85edo: 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:genewardsmith|genewardsmith]] and made on <tt>2012-05-28 14:03:03 UTC</tt>.<br>
 
: The original revision id was <tt>340172196</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
Since 85 {{=}} 5 × 17, 85edo shares the same 3.9-cent-sharp fifth as [[17edo|17]], [[34edo|34]], and [[68edo|68]]. 3/1 is therefore divisible into 5, and the [[patent val]] correspondingly supports [[magic]], [[tempering out]] [[3125/3072]] in the [[5-limit]] and [[225/224]], [[245/243]], and [[875/864]] in the [[7-limit]]. It tempers out [[100/99]] and [[385/384]] in the [[11-limit]], supporting 11-limit magic, and [[847/845]], [[1188/1183]], and [[1575/1573]] in the 13-limit. It provides the [[optimal patent val]] for the 13-limit 36ce & 49f temperament tempering out 100/99, 540/539, 847/845 and 1575/1573. The 85c val, with a very sharp 5/4 of 395.3{{c}}, supports 7-limit [[myna]].
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>
=== Odd harmonics ===
<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">The 85 equal temperament divides the octave in 85 equal parts of 14.458 cents each. 85 = 5 * 17, and it shares the same 3.9 cent sharp fifth as 17, 34, and 68. The patent val tempers out 3125/3072 in the 5-limit and 875/864, 245/243 and 225/224 in the 7-limit, so that it supports magic. It tempers out 100/99 and 245/243 in the 11-limit, supporting 11-limit magic, and 1188/1183, 847/845 and 1575/1573 in the 13-limit. It provides the optimal patent val for the 13-limit temperament tempering out 100/99, 540/539, 847/845 and 1575/1573.</pre></div>
{{Harmonics in equal|85}}
<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;85edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 85 equal temperament divides the octave in 85 equal parts of 14.458 cents each. 85 = 5 * 17, and it shares the same 3.9 cent sharp fifth as 17, 34, and 68. The patent val tempers out 3125/3072 in the 5-limit and 875/864, 245/243 and 225/224 in the 7-limit, so that it supports magic. It tempers out 100/99 and 245/243 in the 11-limit, supporting 11-limit magic, and 1188/1183, 847/845 and 1575/1573 in the 13-limit. It provides the optimal patent val for the 13-limit temperament tempering out 100/99, 540/539, 847/845 and 1575/1573.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
85edo contains [[5edo]] and [[17edo]] as subsets. [[255edo]], which triples it, is a notable tuning.
 
== Interval table ==
{{Interval table}}
 
== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[magic]]): 7 3 7 7 3 7 10 7 7 3 7 10 7
 
[[Category:Magic]]
 
== Instruments ==
 
A [[Lumatone mapping for 85edo]] is available.
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/DbVsxzfKr7I ''microtonal improvisation in 85edo''] (2025)

Latest revision as of 00:46, 14 May 2026

← 84edo 85edo 86edo →
Prime factorization 5 × 17
Step size 14.1176 ¢ 
Fifth 50\85 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 10:5 (141.2 ¢ : 70.59 ¢)
Consistency limit 3
Distinct consistency limit 3

85 equal divisions of the octave (abbreviated 85edo or 85ed2), also called 85-tone equal temperament (85tet) or 85 equal temperament (85et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 85 equal parts of about 14.1 ¢ each. Each step represents a frequency ratio of 21/85, or the 85th root of 2.

Theory

Since 85 = 5 × 17, 85edo shares the same 3.9-cent-sharp fifth as 17, 34, and 68. 3/1 is therefore divisible into 5, and the patent val correspondingly supports magic, tempering out 3125/3072 in the 5-limit and 225/224, 245/243, and 875/864 in the 7-limit. It tempers out 100/99 and 385/384 in the 11-limit, supporting 11-limit magic, and 847/845, 1188/1183, and 1575/1573 in the 13-limit. It provides the optimal patent val for the 13-limit 36ce & 49f temperament tempering out 100/99, 540/539, 847/845 and 1575/1573. The 85c val, with a very sharp 5/4 of 395.3 ¢, supports 7-limit myna.

Odd harmonics

Approximation of odd harmonics in 85edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.93 -5.14 +5.29 -6.26 -0.73 +6.53 -1.21 -6.13 -1.04 -4.90 +7.02
Relative (%) +27.8 -36.4 +37.5 -44.4 -5.2 +46.3 -8.6 -43.4 -7.4 -34.7 +49.7
Steps
(reduced) 		135
(50) 		197
(27) 		239
(69) 		269
(14) 		294
(39) 		315
(60) 		332
(77) 		347
(7) 		361
(21) 		373
(33) 		385
(45)

Subsets and supersets

85edo contains 5edo and 17edo as subsets. 255edo, which triples it, is a notable tuning.

