102edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-08-01 14:52:49 UTC</tt>.<br>
: The original revision id was <tt>588501714</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five [[#|temperament]] tempering out 91/90.
===13-limit Echidnic===
|| 2 || 23.529 ||
|| 4 || 47.059 ||
|| 7 || 82.353 ||
|| 9 || 105.882 ||
|| 11 || 129.412 ||
|| 13 || 152.941 ||
|| 16 || 188.235 ||
|| 18 || 211.765 ||
|| 20 || 235.294 ||
|| 22 || 258.824 ||
|| 24 || 282.353 ||
|| 27 || 317.647 ||
|| 29 || 341.1765 ||
|| 31 || 364.706 ||
|| 33 || 388.235 ||
|| 35 || 411.765 ||
|| 38 || 447.059 ||
|| 40 || 470.588 ||
|| 42 || 494.117 ||
|| 44 || 517.647 ||
|| 47 || 552.941 ||
|| 49 || 576.471 ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;102edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;102edo&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into 102 steps of size 11.765 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. In the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; it &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the same &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s (2048/2025, 15625/15552, 20000/19683) as &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;. In the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; it tempers out 686/675 and 1029/1024; in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; 385/384, 441/440 and 4000/3993; in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; 91/90 and 169/168; in the &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt; 136/135 and 154/153; and in the &lt;a class="wiki_link" href="/19-limit"&gt;19-limit&lt;/a&gt; 133/132 and 190/189. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 13-limit &lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&gt;echidnic temperament&lt;/a&gt;, and the rank five [[#|temperament]] tempering out 91/90.&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--13-limit Echidnic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;13-limit Echidnic&lt;/h3&gt;


&lt;table class="wiki_table"&gt;
== Theory ==
    &lt;tr&gt;
102edo is [[enfactoring|enfactored]] in the [[5-limit]], where it [[tempering out|tempers out]] the same [[comma]]s ([[2048/2025]], [[15625/15552]], [[20000/19683]]) as [[34edo]]. In the [[7-limit]] it tempers out [[686/675]] and [[1029/1024]]; in the [[11-limit]] [[385/384]], [[441/440]] and [[4000/3993]]; in the [[13-limit]] [[91/90]] and [[169/168]]; in the [[17-limit]] [[136/135]] and [[154/153]]; and in the [[19-limit]] [[133/132]] and [[190/189]]. It is the [[optimal patent val]] for 13-limit [[Diaschismic family #Echidnic|echidnic]] temperament, and the rank-5 temperament tempering out 91/90.
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;23.529&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;47.059&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;82.353&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;105.882&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;129.412&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;152.941&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;188.235&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;211.765&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;235.294&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;258.824&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282.353&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;317.647&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;341.1765&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;364.706&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;388.235&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;411.765&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;447.059&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;470.588&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;494.117&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;44&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;517.647&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;552.941&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;576.471&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Odd harmonics ===
{{Harmonics in equal|102}}
 
== Intervals ==
{{Interval table}}
 
[[Category:Echidnic]]

Latest revision as of 16:46, 18 February 2025

← 101edo 102edo 103edo →
Prime factorization 2 × 3 × 17
Step size 11.7647 ¢ 
Fifth 60\102 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 12:6 (141.2 ¢ : 70.59 ¢)
Dual sharp fifth 60\102 (705.882 ¢) (→ 10\17)
Dual flat fifth 59\102 (694.118 ¢)
Dual major 2nd 17\102 (200 ¢) (→ 1\6)
Consistency limit 5
Distinct consistency limit 5

102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 21/102, or the 102nd root of 2.

Theory

102edo is enfactored in the 5-limit, where it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank-5 temperament tempering out 91/90.

