18edf: Difference between revisions
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== Theory == | == Theory == | ||
18edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth. | |||
In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds. | |||
=== Regular temperaments === | |||
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]]. | 18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]]. | ||
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{{Harmonics in equal|18|3|2|intervals=integer|columns=11}} | {{Harmonics in equal|18|3|2|intervals=integer|columns=11}} | ||
{{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}} | {{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}} | ||
=== Subsets and supersets === | |||
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}. | |||
== Intervals == | == Intervals == | ||
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! Cents | ! Cents | ||
! Approximate ratios | ! Approximate ratios | ||
|- | |- | ||
| 0 | | 0 | ||
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== See also == | == See also == | ||
* [[31edo]] – relative edo | * [[31edo]] – relative edo | ||
* [[49edt]] – relative | * [[49edt]] – relative edt | ||
* [[72ed5]] – relative ed5 | * [[72ed5]] – relative ed5 | ||
* [[80ed6]] – relative ed6 | * [[80ed6]] – relative ed6 | ||
* [[87ed7]] – relative ed7 | * [[87ed7]] – relative ed7 | ||
* [[107ed11]] – relative ed11 | |||
* [[111ed12]] – relative ed12 | * [[111ed12]] – relative ed12 | ||
* [[138ed22]] – relative ed22 | |||
* [[204ed96]] – close to the zeta-optimized tuning for 31edo | |||
* [[39cET]] | * [[39cET]] | ||
[[Category:31edo]] | |||
Latest revision as of 05:43, 1 August 2025
| ← 17edf | 18edf | 19edf → |
18 equal divisions of the perfect fifth (abbreviated 18edf or 18ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 18 equal parts of about 39 ¢ each. Each step represents a frequency ratio of (3/2)1/18, or the 18th root of 3/2.
Theory
18edf corresponds to 31edo with an octave stretching of about 9 cents. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor 7/6 (+6 ¢), minor 6/5 (-3 ¢), neutral 11/9 (+4 ¢), major 5/4 (+4 ¢), and supermajor 9/7 (-6 ¢). These intervals may be used to form a variety of triads and tetrads in close harmony along with the tuning's pure fifth.
In comparison, 20edf (and Carlos Gamma) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.
Regular temperaments
18edf is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; with 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.9 | +8.9 | +17.8 | -17.5 | +17.8 | -15.0 | -12.2 | +17.8 | -8.6 | -17.6 | -12.2 |
| Relative (%) | +22.9 | +22.9 | +45.8 | -44.9 | +45.8 | -38.6 | -31.4 | +45.8 | -22.0 | -45.1 | -31.4 | |
| Steps (reduced) |
31 (13) |
49 (13) |
62 (8) |
71 (17) |
80 (8) |
86 (14) |
92 (2) |
98 (8) |
102 (12) |
106 (16) |
110 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | -6.1 | -8.6 | -3.3 | +8.7 | -12.2 | +11.2 | +0.4 | -6.1 | -8.7 | -7.6 | -3.3 |
| Relative (%) | +13.3 | -15.7 | -22.0 | -8.5 | +22.4 | -31.4 | +28.6 | +0.9 | -15.7 | -22.2 | -19.5 | -8.5 | |
| Steps (reduced) |
114 (6) |
117 (9) |
120 (12) |
123 (15) |
126 (0) |
128 (2) |
131 (5) |
133 (7) |
135 (9) |
137 (11) |
139 (13) |
141 (15) | |
Subsets and supersets
Since 18 factors into primes as 2 × 32, 18edf has subset edfs 2, 3, 6, and 9.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 39.0 | 33/32, 36/35, 49/48, 50/49, 64/63 |
| 2 | 78.0 | 21/20, 22/21, 25/24, 28/27 |
| 3 | 117.0 | 15/14, 16/15 |
| 4 | 156.0 | 11/10, 12/11 |
| 5 | 195.0 | 9/8, 10/9 |
| 6 | 234.0 | 8/7 |
| 7 | 273.0 | 7/6 |
| 8 | 312.0 | 6/5 |
| 9 | 351.0 | 11/9, 16/13 |
| 10 | 390.0 | 5/4 |
| 11 | 429.0 | 9/7, 14/11 |
| 12 | 468.0 | 13/10, 21/16 |
| 13 | 507.0 | 4/3 |
| 14 | 546.0 | 11/8, 15/11 |
| 15 | 585.0 | 7/5 |
| 16 | 624.0 | 10/7 |
| 17 | 663.0 | 16/11, 22/15 |
| 18 | 702.0 | 3/2 |
| 19 | 741.0 | 20/13, 32/21 |
| 20 | 780.0 | 11/7, 14/9 |
| 21 | 818.9 | 8/5 |
| 22 | 857.9 | 18/11 |
| 23 | 896.9 | 5/3 |
| 24 | 935.9 | 12/7 |
| 25 | 974.9 | 7/4 |
| 26 | 1013.9 | 9/5 |
| 27 | 1052.9 | 11/6 |
| 28 | 1091.9 | 15/8 |
| 29 | 1130.9 | 27/14 |
| 30 | 1169.9 | 35/18, 49/25, 63/32 |
| 31 | 1208.9 | 2/1 |
| 32 | 1247.9 | 33/16, 45/22, 49/24, 55/27 |
| 33 | 1286.9 | 21/10, 25/12 |
| 34 | 1325.9 | 15/7 |
| 35 | 1364.9 | 11/5 |
| 36 | 1403.9 | 9/4 |
Related regular temperaments
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
7-limit 31 & 369
Commas: 2401/2400, 8589934592/8544921875
POTE generator: ~5/4 = 386.997
Mapping: [⟨1 19 2 7], ⟨0 -54 1 -13]]
11-limit 31 & 369
Commas: 2401/2400, 5632/5625, 46656/46585
POTE generator: ~5/4 = 386.999
Mapping: [⟨1 19 2 7 37], ⟨0 -54 1 -13 -104]]
EDOs: 31, 369, 400, 431, 462
13-limit 31 & 369
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
POTE generator: ~5/4 = 387.003
Mapping: [⟨1 19 2 7 37 -35], ⟨0 -54 1 -13 -104 120]]
EDOs: 31, 369, 400, 431, 462
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Todo: cleanup , expand say what the temperaments are like and why one would want to use them, and for what |