20edf: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''20edf''' is very close to the [[Carlos Gamma]] tuning.
{{ED intro}}


==Intervals==
== Theory ==
20edf corresponds to 34.1902edo. It is closely related to [[Carlos Gamma]], where Carlos Gamma, and similarly the [[gammic]] temperament, can be seen as 20edf with an independent dimension for 2 (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.
 
=== Harmonics ===
{{Harmonics in equal|20|3|2|columns=11}}
{{Harmonics in equal|20|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 20edf (continued)}}
 
== Intervals ==
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
Line 13: Line 20:
|35.1
|35.1
|-
|-
| |2
|2
| |70.2
|70.2
|-
|-
| |3
|3
| |105.29
|105.29
|-
|-
| |4
|4
| |140.39
|140.39
|-
|-
| |5
|5
| |175.49
|175.49
|-
|-
| |6
|6
| |210.59
|210.59
|-
|-
| |7
|7
| |245.68
|245.68
|-
|-
| |8
|8
| |280.78
|280.78
|-
|-
| |9
|9
| |315.88
|315.88
|-
|-
| |10
|10
| |350.98
|350.98
|-
|-
| |11
|11
| |386.075
|386.075
|-
|-
| |12
|12
| |421.17
|421.17
|-
|-
| |13
|13
| |456.27
|456.27
|-
|-
| |14
|14
| |491.37
|491.37
|-
|-
| |15
|15
| |526.47
|526.47
|-
|-
| |16
|16
| |561.56
|561.56
|-
|-
| |17
|17
| |596.66
|596.66
|-
|-
| |18
|18
| |631.76
|631.76
|-
|-
| |19
|19
| |666.86
|666.86
|-
|-
| |20
|20
| |701.955
|701.955
|-
|-
|21
|21
Line 130: Line 137:
|1403.91
|1403.91
|}
|}
== Harmonics ==
{{Harmonics in equal|20|3|2|intervals=prime}}
{{Harmonics in equal|20|3|2|start=12|collapsed=1|intervals=prime}}


{{stub}}
{{stub}}

Revision as of 20:00, 5 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, where Carlos Gamma, and similarly the gammic temperament, can be seen as 20edf with an independent dimension for 2 (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 Cents
1 35.1
2 70.2
3 105.29
4 140.39
5 175.49
6 210.59
7 245.68
8 280.78
9 315.88
10 350.98
11 386.075
12 421.17
13 456.27
14 491.37
15 526.47
16 561.56
17 596.66
18 631.76
19 666.86
20 701.955
21 737.05
22 772.15
23 807.25
24 842.35
25 877.44
26 912.54
27 947.64
28 982.74
29 1017.835
30 1052.93
31 1088.03
32 1123.13
33 1158.23
34 1193.32
35 1228.42
36 1263.52
37 1298.62
38 1333.715
39 1368.81
40 1403.91
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