Odd harmonics

Approximation of odd harmonics in 102edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.93 +1.92 -4.12 -3.91 +1.62 -5.23 +5.85 +0.93 -3.40 -0.19 -4.74
Relative (%) +33.4 +16.3 -35.0 -33.2 +13.8 -44.5 +49.7 +7.9 -28.9 -1.6 -40.3
Steps
(reduced) 		162
(60) 		237
(33) 		286
(82) 		323
(17) 		353
(47) 		377
(71) 		399
(93) 		417
(9) 		433
(25) 		448
(40) 		461
(53)

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 59\102) 		Ups and downs notation
(Dual sharp fifth 60\102)
0 0 1/1 D D
1 11.8 ^D, E♭♭♭ ^D, v5E♭
2 23.5 ^^D, ^E♭♭♭ ^^D, v4E♭
3 35.3 vvD♯, ^^E♭♭♭ ^3D, v3E♭
4 47.1 38/37, 39/38 vD♯, vvE♭♭ ^4D, vvE♭
5 58.8 30/29 D♯, vE♭♭ ^5D, vE♭
6 70.6 25/24 ^D♯, E♭♭ ^6D, E♭
7 82.4 21/20, 22/21, 43/41 ^^D♯, ^E♭♭ v5D♯, ^E♭
8 94.1 37/35 vvD𝄪, ^^E♭♭ v4D♯, ^^E♭
9 105.9 17/16 vD𝄪, vvE♭ v3D♯, ^3E♭
10 117.6 D𝄪, vE♭ vvD♯, ^4E♭
11 129.4 14/13, 41/38 ^D𝄪, E♭ vD♯, ^5E♭
12 141.2 38/35 ^^D𝄪, ^E♭ D♯, v6E
13 152.9 12/11, 35/32 vvD♯𝄪, ^^E♭ ^D♯, v5E
14 164.7 11/10 vD♯𝄪, vvE ^^D♯, v4E
15 176.5 31/28, 41/37 D♯𝄪, vE ^3D♯, v3E
16 188.2 39/35 E ^4D♯, vvE
17 200 ^E, F♭♭ ^5D♯, vE
18 211.8 26/23, 35/31, 43/38 ^^E, ^F♭♭ E
19 223.5 25/22, 33/29 vvE♯, ^^F♭♭ ^E, v5F
20 235.3 39/34 vE♯, vvF♭ ^^E, v4F
21 247.1 E♯, vF♭ ^3E, v3F
22 258.8 29/25, 43/37 ^E♯, F♭ ^4E, vvF
23 270.6 ^^E♯, ^F♭ ^5E, vF
24 282.4 20/17 vvE𝄪, ^^F♭ F
25 294.1 vE𝄪, vvF ^F, v5G♭
26 305.9 31/26, 37/31 E𝄪, vF ^^F, v4G♭
27 317.6 6/5 F ^3F, v3G♭
28 329.4 23/19, 29/24 ^F, G♭♭♭ ^4F, vvG♭
29 341.2 28/23, 39/32 ^^F, ^G♭♭♭ ^5F, vG♭
30 352.9 38/31 vvF♯, ^^G♭♭♭ ^6F, G♭
31 364.7 21/17 vF♯, vvG♭♭ v5F♯, ^G♭
32 376.5 36/29 F♯, vG♭♭ v4F♯, ^^G♭
33 388.2 5/4 ^F♯, G♭♭ v3F♯, ^3G♭
34 400 ^^F♯, ^G♭♭ vvF♯, ^4G♭
35 411.8 vvF𝄪, ^^G♭♭ vF♯, ^5G♭
36 423.5 vF𝄪, vvG♭ F♯, v6G
37 435.3 F𝄪, vG♭ ^F♯, v5G
38 447.1 22/17 ^F𝄪, G♭ ^^F♯, v4G
39 458.8 ^^F𝄪, ^G♭ ^3F♯, v3G
40 470.6 21/16 vvF♯𝄪, ^^G♭ ^4F♯, vvG
41 482.4 33/25, 37/28, 41/31 vF♯𝄪, vvG ^5F♯, vG
42 494.1 F♯𝄪, vG G
