256ed5: Difference between revisions
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'''256 equal divisions of the 5th harmonic''' is an equal-step tuning of 10.884 cents | '''256 equal divisions of the 5th harmonic''' is an equal-step tuning where each step represents a frequency ratio of 256th root of 5, which amounts to 3.90625 millipentaves or about 10.884 cents. It is equivalent to 110.2532 EDO. | ||
256ed5 combines [[dual-fifth temperaments]] with [[quarter-comma meantone]]. | 256ed5 combines [[dual-fifth temperaments]] with [[quarter-comma meantone]]. | ||
| Line 8: | Line 8: | ||
Uniquely, 6/5 is nearly perfect. | Uniquely, 6/5 is nearly perfect. | ||
== Table of intervals == | |||
{| class="wikitable" | |||
|+ | |||
!Step | |||
!Name | |||
!Size (cents) | |||
!Size (millipentaves) | |||
!Associated ratio | |||
|- | |||
|0 | |||
|prime, unison | |||
|0 | |||
|0 | |||
|exact 1/1 | |||
|- | |||
|29 | |||
|classical minor third | |||
|315.63710 | |||
|113.28125 | |||
|6/5 | |||
|- | |||
|64 | |||
|minor fifth | |||
|[[Quarter-comma meantone|696.57843]] | |||
|250 | |||
|3/2 I, exact 4th root of(5) | |||
|- | |||
|65 | |||
|major fifth | |||
| | |||
|253.90625 | |||
| | |||
|- | |||
|128 | |||
|octitone, symmetric ninth | |||
|1393.15686 | |||
|500 | |||
| | |||
|- | |||
|256 | |||
|pentave, fifth harmonic | |||
|2786.31371 | |||
|1000 | |||
|exact 5/1 | |||
|} | |||
== See also == | == See also == | ||
Revision as of 09:11, 13 March 2022
256 equal divisions of the 5th harmonic is an equal-step tuning where each step represents a frequency ratio of 256th root of 5, which amounts to 3.90625 millipentaves or about 10.884 cents. It is equivalent to 110.2532 EDO.
256ed5 combines dual-fifth temperaments with quarter-comma meantone.
Theory
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.76 | +2.75 | +5.37 | +0.00 | -0.00 | +5.23 | +2.62 | -5.38 | -2.76 | -4.50 | -2.76 |
| Relative (%) | -25.3 | +25.3 | +49.4 | +0.0 | -0.0 | +48.0 | +24.0 | -49.4 | -25.3 | -41.3 | -25.4 | |
| Steps (reduced) |
110 (110) |
175 (175) |
221 (221) |
256 (0) |
285 (29) |
310 (54) |
331 (75) |
349 (93) |
366 (110) |
381 (125) |
395 (139) | |
In 256ed5, the just perfect fifth of 3/2, corresponds to approximately 64.5 steps, thus occurring almost halfway between the quarter-comma meantone fifth and it's next step.
Uniquely, 6/5 is nearly perfect.
Table of intervals
| Step | Name | Size (cents) | Size (millipentaves) | Associated ratio |
|---|---|---|---|---|
| 0 | prime, unison | 0 | 0 | exact 1/1 |
| 29 | classical minor third | 315.63710 | 113.28125 | 6/5 |
| 64 | minor fifth | 696.57843 | 250 | 3/2 I, exact 4th root of(5) |
| 65 | major fifth | 253.90625 | ||
| 128 | octitone, symmetric ninth | 1393.15686 | 500 | |
| 256 | pentave, fifth harmonic | 2786.31371 | 1000 | exact 5/1 |