25edt: Difference between revisions

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**Imported revision 543142144 - Original comment: **
 
+subsets and supersets; +see also
 
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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>2015-03-05 20:34:53 UTC</tt>.<br>
: The original revision id was <tt>543142144</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">This scale coincidentally turns out to be 16 equal divisions of a stretched octave (1217.25 cents) and a tritave twin of the Armodue/Hornbostel flat third-tone system (6th=1065.095 cents, squared=2130.19 cents=228.235 cents, cubed=1293.33 cents, fourth power=2358.425 cents=456.47 cents).
||~ Degree ||~ cents ||~ Armodue name ||
|| 1\25 || 76.08 || 1#/2bb ||
|| 2\25 || 152.16 || 1x/2b ||
|| 3\25 || 228.235 || 2 ||
|| 4\25 || 304.31 || 2#/3bb ||
|| 5\25 || 380.39 || 2x/3b ||
|| 6\25 || 456.47 || 3 ||
|| 7\25 || 532.55 || 3#/4b ||
|| 8\25 || 608.625 || 4 ||
|| 9\25 || 684.70 || 4#/5bb ||
|| 10\25 || 760.78 || 4x/5b ||
|| 11\25 || 836.86 || 5 ||
|| 12\25 || 912.94 || 5#/6bb ||
|| 13\25 || 989.02 || 5x/6b ||
|| 14\25 || 1065.095 || 6 ||
|| 15\25 || 1141.17 || 6#/7bb ||
|| 16\25 || 1217.25 || 6x/7b ||
|| 17\25 || 1293.33 || 7 ||
|| 18\25 || 1369.41 || 7#/8b ||
|| 19\25 || 1445.485 || 8 ||
|| 20\25 || 1521.56 || 8#/9bb ||
|| 21\25 || 1597.64 || 8x/9b ||
|| 22\25 || 1673.72 || 9 ||
|| 23\25 || 1749.80 || 9#/1bb ||
|| 24\25 || 1825.88 || 9x/1b ||
|| 25\25 || 1901.955 || 1 ||</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;25edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This scale coincidentally turns out to be 16 equal divisions of a stretched octave (1217.25 cents) and a tritave twin of the Armodue/Hornbostel flat third-tone system (6th=1065.095 cents, squared=2130.19 cents=228.235 cents, cubed=1293.33 cents, fourth power=2358.425 cents=456.47 cents).&lt;br /&gt;


== Theory ==
25edt corresponds to 15.7732…[[edo]], or 16 equal divisions of a stretched octave (1217.25{{c}}) and a tritave twin of the Armodue/Hornbostel flat third-tone system:
* 6th = 1065.095{{c}}
* squared = 2130.19{{c}} → 228.235{{c}}
* cubed = 1293.33{{c}}
* fourth power = 2358.425{{c}} → 456.47{{c}}


&lt;table class="wiki_table"&gt;
It can be used as a tuning for [[mavila]] and has an antidiatonic ([[2L&nbsp;5s]]) scale which approximates [[Pelog]] tunings in Indonesian gamelan music.
    &lt;tr&gt;
        &lt;th&gt;Degree&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Armodue name&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;76.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1#/2bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;152.16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1x/2b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;228.235&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;304.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2#/3bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;380.39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2x/3b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;456.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;532.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3#/4b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;608.625&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;684.70&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4#/5bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;760.78&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4x/5b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;836.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;912.94&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5#/6bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;989.02&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5x/6b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1065.095&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1141.17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6#/7bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1217.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6x/7b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1293.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1369.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7#/8b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1445.485&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1521.56&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8#/9bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1597.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8x/9b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1673.72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1749.80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9#/1bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1825.88&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9x/1b&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.955&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Harmonics ===
{{Harmonics in equal|25|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|25|3|1|intervals=integer|columns=11|start=12|collapsed=true|title=Approximation of harmonics in 25edt (continued)}}
 
