105edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,382 edits
CompactStar (talk | contribs)
autopatrolled
5,084 edits
No edit summary
Line 13: Line 13:
== Intervals ==
== Intervals ==
{{Main|Table of 105edo intervals}}
{{Main|Table of 105edo intervals}}
 
{{Interval table}}
=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
{{15-odd-limit|105}}
{{15-odd-limit|105}}

Revision as of 02:10, 28 February 2024

← 104edo 105edo 106edo →
Prime factorization 3 × 5 × 7
Step size 11.4286 ¢ 
Fifth 61\105 (697.143 ¢)
Semitones (A1:m2) 7:10 (80 ¢ : 114.3 ¢)
Dual sharp fifth 62\105 (708.571 ¢)
Dual flat fifth 61\105 (697.143 ¢)
Dual major 2nd 18\105 (205.714 ¢) (→ 6\35)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

Theory

105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone, such as grosstone and Huygens. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the optimal patent val, and provides a good tuning for the 13-limit extension, though 74edo is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the CTE generator for meantone.

Odd harmonics

Approximation of odd harmonics in 105edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.81 +2.26 +2.60 +1.80 -2.75 +5.19 -2.55 -2.10 -0.37 -2.21 +0.30
Relative (%) -42.1 +19.8 +22.8 +15.8 -24.0 +45.4 -22.4 -18.4 -3.2 -19.3 +2.6
Steps
(reduced) 		166
(61) 		244
(34) 		295
(85) 		333
(18) 		363
(48) 		389
(74) 		410
(95) 		429
(9) 		446
(26) 		461
(41) 		475
(55)