43 505.9 G ^G, v5A♭
44 517.6 31/23 ^G, A♭♭♭ ^^G, v4A♭
45 529.4 19/14 ^^G, ^A♭♭♭ ^3G, v3A♭
46 541.2 26/19 vvG♯, ^^A♭♭♭ ^4G, vvA♭
47 552.9 11/8 vG♯, vvA♭♭ ^5G, vA♭
48 564.7 43/31 G♯, vA♭♭ ^6G, A♭
49 576.5 ^G♯, A♭♭ v5G♯, ^A♭
50 588.2 ^^G♯, ^A♭♭ v4G♯, ^^A♭
51 600 vvG𝄪, ^^A♭♭ v3G♯, ^3A♭
52 611.8 37/26 vG𝄪, vvA♭ vvG♯, ^4A♭
53 623.5 G𝄪, vA♭ vG♯, ^5A♭
54 635.3 ^G𝄪, A♭ G♯, v6A
55 647.1 16/11 ^^G𝄪, ^A♭ ^G♯, v5A
56 658.8 19/13, 41/28 vvG♯𝄪, ^^A♭ ^^G♯, v4A
57 670.6 28/19 vG♯𝄪, vvA ^3G♯, v3A
58 682.4 G♯𝄪, vA ^4G♯, vvA
59 694.1 A ^5G♯, vA
60 705.9 ^A, B♭♭♭ A
61 717.6 ^^A, ^B♭♭♭ ^A, v5B♭
62 729.4 32/21 vvA♯, ^^B♭♭♭ ^^A, v4B♭
63 741.2 43/28 vA♯, vvB♭♭ ^3A, v3B♭
64 752.9 17/11 A♯, vB♭♭ ^4A, vvB♭
65 764.7 ^A♯, B♭♭ ^5A, vB♭
66 776.5 ^^A♯, ^B♭♭ ^6A, B♭
67 788.2 41/26 vvA𝄪, ^^B♭♭ v5A♯, ^B♭
68 800 vA𝄪, vvB♭ v4A♯, ^^B♭
69 811.8 8/5 A𝄪, vB♭ v3A♯, ^3B♭
70 823.5 29/18, 37/23 ^A𝄪, B♭ vvA♯, ^4B♭
71 835.3 34/21 ^^A𝄪, ^B♭ vA♯, ^5B♭
72 847.1 31/19 vvA♯𝄪, ^^B♭ A♯, v6B
73 858.8 23/14 vA♯𝄪, vvB ^A♯, v5B
74 870.6 38/23, 43/26 A♯𝄪, vB ^^A♯, v4B
75 882.4 5/3 B ^3A♯, v3B
76 894.1 ^B, C♭♭ ^4A♯, vvB
77 905.9 ^^B, ^C♭♭ ^5A♯, vB
78 917.6 17/10 vvB♯, ^^C♭♭ B
79 929.4 vB♯, vvC♭ ^B, v5C
80 941.2 B♯, vC♭ ^^B, v4C
81 952.9 ^B♯, C♭ ^3B, v3C
82 964.7 ^^B♯, ^C♭ ^4B, vvC
83 976.5 44/25 vvB𝄪, ^^C♭ ^5B, vC
84 988.2 23/13 vB𝄪, vvC C
85 1000 41/23 B𝄪, vC ^C, v5D♭
86 1011.8 C ^^C, v4D♭
87 1023.5 ^C, D♭♭♭ ^3C, v3D♭
88 1035.3 20/11 ^^C, ^D♭♭♭ ^4C, vvD♭
89 1047.1 11/6 vvC♯, ^^D♭♭♭ ^5C, vD♭
90 1058.8 35/19 vC♯, vvD♭♭ ^6C, D♭
91 1070.6 13/7 C♯, vD♭♭ v5C♯, ^D♭
92 1082.4 43/23 ^C♯, D♭♭ v4C♯, ^^D♭
93 1094.1 32/17 ^^C♯, ^D♭♭ v3C♯, ^3D♭
94 1105.9 vvC𝄪, ^^D♭♭ vvC♯, ^4D♭
95 1117.6 21/11, 40/21 vC𝄪, vvD♭ vC♯, ^5D♭
96 1129.4 C𝄪, vD♭ C♯, v6D
97 1141.2 29/15 ^C𝄪, D♭ ^C♯, v5D
98 1152.9 37/19 ^^C𝄪, ^D♭ ^^C♯, v4D
99 1164.7 vvC♯𝄪, ^^D♭ ^3C♯, v3D
100 1176.5 vC♯𝄪, vvD ^4C♯, vvD
101 1188.2 C♯𝄪, vD ^5C♯, vD
102 1200 2/1 D D
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