9ed5: Difference between revisions
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== Theory == | |||
This tuning tempers out [[9/8]] in the [[3-limit]]; [[25/24]] and [[27/25]] in the [[5-limit]]; [[15/14]], [[35/32]], [[36/35]] and [[21/20]] in the [[7-limit]]; [[11/10]], [[33/28]], and [[22/21]] in the [[11-limit]]; [[13/12]], [[26/25]], and [[27/26]] in the [[13-limit]]; [[17/16]], [[35/34]], and [[18/17]] in the [[17-limit]]; [[19/17]] and [[19/18]] in the [[19-limit]]; [[24/23]], [[25/23]], [[26/23]], and [[27/23]] in the [[23-limit]]; [[29/28]], [[33/29]] and [[30/29]] in the [[29-limit]]; [[31/29]], [[31/28]], [[33/31]], and [[31/30]] in the [[31-limit]]; and [[37/36]], [[37/35]], [[37/34]], and [[37/32]] in the [[37-limit]]. | |||
== Intervals == | == Intervals == | ||
Revision as of 16:45, 6 January 2025
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(semiconvergent)
9 equal divisions of the 5th harmonic (abbreviated 9ed5) is a nonoctave tuning system that divides the interval of 5/1 into 9 equal parts of about 310 ¢ each. Each step represents a frequency ratio of 51/9, or the 9th root of 5.
Theory
This tuning tempers out 9/8 in the 3-limit; 25/24 and 27/25 in the 5-limit; 15/14, 35/32, 36/35 and 21/20 in the 7-limit; 11/10, 33/28, and 22/21 in the 11-limit; 13/12, 26/25, and 27/26 in the 13-limit; 17/16, 35/34, and 18/17 in the 17-limit; 19/17 and 19/18 in the 19-limit; 24/23, 25/23, 26/23, and 27/23 in the 23-limit; 29/28, 33/29 and 30/29 in the 29-limit; 31/29, 31/28, 33/31, and 31/30 in the 31-limit; and 37/36, 37/35, 37/34, and 37/32 in the 37-limit.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 309.6 | 6/5, 11/9, 13/11, 17/14, 20/17 |
| 2 | 619.2 | 7/5, 10/7, 13/9, 17/12, 19/13 |
| 3 | 928.8 | 12/7, 17/10, 19/11, 22/13 |
| 4 | 1238.4 | 2/1 |
| 5 | 1548 | 5/2, 12/5, 17/7, 22/9 |
| 6 | 1857.5 | |
| 7 | 2167.1 | 7/2 |
| 8 | 2476.7 | 17/4, 21/5 |
| 9 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +38 | -44 | +77 | +0 | -6 | +37 | +115 | -89 | +38 | -127 | +32 |
| Relative (%) | +12.4 | -14.3 | +24.8 | +0.0 | -2.0 | +11.8 | +37.2 | -28.7 | +12.4 | -40.9 | +10.4 | |
| Steps (reduced) |
4 (4) |
6 (6) |
8 (8) |
9 (0) |
10 (1) |
11 (2) |
12 (3) |
12 (3) |
13 (4) |
13 (4) |
14 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -106 | +75 | -44 | +153 | +48 | -50 | -144 | +77 | -8 | -88 | +144 |
| Relative (%) | -34.3 | +24.2 | -14.3 | +49.6 | +15.7 | -16.3 | -46.5 | +24.8 | -2.5 | -28.5 | +46.6 | |
| Steps (reduced) |
14 (5) |
15 (6) |
15 (6) |
16 (7) |
16 (7) |
16 (7) |
16 (7) |
17 (8) |
17 (8) |
17 (8) |
18 (0) | |