20edf: Difference between revisions

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Wikispaces>keenanpepper
**Imported revision 288541998 - Original comment: **
 
Carlos Gamma as we know today is non-octave, even tho it was originally intended to include the octaves
 
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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:keenanpepper|keenanpepper]] and made on <tt>2011-12-27 01:13:55 UTC</tt>.<br>
 
: The original revision id was <tt>288541998</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
20edf corresponds to 34.1902edo. It is closely related to [[Carlos Gamma]] and the [[gammic]] temperament, which adds an independent dimension for the [[2/1|octave]] (although strictly speaking, the "canonical" optimized Carlos Gamma tuning is not exactly 20edf, with its fifth stretched by the microscopic amount of 0.016{{c}}). It very accurately represents the intervals [[5/4]], with 11 steps, and [[17/16]], with 3 steps.
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>
=== 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">Redirects to Carlos Gamma.</pre></div>
{{Harmonics in equal|20|3|2|columns=11}}
<h4>Original HTML content:</h4>
{{Harmonics in equal|20|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 20edf (continued)}}
<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;20edf&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Redirects to Carlos Gamma.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Intervals ==
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
{| class="wikitable mw-collapsible"
|+ Intervals of 20edf
|-
!Degrees
!3/2.5/4.17/16 interpretation
!Cents
|-
|1
|51/50
|35.1
|-
|2
|25/24
|70.2
|-
|3
|17/16
|105.29
|-
|4
|625/576, 867/800
|140.39
|-
|5
|320/289, 425/384
|175.49
|-
|6
|96/85
|210.59
|-
|7
|144/125
|245.68
|-
|8
|20/17
|280.78
|-
|9
|6/5
|315.88
|-
|10
|153/125, 125/102
|350.98
|-
|11
|5/4
|386.075
|-
|12
|51/40
|421.17
|-
|13
|125/96
|456.27
|-
|14
|85/64
|491.37
|-
|15
|576/425, 867/640
|526.47
|-
|16
|400/289, 864/625
|561.56
|-
|17
|24/17
|596.66
|-
|18
|36/25
|631.76
|-
|19
|25/17
|666.86
|-
|'''20'''
|'''3/2'''
|'''701.955'''
|-
|21
|153/100
|737.05
|-
|22
|25/16
|772.15
|-
|23
|51/32
|807.25
|-
|24
|625/384
|842.35
|-
|25
|425/256, 480/289
|877.44
|-
|26
|144/85
|912.54
|-
|27
|216/125
|947.64
|-
|28
|30/17
|982.74
|-
|29
|9/5
|1017.835
|-
|30
|125/68
|1052.93
|-
|31
|15/8
|1088.03
|-
|32
|153/80
|1123.13
|-
|33
|125/64
|1158.23
|-
|34
|255/128
|1193.32
|-
|35
|864/425
|1228.42
|-
|36
|600/289
|1263.52
|-
|37
|36/17
|1298.62
|-
|38
|54/25
|1333.715
|-
|39
|75/34
|1368.81
|-
|'''40'''
|'''9/4'''
|'''1403.91'''
|}
 
{{stub}}

Latest revision as of 11:40, 6 August 2025

← 19edf 20edf 21edf →
Prime factorization 22 × 5
Step size 35.0978 ¢ 
Octave 34\20edf (1193.32 ¢) (→ 17\10edf)
Twelfth 54\20edf (1895.28 ¢) (→ 27\10edf)
Consistency limit 7
Distinct consistency limit 7

20 equal divisions of the perfect fifth (abbreviated 20edf or 20ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 20 equal parts of about 35.1 ¢ each. Each step represents a frequency ratio of (3/2)1/20, or the 20th root of 3/2.

Theory

20edf corresponds to 34.1902edo. It is closely related to Carlos Gamma and the gammic temperament, which adds an independent dimension for the octave (although strictly speaking, the "canonical" optimized Carlos Gamma tuning is not exactly 20edf, with its fifth stretched by the microscopic amount of 0.016 ¢). It very accurately represents the intervals 5/4, with 11 steps, and 17/16, with 3 steps.

Harmonics

Approximation of harmonics in 20edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.7 -6.7 -13.4 -13.6 -13.4 +0.6 +15.1 -13.4 +14.8 -9.8 +15.1
Relative (%) -19.0 -19.0 -38.0 -38.7 -38.0 +1.6 +42.9 -38.0 +42.3 -27.9 +42.9
Steps
(reduced) 		34
(14) 		54
(14) 		68
(8) 		79
(19) 		88
(8) 		96
(16) 		103
(3) 		108
(8) 		114
(14) 		118
(18) 		123
(3)
Approximation of harmonics in 20edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.9 -6.1 +14.8 +8.4 +8.7 +15.1 -8.3 +8.2 -6.1 -16.5 +11.9 +8.4
Relative (%) +48.1 -17.4 +42.3 +23.9 +24.9 +42.9 -23.8 +23.2 -17.4 -46.9 +33.8 +23.9
Steps
(reduced) 		127
(7) 		130
(10) 		134
(14) 		137
(17) 		140
(0) 		143
(3) 		145
(5) 		148
(8) 		150
(10) 		152
(12) 		155
(15) 		157
(17)

Intervals

The first steps up to two just perfect fifths should give a feeling of the granularity of this system…

Intervals of 20edf
Degrees 3/2.5/4.17/16 interpretation Cents
1 51/50 35.1
2 25/24 70.2
3 17/16 105.29
4 625/576, 867/800 140.39
5 320/289, 425/384 175.49
6 96/85 210.59
7 144/125 245.68
8 20/17 280.78
9 6/5 315.88
10 153/125, 125/102 350.98
11 5/4 386.075
12 51/40 421.17
13 125/96 456.27
14 85/64 491.37
15 576/425, 867/640 526.47
16 400/289, 864/625 561.56
17 24/17 596.66
18 36/25 631.76
19 25/17 666.86
20 3/2 701.955
21 153/100 737.05
22 25/16 772.15
23 51/32 807.25
24 625/384 842.35
25 425/256, 480/289 877.44
26 144/85 912.54
27 216/125 947.64
28 30/17 982.74
29 9/5 1017.835
30 125/68 1052.93
31 15/8 1088.03
32 153/80 1123.13
33 125/64 1158.23
34 255/128 1193.32
35 864/425 1228.42
36 600/289 1263.52
37 36/17 1298.62
38 54/25 1333.715
39 75/34 1368.81
40 9/4 1403.91
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