28edt: 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>
'''28edt''' divides the [[3/1|third harmonic]] into 28 [[equal]] parts of 67.9 [[cent]]s each. It is the simplest [[edt]] which intelligibly doubles the Anti-BP harmonic chain{{clarify}}. One step is very close to every third step of [[53edo]]. It tempers out the [[schisma]] in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP.  It also supports [[deneb]] temperament in the 3.5.11 subgroup, equating 4 [[11/9]]s with [[11/5]].
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-20 18:11:30 UTC</tt>.<br>
: The original revision id was <tt>596295840</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 is the simplest edt which intelligibly doubles the Anti-BP harmonic chain. It is a schematic temperament, albeit a really strange one due to being a multiple of 4edt and being index-2 in BP.


||~ Degree ||~ cents ||~ notation ||
== Harmonics ==
|| 1/28 || 67.93 || Cq/D@y ||
{{Harmonics in equal
|| 2/28 || 135.85 || Cp/D@ ||
| steps = 28
|| 3/28 || 203.78 || Cp/Dy ||
| num = 3
|| 4/28 || 271.71 || D ||
| denom = 1
|| 5/28 || 339.635 || Dp/Ey ||
| intervals = integer
|| 6/28 || 407.56 || E ||
}}
|| 7/28 || 475.49 || Epq/F@y ||
{{Harmonics in equal
|| 8/28 || 543.42 || Ep/F@ ||
| steps = 28
|| 9/28 || 611.34 || Eq/Fy ||
| num = 3
|| 10/28 || 679.27 || F ||
| denom = 1
|| 11/28 || 747.2 || Fq/Gy ||
| start = 12
|| 12/28 || 815.12 || G ||
| collapsed = 1
|| 13/28 || 883.05 || Gq/Hy ||
| intervals = integer
|| 14/28 || 950.98 || Gp/H@ ||
}}
|| 15/28 || 1018.9 || Gpq/H@y ||
|| 16/28 || 1086.83 || H ||
|| 17/28 || 1154.76 || Hq/Jy ||
|| 18/28 || 1222.685 || J ||
|| 19/28 || 1290.61 || Jq/A@y ||
|| 20/28 || 1358.54 || Jp/A@ ||
|| 21/28 || 1426.47 || Jpq/Ay ||
|| 22/28 || 1494.39 || A ||
|| 23/28 || 1562.32 || Aq/B@y ||
|| 24/28 || 1630.25 || Ap/B@ ||
|| 25/28 || 1698.17 || Apq/By ||
|| 26/28 || 1766.1 || B ||
|| 27/28 || 1834.03 || Bq/Cy ||
|| 28/28 || 1901.955 || C ||</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;28edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This is the simplest edt which intelligibly doubles the Anti-BP harmonic chain. It is a schematic temperament, albeit a really strange one due to being a multiple of 4edt and being index-2 in BP.&lt;br /&gt;
&lt;br /&gt;


== Intervals ==
{| class="wikitable right-all left-4"
|-
! Steps
! [[cent]]s
! [[hekt]]s
! notation
|-
| 1
| 67.927
| 46.429
| Cq/D\\y
|-
| 2
| 135.854
| 92.857
| Cp/D\\
|-
| 3
| 203.781
| 139.286
| Cpq/Dy
|-
| 4
| 271.708
| 185.714
| D
|-
| 5
| 339.635
| 232.143
| Dp/Ey
|-
| 6
| 407.562
| 278.571
| E
|-
| 7
| 475.489
| 325.000
| Epq/F\\y
|-
| 8
| 543.416
| 371.429
| Ep/F\\
|-
| 9
| 611.343
| 417.857
| Eq/Fy
|-
| 10
| 679.270
| 464.286
| F
|-
| 11
| 747.197
| 510.714
| Fq/Gy
|-
| 12
| 815.124
| 557.143
| G
|-
| 13
| 883.051
| 603.571
| Gq/H\\y
|-
| 14
| 950.978
| 650.000
| Gp/H\\
|-
| 15
| 1018.904
| 696.429
| Gpq/Hy
|-
| 16
| 1086.831
| 742.857
| H
|-
| 17
| 1154.758
| 789.286
| Hq/Jy
|-
| 18
| 1222.685
| 835.714
| J
|-
| 19
| 1290.612
| 882.143
| Jq/A\\y
|-
| 20
| 1358.539
| 928.571
| Jp/A\\
|-
| 21
| 1426.466
| 975.000
| Jpq/Ay
|-
| 22
| 1494.393
| 1021.429
| A
|-
| 23
| 1562.320
| 1067.857
| Aq/B\\y
|-
| 24
| 1630.247
| 1114.286
| Ap/B\\
|-
| 25
| 1698.174
| 1160.714
| Apq/By
|-
| 26
| 1766.101
| 1207.143
| B
|-
| 27
| 1834.028
| 1253.571
| Bq/Cy
|-
| 28
| 1901.955
| 1300.000
| C
|}


