8ed11/5: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''8ed11/5''' is the [[ed11/5|equal division of the neutral ninth]] into eight parts of 170.6 cents each. It is [[7edo]] with ~7 cent compressed octaves. | '''8ed11/5''' is the [[ed11/5|equal division of the neutral ninth]] into eight parts of 170.6 cents each. It is [[7edo]] with ~7 cent [[Octave shrinking|compressed octaves]]. This tuning tempers out the [[syntonic comma]]. | ||
== Intervals == | == Intervals == | ||
{|class="wikitable" | {|class="wikitable" | ||
| Line 44: | Line 45: | ||
|[[11/5]] | |[[11/5]] | ||
|} | |} | ||
== Harmonics == | |||
Compared to 7edo, 8ed11/5 offers worse approximations of 2, 3, 5, 11 and 29. But it offers better approximations of 7, 13, 17, 19, 23, 31 and 37. Its performance on those higher harmonics lends it a less dissonant sound than one might expect from its poor approximations of simple ratios. | |||
Most of its errors are also in the same direction: it tunes the 2nd, 3rd, 5th and 11th harmonics all flat. So this may further contribute to making the error sound less nasty in practice than it looks on paper. | |||
{{Harmonics in equal | |||
| steps = 8 | |||
| num = 11 | |||
| denom = 5 | |||
| start = 1 | |||
| columns = 12 | |||
| intervals = prime | |||
}} | |||
{{Harmonics in equal | |||
| steps = 7 | |||
| num = 2 | |||
| denom = 1 | |||
| start = 1 | |||
| columns = 12 | |||
| intervals = prime | |||
}} | |||
== Scales == | |||
=== 11/5-repeating === | |||
=== Pseudo-octave-repeating === | |||
Groovy Pentatonic | |||
* 341.251 | |||
* 511.877 | |||
* 682.502 | |||
* 1023.753 | |||
* 1194.379 | |||
=== Other === | |||
[[Category:Nonoctave]] | [[Category:Nonoctave]] | ||
Latest revision as of 21:11, 12 December 2024
| ← 7ed11/5 | 8ed11/5 | 9ed11/5 → |
(convergent)
(semiconvergent)
8ed11/5 is the equal division of the neutral ninth into eight parts of 170.6 cents each. It is 7edo with ~7 cent compressed octaves. This tuning tempers out the syntonic comma.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 170.6 | 11/10, 54/49, 10/9 |
| 2 | 341.3 | 17/14, 39/32, 11/9 |
| 3 | 511.9 | 4/3, 27/20 |
| 4 | 682.5 | 3/2, 40/27 |
| 5 | 853.1 | 18/11, 64/39 |
| 6 | 1023.8 | 9/5, 29/16, 20/11 |
| 7 | 1194.4 | 2/1 |
| 8 | 1365.0 | 11/5 |
Harmonics
Compared to 7edo, 8ed11/5 offers worse approximations of 2, 3, 5, 11 and 29. But it offers better approximations of 7, 13, 17, 19, 23, 31 and 37. Its performance on those higher harmonics lends it a less dissonant sound than one might expect from its poor approximations of simple ratios.
Most of its errors are also in the same direction: it tunes the 2nd, 3rd, 5th and 11th harmonics all flat. So this may further contribute to making the error sound less nasty in practice than it looks on paper.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.6 | -25.1 | -56.3 | +43.7 | -56.3 | -4.3 | +43.2 | +21.3 | +31.7 | -28.3 | +26.9 | +61.8 |
| Relative (%) | -3.3 | -14.7 | -33.0 | +25.6 | -33.0 | -2.5 | +25.3 | +12.5 | +18.6 | -16.6 | +15.7 | +36.2 | |
| Steps (reduced) |
7 (7) |
11 (3) |
16 (0) |
20 (4) |
24 (0) |
26 (2) |
29 (5) |
30 (6) |
32 (0) |
34 (2) |
35 (3) |
37 (5) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -16.2 | -43.5 | +59.7 | -37.0 | +16.6 | +66.5 | +45.3 | +57.4 | -1.0 | +55.0 | -79.9 |
| Relative (%) | +0.0 | -9.5 | -25.3 | +34.9 | -21.6 | +9.7 | +38.8 | +26.5 | +33.5 | -0.6 | +32.1 | -46.6 | |
| Steps (reduced) |
7 (0) |
11 (4) |
16 (2) |
20 (6) |
24 (3) |
26 (5) |
29 (1) |
30 (2) |
32 (4) |
34 (6) |
35 (0) |
36 (1) | |
Scales
11/5-repeating
Pseudo-octave-repeating
Groovy Pentatonic
- 341.251
- 511.877
- 682.502
- 1023.753
- 1194.379