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 14.1 ^D, v4E♭
2 28.2 ^^D, v3E♭
3 42.4 41/40 ^3D, vvE♭
4 56.5 30/29, 31/30, 32/31 ^4D, vE♭
5 70.6 ^5D, E♭
6 84.7 v4D♯, ^E♭
7 98.8 37/35 v3D♯, ^^E♭
8 112.9 16/15, 31/29 vvD♯, ^3E♭
9 127.1 14/13 vD♯, ^4E♭
10 141.2 13/12, 38/35 D♯, v5E
11 155.3 23/21, 35/32 ^D♯, v4E
12 169.4 32/29 ^^D♯, v3E
13 183.5 ^3D♯, vvE
14 197.6 37/33 ^4D♯, vE
15 211.8 26/23, 35/31 E
16 225.9 33/29 ^E, v4F
17 240 ^^E, v3F
18 254.1 22/19, 37/32 ^3E, vvF
19 268.2 7/6 ^4E, vF
20 282.4 20/17, 33/28 F
21 296.5 19/16 ^F, v4G♭
22 310.6 ^^F, v3G♭
23 324.7 29/24, 35/29, 41/34 ^3F, vvG♭
24 338.8 28/23 ^4F, vG♭
25 352.9 38/31 ^5F, G♭
26 367.1 26/21 v4F♯, ^G♭
27 381.2 v3F♯, ^^G♭
28 395.3 vvF♯, ^3G♭
29 409.4 19/15 vF♯, ^4G♭
30 423.5 23/18, 37/29 F♯, v5G
31 437.6 9/7 ^F♯, v4G
32 451.8 ^^F♯, v3G
33 465.9 38/29 ^3F♯, vvG
34 480 29/22, 37/28 ^4F♯, vG
35 494.1 G
36 508.2 ^G, v4A♭
37 522.4 ^^G, v3A♭
38 536.5 15/11 ^3G, vvA♭
39 550.6 11/8 ^4G, vA♭
40 564.7 18/13 ^5G, A♭
41 578.8 v4G♯, ^A♭
42 592.9 31/22 v3G♯, ^^A♭
43 607.1 vvG♯, ^3A♭
44 621.2 vG♯, ^4A♭
45 635.3 13/9 G♯, v5A
46 649.4 16/11 ^G♯, v4A
47 663.5 22/15 ^^G♯, v3A
48 677.6 ^3G♯, vvA
49 691.8 ^4G♯, vA
50 705.9 A
51 720 ^A, v4B♭
52 734.1 29/19 ^^A, v3B♭
53 748.2 37/24 ^3A, vvB♭
54 762.4 14/9 ^4A, vB♭
55 776.5 36/23 ^5A, B♭
56 790.6 30/19 v4A♯, ^B♭
57 804.7 35/22 v3A♯, ^^B♭
58 818.8 vvA♯, ^3B♭
59 832.9 21/13 vA♯, ^4B♭
60 847.1 31/19 A♯, v5B
61 861.2 23/14 ^A♯, v4B
62 875.3 ^^A♯, v3B
63 889.4 ^3A♯, vvB
64 903.5 32/19 ^4A♯, vB
65 917.6 17/10 B
66 931.8 12/7 ^B, v4C
67 945.9 19/11 ^^B, v3C
68 960 ^3B, vvC
69 974.1 ^4B, vC
70 988.2 23/13 C
71 1002.4 ^C, v4D♭
72 1016.5 ^^C, v3D♭
73 1030.6 29/16 ^3C, vvD♭
74 1044.7 ^4C, vD♭
75 1058.8 24/13, 35/19 ^5C, D♭
76 1072.9 13/7 v4C♯, ^D♭
77 1087.1 15/8 v3C♯, ^^D♭
78 1101.2 vvC♯, ^3D♭
79 1115.3 vC♯, ^4D♭
80 1129.4 C♯, v5D
81 1143.5 29/15, 31/16 ^C♯, v4D
82 1157.6 ^^C♯, v3D
83 1171.8 ^3C♯, vvD
84 1185.9 ^4C♯, vD
85 1200 2/1 D

Scales

Instruments

A Lumatone mapping for 85edo is available.

Music

Bryan Deister
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