&lt;table class="wiki_table"&gt;
== Music ==
    &lt;tr&gt;
* [http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry])
        &lt;th&gt;Degree&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;notation&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;67.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cq/D@y&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135.85&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cp/D@&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;203.78&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cp/Dy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;339.635&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Dp/Ey&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;407.56&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;475.49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Epq/F@y&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;543.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ep/F@&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;611.34&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Eq/Fy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;679.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;747.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Fq/Gy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;815.12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;883.05&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gq/Hy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;950.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gp/H@&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1018.9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gpq/H@y&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1086.83&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;H&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1154.76&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Hq/Jy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1222.685&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;J&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1290.61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Jq/A@y&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1358.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Jp/A@&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1426.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Jpq/Ay&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1494.39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1562.32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Aq/B@y&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1630.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ap/B@&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1698.17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Apq/By&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;26/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1766.1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1834.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Bq/Cy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;28/28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.955&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
{{todo|expand}}

Latest revision as of 15:31, 31 July 2025

← 27edt 28edt 29edt →
Prime factorization 22 × 7
Step size 67.927 ¢ 
Octave 18\28edt (1222.69 ¢) (→ 9\14edt)
Consistency limit 3
Distinct consistency limit 3

28edt divides the third harmonic into 28 equal parts of 67.9 cents each. It is the simplest edt which intelligibly doubles the Anti-BP harmonic chain[clarification needed]. One step is very close to every third step of 53edo. It tempers out the schisma in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP. It also supports deneb temperament in the 3.5.11 subgroup, equating 4 11/9s with 11/5.

Harmonics

Approximation of harmonics in 28edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +22.7 +0.0 -22.6 -1.3 +22.7 +27.5 +0.1 +0.0 +21.4 -7.8 -22.6
Relative (%) +33.4 +0.0 -33.2 -1.9 +33.4 +40.5 +0.2 +0.0 +31.5 -11.4 -33.2
Steps
(reduced) 		18
(18) 		28
(0) 		35
(7) 		41
(13) 		46
(18) 		50
(22) 		53
(25) 		56
(0) 		59
(3) 		61
(5) 		63
(7)
Approximation of harmonics in 28edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -25.3 -17.7 -1.3 +22.8 -14.2 +22.7 -3.0 -23.9 +27.5 +14.9 +5.9
Relative (%) -37.2 -26.1 -1.9 +33.6 -20.9 +33.4 -4.4 -35.1 +40.5 +22.0 +8.7
Steps
(reduced) 		65
(9) 		67
(11) 		69
(13) 		71
(15) 		72
(16) 		74
(18) 		75
(19) 		76
(20) 		78
(22) 		79
(23) 		80
(24)

Intervals

Steps cents hekts notation
1 67.927 46.429 Cq/D\\y
2 135.854 92.857 Cp/D\\
3 203.781 139.286 Cpq/Dy
4 271.708 185.714 D
5 339.635 232.143 Dp/Ey
6 407.562 278.571 E
7 475.489 325.000 Epq/F\\y
8 543.416 371.429 Ep/F\\
9 611.343 417.857 Eq/Fy
10 679.270 464.286 F
11 747.197 510.714 Fq/Gy
12 815.124 557.143 G
13 883.051 603.571 Gq/H\\y
14 950.978 650.000 Gp/H\\
15 1018.904 696.429 Gpq/Hy
16 1086.831 742.857 H
17 1154.758 789.286 Hq/Jy
18 1222.685 835.714 J
19 1290.612 882.143 Jq/A\\y
20 1358.539 928.571 Jp/A\\
21 1426.466 975.000 Jpq/Ay
22 1494.393 1021.429 A
23 1562.320 1067.857 Aq/B\\y
24 1630.247 1114.286 Ap/B\\
25 1698.174 1160.714 Apq/By
26 1766.101 1207.143 B
27 1834.028 1253.571 Bq/Cy
28 1901.955 1300.000 C

Music

Retrieved from "https://en.xen.wiki/index.php?title=28edt&oldid=205589"