Subsets and supersets

105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

Intervals

Main article: Table of 105edo intervals
Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 61\105) 		Ups and downs notation
(Dual sharp fifth 62\105)
0 0 1/1 D D
1 11.4 ^D, vvE♭♭ ^D, v4E♭
2 22.9 ^^D, vE♭♭ ^^D, v3E♭
3 34.3 ^3D, E♭♭ ^3D, vvE♭
4 45.7 38/37, 39/38, 40/39 v3D♯, ^E♭♭ ^4D, vE♭
5 57.1 30/29, 31/30, 32/31 vvD♯, ^^E♭♭ ^5D, E♭
6 68.6 26/25 vD♯, ^3E♭♭ ^6D, ^E♭
7 80 22/21 D♯, v3E♭ ^7D, ^^E♭
8 91.4 39/37 ^D♯, vvE♭ v6D♯, ^3E♭
9 102.9 17/16 ^^D♯, vE♭ v5D♯, ^4E♭
10 114.3 31/29 ^3D♯, E♭ v4D♯, ^5E♭
11 125.7 43/40 v3D𝄪, ^E♭ v3D♯, ^6E♭
12 137.1 40/37 vvD𝄪, ^^E♭ vvD♯, v7E
13 148.6 12/11, 37/34 vD𝄪, ^3E♭ vD♯, v6E
14 160 34/31 D𝄪, v3E D♯, v5E
15 171.4 21/19, 32/29 ^D𝄪, vvE ^D♯, v4E
16 182.9 ^^D𝄪, vE ^^D♯, v3E
17 194.3 19/17, 28/25 E ^3D♯, vvE
18 205.7 ^E, vvF♭ ^4D♯, vE
19 217.1 17/15, 42/37 ^^E, vF♭ E
20 228.6 ^3E, F♭ ^E, v4F
21 240 23/20 v3E♯, ^F♭ ^^E, v3F
22 251.4 37/32 vvE♯, ^^F♭ ^3E, vvF
23 262.9 vE♯, ^3F♭ ^4E, vF
24 274.3 34/29, 41/35 E♯, v3F F
25 285.7 46/39 ^E♯, vvF ^F, v4G♭
26 297.1 19/16 ^^E♯, vF ^^F, v3G♭
27 308.6 37/31 F ^3F, vvG♭
28 320 ^F, vvG♭♭ ^4F, vG♭
29 331.4 23/19 ^^F, vG♭♭ ^5F, G♭
30 342.9 28/23, 39/32 ^3F, G♭♭ ^6F, ^G♭
31 354.3 38/31, 43/35 v3F♯, ^G♭♭ ^7F, ^^G♭
32 365.7 21/17 vvF♯, ^^G♭♭ v6F♯, ^3G♭
33 377.1 46/37 vF♯, ^3G♭♭ v5F♯, ^4G♭
34 388.6 5/4 F♯, v3G♭ v4F♯, ^5G♭
35 400 29/23 ^F♯, vvG♭ v3F♯, ^6G♭
36 411.4 19/15 ^^F♯, vG♭ vvF♯, v7G
37 422.9 37/29 ^3F♯, G♭ vF♯, v6G
38 434.3 v3F𝄪, ^G♭ F♯, v5G
39 445.7 22/17 vvF𝄪, ^^G♭ ^F♯, v4G
40 457.1 vF𝄪, ^3G♭ ^^F♯, v3G
41 468.6 21/16, 38/29 F𝄪, v3G ^3F♯, vvG
42 480 29/22 ^F𝄪, vvG ^4F♯, vG
43 491.4 ^^F𝄪, vG G
44 502.9 G ^G, v4A♭
45 514.3 35/26, 39/29 ^G, vvA♭♭ ^^G, v3A♭
46 525.7 42/31 ^^G, vA♭♭ ^3G, vvA♭
47 537.1 15/11 ^3G, A♭♭ ^4G, vA♭
48 548.6 v3G♯, ^A♭♭ ^5G, A♭
49 560 29/21 vvG♯, ^^A♭♭ ^6G, ^A♭
50 571.4 32/23, 39/28 vG♯, ^3A♭♭ ^7G, ^^A♭
51 582.9 7/5 G♯, v3A♭ v6G♯, ^3A♭
52 594.3 31/22 ^G♯, vvA♭ v5G♯, ^4A♭
53 605.7 44/31 ^^G♯, vA♭ v4G♯, ^5A♭
54 617.1 10/7 ^3G♯, A♭ v3G♯, ^6A♭
55 628.6 23/16 v3G𝄪, ^A♭ vvG♯, v7A
56 640 42/29 vvG𝄪, ^^A♭ vG♯, v6A
57 651.4 vG𝄪, ^3A♭ G♯, v5A
58 662.9 22/15 G𝄪, v3A ^G♯, v4A
59 674.3 31/21 ^G𝄪, vvA ^^G♯, v3A
60 685.7 ^^G𝄪, vA ^3G♯, vvA
61 697.1 A ^4G♯, vA
62 708.6 ^A, vvB♭♭ A
63 720 44/29 ^^A, vB♭♭ ^A, v4B♭
64 731.4 29/19, 32/21 ^3A, B♭♭ ^^A, v3B♭
65 742.9 43/28 v3A♯, ^B♭♭ ^3A, vvB♭
66 754.3 17/11 vvA♯, ^^B♭♭ ^4A, vB♭
67 765.7 vA♯, ^3B♭♭ ^5A, B♭
68 777.1 A♯, v3B♭ ^6A, ^B♭
69 788.6 30/19, 41/26 ^A♯, vvB♭ ^7A, ^^B♭
70 800 46/29 ^^A♯, vB♭ v6A♯, ^3B♭
71 811.4 8/5 ^3A♯, B♭ v5A♯, ^4B♭
72 822.9 37/23 v3A𝄪, ^B♭ v4A♯, ^5B♭
73 834.3 34/21 vvA𝄪, ^^B♭ v3A♯, ^6B♭
74 845.7 31/19 vA𝄪, ^3B♭ vvA♯, v7B
75 857.1 23/14, 41/25 A𝄪, v3B vA♯, v6B
76 868.6 38/23 ^A𝄪, vvB A♯, v5B
77 880 ^^A𝄪, vB ^A♯, v4B
78 891.4 B ^^A♯, v3B
79 902.9 32/19 ^B, vvC♭ ^3A♯, vvB
80 914.3 39/23 ^^B, vC♭ ^4A♯, vB
81 925.7 29/17 ^3B, C♭ B
82 937.1 43/25 v3B♯, ^C♭ ^B, v4C
83 948.6 vvB♯, ^^C♭ ^^B, v3C
84 960 40/23 vB♯, ^3C♭ ^3B, vvC
85 971.4 B♯, v3C ^4B, vC
86 982.9 30/17, 37/21 ^B♯, vvC C
87 994.3 ^^B♯, vC ^C, v4D♭
88 1005.7 25/14, 34/19 C ^^C, v3D♭
89 1017.1 ^C, vvD♭♭ ^3C, vvD♭
90 1028.6 29/16, 38/21 ^^C, vD♭♭ ^4C, vD♭
91 1040 31/17 ^3C, D♭♭ ^5C, D♭
92 1051.4 11/6 v3C♯, ^D♭♭ ^6C, ^D♭
93 1062.9 37/20 vvC♯, ^^D♭♭ ^7C, ^^D♭
94 1074.3 vC♯, ^3D♭♭ v6C♯, ^3D♭
95 1085.7 C♯, v3D♭ v5C♯, ^4D♭
96 1097.1 32/17 ^C♯, vvD♭ v4C♯, ^5D♭
97 1108.6 ^^C♯, vD♭ v3C♯, ^6D♭
98 1120 21/11 ^3C♯, D♭ vvC♯, v7D
99 1131.4 25/13 v3C𝄪, ^D♭ vC♯, v6D
100 1142.9 29/15, 31/16 vvC𝄪, ^^D♭ C♯, v5D
101 1154.3 37/19, 39/20 vC𝄪, ^3D♭ ^C♯, v4D
102 1165.7 C𝄪, v3D ^^C♯, v3D
103 1177.1 ^C𝄪, vvD ^3C♯, vvD
104 1188.6 ^^C𝄪, vD ^4C♯, vD
105 1200 2/1 D D

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 105edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 105edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
9/5, 10/9 0.453 4.0
9/7, 14/9 0.798 7.0
13/12, 24/13 1.430 12.5
9/8, 16/9 1.804 15.8
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
13/9, 18/13 3.382 29.6
13/11, 22/13 3.495 30.6
15/13, 26/15 3.688 32.3
7/6, 12/7 4.014 35.1
5/3, 6/5 4.359 38.1
11/9, 18/11 4.551 39.8
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
15-odd-limit intervals in 105edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
11/9, 18/11 6.878 60.2
5/3, 6/5 7.070 61.9
7/6, 12/7 7.415 64.9
15/13, 26/15 7.741 67.7
13/11, 22/13 7.933 69.4
9/8, 16/9 9.624 84.2
13/12, 24/13 9.999 87.5
9/5, 10/9 11.882 104.0
9/7, 14/9 12.227 107.0
13/9, 18/13 14.811 129.6
Retrieved from "https://en.xen.wiki/index.php?title=105edo&oldid=137413"