103edo: Difference between revisions

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**Imported revision 214557916 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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:xenwolf|xenwolf]] and made on <tt>2011-03-28 03:12:09 UTC</tt>.<br>
 
: The original revision id was <tt>214557916</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.
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>
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.
<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 //103 equal division// divides the [[octave]] into 103 equal parts of 11.650 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It tempers out 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.</pre></div>
 
<h4>Original HTML content:</h4>
=== Prime 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;103edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;103 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 103 equal parts of 11.650 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is a good &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle&lt;/a&gt; tuning, especially for the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;benediction&lt;/a&gt;, one of the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; extensions of miracle. It tempers out 78732/78125 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;; 351/350 in the 13-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|103|intervals=prime}}
 
=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].
 
== Intervals ==
{{Main|Table of 103edo intervals}}
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
 
=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}
 
* 103et (103h val) has lower absolute errors than any smaller equal temperaments in the [[13-limit|13-]], [[17-limit|17-]], and [[19-limit]]s, being beaten by [[111edo|111]] in all of them.
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]
 
[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]

Latest revision as of 13:32, 13 March 2026

← 102edo 103edo 104edo →
Prime factorization 103 (prime)
Step size 11.6505 ¢ 
Fifth 60\103 (699.029 ¢)
Semitones (A1:m2) 8:9 (93.2 ¢ : 104.9 ¢)
Consistency limit 7
Distinct consistency limit 7

103 equal divisions of the octave (abbreviated 103edo or 103ed2), also called 103-tone equal temperament (103tet) or 103 equal temperament (103et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 103 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of 21/103, or the 103rd root of 2.

Theory

In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%). Its closest zeta peak index, 596zpi, stretches the octave by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good miracle tuning, especially for the 7-limit, and for benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024, and 2401/2400 in the 7-limit; 243/242, 441/440, and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.

Prime harmonics

Approximation of prime harmonics in 103edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -2.93 -1.85 -1.84 -3.75 -1.69 -0.10 +5.40 +0.85 -4.33 -3.29
Relative (%) +0.0 -25.1 -15.9 -15.8 -32.1 -14.5 -0.9 +46.3 +7.3 -37.2 -28.2
Steps
(reduced) 		103
(0) 		163
(60) 		239
(33) 		289
(83) 		356
(47) 		381
(72) 		421
(9) 		438
(26) 		466
(54) 		500
(88) 		510
(98)

Subsets and supersets

103edo is the 27th prime edo, following 101edo and before 107edo.

Intervals

Main article: Table of 103edo intervals

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 103edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 103edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.012 0.1
13/7, 14/13 0.143 1.2
13/10, 20/13 0.155 1.3
11/6, 12/11 0.819 7.0
15/11, 22/15 1.028 8.8
5/3, 6/5 1.078 9.3
7/6, 12/7 1.090 9.4
13/12, 24/13 1.233 10.6
13/8, 16/13 1.693 14.5
7/4, 8/7 1.836 15.8
5/4, 8/5 1.848 15.9
11/10, 20/11 1.897 16.3
11/7, 14/11 1.910 16.4
13/11, 22/13 2.052 17.6
11/9, 18/11 2.107 18.1
3/2, 4/3 2.926 25.1
15/14, 28/15 2.938 25.2
15/13, 26/15 3.081 26.4
11/8, 16/11 3.745 32.1
9/5, 10/9 4.004 34.4
9/7, 14/9 4.016 34.5
13/9, 18/13 4.159 35.7
15/8, 16/15 4.774 41.0
9/8, 16/9 5.799 49.8
15-odd-limit intervals in 103edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.012 0.1
13/7, 14/13 0.143 1.2
13/10, 20/13 0.155 1.3
11/6, 12/11 0.819 7.0
15/11, 22/15 1.028 8.8
5/3, 6/5 1.078 9.3
7/6, 12/7 1.090 9.4
13/12, 24/13 1.233 10.6
13/8, 16/13 1.693 14.5
7/4, 8/7 1.836 15.8
5/4, 8/5 1.848 15.9
11/10, 20/11 1.897 16.3
11/7, 14/11 1.910 16.4
13/11, 22/13 2.052 17.6
11/9, 18/11 2.107 18.1
3/2, 4/3 2.926 25.1
15/14, 28/15 2.938 25.2
15/13, 26/15 3.081 26.4
11/8, 16/11 3.745 32.1
9/5, 10/9 4.004 34.4
9/7, 14/9 4.016 34.5
13/9, 18/13 4.159 35.7
15/8, 16/15 4.774 41.0
9/8, 16/9 5.852 50.2

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
596zpi 102.93663 11.657658 8.54351 5.620365 1.340775 18.270998 1200.738751 0.738751 15 15

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-163 103⟩ [⟨103 166]] +0.923 0.924 7.92
2.3.5 78732/78125, 34171875/33554432 [⟨103 166 239]] +0.881 0.757 6.49
2.3.5.7 225/224, 1029/1024, 78732/78125 [⟨103 166 239 289]] +0.824 0.663 5.68
2.3.5.7.11 225/224, 243/242, 385/384, 43923/43750 [⟨103 166 239 289 356]] +0.876 0.602 5.16
2.3.5.7.11.13 225/224, 243/242, 351/350, 385/384, 847/845 [⟨103 166 239 289 356 381]] +0.806 0.571 4.90
2.3.5.7.11.13.17 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 [⟨103 166 239 289 356 381 421]] +0.694 0.595 5.10
  • 103et (103h val) has lower absolute errors than any smaller equal temperaments in the 13-, 17-, and 19-limits, being beaten by 111 in all of them.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 3\103 34.951 1990656/1953125 Gammy
1 5\103 58.252 27/26 Hemisecordite
1 9\103 104.854 17/16 Septendesemi
1 10\103 116.505 15/14~16/15 Miracle / benediction
1 16\103 186.408 10/9 Mintone
1 20\103 233.010 8/7 Slendric
1 21\103 244.660 15/13 Subsemifourth
1 26\103 303.013 25/21 Quinmite
1 31\103 361.165 16/13 Phicordial
1 37\103 431.06 77/60 Lockerbie
1 38\103 442.708 162/125 Sensei
1 39\103 454.369 13/10 Fibo
1 40\103 466.019 55/42 Hemiseptisix
1 42\103 489.320 65/49 Catafourth
1 45\103 524.272 65/48 Widefourth
1 47\103 547.573 11/8 Heinz
1 48\103 559.223 242/175 Tritriple
1 50\103 582.524 7/5 Neptune

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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