=== Subsets and supersets ===
Since 25 factors into primes as 5<sup>2</sup>, 25edt contains [[5edt]] as its only nontrivial subset edt.
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Armodue name
|-
| 0
| 0.0
| 0.0
| 1
|-
| 1
| 76.1
| 52.0
| 1#/2bb
|-
| 2
| 152.2
| 104.0
| 1x/2b
|-
| 3
| 228.2
| 156.0
| 2
|-
| 4
| 304.3
| 208.0
| 2#/3bb
|-
| 5
| 380.4
| 260.0
| 2x/3b
|-
| 6
| 456.5
| 312.0
| 3
|-
| 7
| 532.5
| 364.0
| 3#/4b
|-
| 8
| 608.6
| 416.0
| 4
|-
| 9
| 684.7
| 468.0
| 4#/5bb
|-
| 10
| 760.8
| 520.0
| 4x/5b
|-
| 11
| 836.9
| 572.0
| 5
|-
| 12
| 912.9
| 624.0
| 5#/6bb
|-
| 13
| 989.0
| 676.0
| 5x/6b
|-
| 14
| 1065.1
| 728.0
| 6
|-
| 15
| 1141.2
| 780.0
| 6#/7bb
|-
| 16
| 1217.3
| 832.0
| 6x/7b
|-
| 17
| 1293.3
| 884.0
| 7
|-
| 18
| 1369.4
| 936.0
| 7#/8b
|-
| 19
| 1445.5
| 988.0
| 8
|-
| 20
| 1521.6
| 1040.0
| 8#/9bb
|-
| 21
| 1597.6
| 1092.0
| 8x/9b
|-
| 22
| 1673.7
| 1144.0
| 9
|-
| 23
| 1749.8
| 1196.0
| 9#/1bb
|-
| 24
| 1825.9
| 1248.0
| 9x/1b
|-
| 25
| 1902.0
| 1300.0
| 1
|}
 
== See also ==
* [[16edo]] – relative edo
* [[41ed6]] – relative ed6
* [[57ed12]] – relative ed12
 
{{Todo|expand}}
[[Category:Armodue]]

Latest revision as of 09:07, 27 May 2025

← 24edt 25edt 26edt →
Prime factorization 52
Step size 76.0782 ¢ 
Octave 16\25edt (1217.25 ¢)
Consistency limit 6
Distinct consistency limit 6

25 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 25edt or 25ed3), is a nonoctave tuning system that divides the interval of 3/1 into 25 equal parts of about 76.1 ¢ each. Each step represents a frequency ratio of 31/25, or the 25th root of 3.

Theory

25edt corresponds to 15.7732…edo, or 16 equal divisions of a stretched octave (1217.25 ¢) and a tritave twin of the Armodue/Hornbostel flat third-tone system:

  • 6th = 1065.095 ¢
  • squared = 2130.19 ¢ → 228.235 ¢
  • cubed = 1293.33 ¢
  • fourth power = 2358.425 ¢ → 456.47 ¢

It can be used as a tuning for mavila and has an antidiatonic (2L 5s) scale which approximates Pelog tunings in Indonesian gamelan music.

Harmonics

Approximation of harmonics in 25edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +17.3 +0.0 +34.5 +28.6 +17.3 -21.4 -24.3 +0.0 -30.2 +33.0 +34.5
Relative (%) +22.7 +0.0 +45.4 +37.6 +22.7 -28.1 -32.0 +0.0 -39.8 +43.4 +45.4
Steps
(reduced) 		16
(16) 		25
(0) 		32
(7) 		37
(12) 		41
(16) 		44
(19) 		47
(22) 		50
(0) 		52
(2) 		55
(5) 		57
(7)
Approximation of harmonics in 25edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -28.0 -4.1 +28.6 -7.1 -36.0 +17.3 -0.3 -13.0 -21.4 -25.8 -26.7
Relative (%) -36.8 -5.4 +37.6 -9.3 -47.3 +22.7 -0.4 -17.1 -28.1 -34.0 -35.1
Steps
(reduced) 		58
(8) 		60
(10) 		62
(12) 		63
(13) 		64
(14) 		66
(16) 		67
(17) 		68
(18) 		69
(19) 		70
(20) 		71
(21)

Subsets and supersets

Since 25 factors into primes as 52, 25edt contains 5edt as its only nontrivial subset edt.

Intervals

# Cents Hekts Armodue name
0 0.0 0.0 1
1 76.1 52.0 1#/2bb
2 152.2 104.0 1x/2b
3 228.2 156.0 2
4 304.3 208.0 2#/3bb
5 380.4 260.0 2x/3b
6 456.5 312.0 3
7 532.5 364.0 3#/4b
8 608.6 416.0 4
9 684.7 468.0 4#/5bb
10 760.8 520.0 4x/5b
11 836.9 572.0 5
12 912.9 624.0 5#/6bb
13 989.0 676.0 5x/6b
14 1065.1 728.0 6
15 1141.2 780.0 6#/7bb
16 1217.3 832.0 6x/7b
17 1293.3 884.0 7
18 1369.4 936.0 7#/8b
19 1445.5 988.0 8
20 1521.6 1040.0 8#/9bb
21 1597.6 1092.0 8x/9b
22 1673.7 1144.0 9
23 1749.8 1196.0 9#/1bb
24 1825.9 1248.0 9x/1b
25 1902.0 1300.0 1

See also

Retrieved from "https://en.xen.wiki/index.php?title=25edt&oldid=197813"