, vB♭
70 792.5 30/19 vvA♯, B♭
71 803.8 35/22, 43/27 vA♯, ^B♭
72 815.1 8/5 A♯, ^^B♭
73 826.4 29/18 ^A♯, ^3B♭
74 837.7 ^^A♯, ^4B♭
75 849.1 31/19 ^3A♯, v5B
76 860.4 ^4A♯, v4B
77 871.7 43/26 ^5A♯, v3B
78 883 5/3 v4A𝄪, vvB
79 894.3 v3A𝄪, vB
80 905.7 27/16 B
81 917 17/10 ^B, ^3C♭
82 928.3 41/24 ^^B, ^4C♭
83 939.6 31/18, 43/25 ^3B, v5C
84 950.9 26/15, 45/26 ^4B, v4C
85 962.3 ^5B, v3C
86 973.6 v4B♯, vvC
87 984.9 30/17 v3B♯, vC
88 996.2 16/9 C
89 1007.5 34/19, 43/24 ^C, ^3D♭♭
90 1018.9 9/5 ^^C, ^4D♭♭
91 1030.2 29/16 ^3C, v5D♭
92 1041.5 31/17 ^4C, v4D♭
93 1052.8 ^5C, v3D♭
94 1064.2 37/20 v4C♯, vvD♭
95 1075.5 41/22 v3C♯, vD♭
96 1086.8 15/8 vvC♯, D♭
97 1098.1 vC♯, ^D♭
98 1109.4 19/10 C♯, ^^D♭
99 1120.8 21/11 ^C♯, ^3D♭
100 1132.1 25/13 ^^C♯, ^4D♭
101 1143.4 29/15, 31/16 ^3C♯, v5D
102 1154.7 37/19, 39/20 ^4C♯, v4D
103 1166 ^5C♯, v3D
104 1177.4 v4C𝄪, vvD
105 1188.7 v3C𝄪, vD
106 1200 2/1 D

Instruments

See also

Artists using 106 et:

Retrieved from "https://en.xen.wiki/index.php?title=106edo&oldid=217293"