23ed6: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
mNo edit summary |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
23ed6 is similar to [[9edo]], but has the 6th harmonic tuned just instead of the [[2/1|octave]], which stretches the octave by about 13.8{{c}}. It also approximates [[Pelog]] tunings in Indonesian gamelan music very well, since Pelog is well-approximated by [[9edo]]. | |||
=== Harmonics === | |||
{{Harmonics in equal|23|6|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|23|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23ed6 (continued)}} | |||
=== Subsets and supersets === | |||
23ed6 is the 9th [[prime equal division|prime ed6]], following [[19ed6]] and before [[29ed6]], so it does not contain any nontrivial subset ed6's. | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
== | == Scales == | ||
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] | |||
== See also == | |||
* [[9edo]] – relative edo | |||
* [[14edt]] – relative edt | |||
* [[32ed12]] – relative ed12 | |||
{{Todo|expand}} | |||
[[Category:Pelog]] | |||
| | |||
}} | |||
Latest revision as of 09:00, 27 September 2025
| ← 22ed6 | 23ed6 | 24ed6 → |
23 equal divisions of the 6th harmonic (abbreviated 23ed6) is a nonoctave tuning system that divides the interval of 6/1 into 23 equal parts of about 135 ¢ each. Each step represents a frequency ratio of 61/23, or the 23rd root of 6.
Theory
23ed6 is similar to 9edo, but has the 6th harmonic tuned just instead of the octave, which stretches the octave by about 13.8 ¢. It also approximates Pelog tunings in Indonesian gamelan music very well, since Pelog is well-approximated by 9edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +13.8 | -13.8 | +27.6 | +45.9 | +0.0 | +2.9 | +41.4 | -27.6 | +59.7 | +29.6 | +13.8 |
| Relative (%) | +10.2 | -10.2 | +20.5 | +34.0 | +0.0 | +2.1 | +30.7 | -20.5 | +44.3 | +21.9 | +10.2 | |
| Steps (reduced) |
9 (9) |
14 (14) |
18 (18) |
21 (21) |
23 (0) |
25 (2) |
27 (4) |
28 (5) |
30 (7) |
31 (8) |
32 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.1 | +16.7 | +32.1 | +55.2 | -49.7 | -13.8 | +27.5 | -61.3 | -10.9 | +43.4 | -33.6 | +27.6 |
| Relative (%) | +7.5 | +12.4 | +23.8 | +41.0 | -36.9 | -10.2 | +20.4 | -45.5 | -8.1 | +32.2 | -24.9 | +20.5 | |
| Steps (reduced) |
33 (10) |
34 (11) |
35 (12) |
36 (13) |
36 (13) |
37 (14) |
38 (15) |
38 (15) |
39 (16) |
40 (17) |
40 (17) |
41 (18) | |
Subsets and supersets
23ed6 is the 9th prime ed6, following 19ed6 and before 29ed6, so it does not contain any nontrivial subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 134.9 | 13/12, 14/13 |
| 2 | 269.7 | 7/6 |
| 3 | 404.6 | 14/11, 19/15, 24/19 |
| 4 | 539.5 | 11/8, 15/11, 19/14 |
| 5 | 674.3 | 22/15 |
| 6 | 809.2 | 8/5, 19/12 |
| 7 | 944.1 | 12/7, 19/11 |
| 8 | 1078.9 | 13/7, 15/8 |
| 9 | 1213.8 | |
| 10 | 1348.7 | 13/6, 24/11 |
| 11 | 1483.5 | |
| 12 | 1618.4 | 23/9 |
| 13 | 1753.3 | 11/4 |
| 14 | 1888.1 | |
| 15 | 2023 | 16/5 |
| 16 | 2157.9 | 7/2 |
| 17 | 2292.7 | 15/4 |
| 18 | 2427.6 | |
| 19 | 2562.5 | 22/5 |
| 20 | 2697.4 | 19/4 |
| 21 | 2832.2 | |
| 22 | 2967.1 | |
| 23 | 3102 | 6/1 |