15/14: Difference between revisions
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In combination with [[19/17]] it forms a good approximation of [[golden meantone]]. The untempered combination of five 19/17's and two 15/14's leads to an interval that is sharp to an octave by the [[mercurial comma]]: (19/17)<sup>5</sup> × (15/14)<sup>2</sup> = 2 / (mercurial comma). | In combination with [[19/17]] it forms a good approximation of [[golden meantone]]. The untempered combination of five 19/17's and two 15/14's leads to an interval that is sharp to an octave by the [[mercurial comma]]: (19/17)<sup>5</sup> × (15/14)<sup>2</sup> = 2 / (mercurial comma). | ||
{{Interval edo approximation|15/14}} | {{Interval edo approximation|max_edo=131|15/14}} | ||
== Temperaments == | == Temperaments == | ||
Revision as of 12:41, 30 December 2025
| Interval information |
septimal major semitone
reduced
[sound info]
15/14 is a superparticular ratio with a numerator which is the fifth triangular number. It may be found as the interval between many 7-limit ratios, including:
- 16/15 and 8/7
- 7/6 and 5/4
- 6/5 and 9/7
- 4/3 and 10/7
- 7/5 and 3/2
- 14/9 and 5/3
- 8/5 and 12/7
- 7/4 and 15/8
It also arises in higher limits as:
Terminology
15/14 is traditionally called a diatonic semitone, perhaps for its proximity (and conflation in systems such as septimal meantone and marvel) with the classic diatonic semitone 16/15. However, 15/14 is a chromatic semitone in both Helmholtz–Ellis notation and the Functional Just System, viewed as the apotome 2187/2048 altered by 5120/5103. Marc Sabat has taken to call it the major chromatic semitone in the same material where 21/20 is also named as the minor diatonic semitone[1].
Approximation
15/14 is very accurately approximated by 10edo (1\10) and all linus temperaments. The linus comma, 5.6¢, is the amount by which a stack of ten 15/14's falls short of the octave.
In combination with 19/17 it forms a good approximation of golden meantone. The untempered combination of five 19/17's and two 15/14's leads to an interval that is sharp to an octave by the mercurial comma: (19/17)5 × (15/14)2 = 2 / (mercurial comma).
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 10 | 1\10 | 120.00 | +0.56 | +0.46 |
| 11 | 1\11 | 109.09 | -10.35 | -9.49 |
| 20 | 2\20 | 120.00 | +0.56 | +0.93 |
| 21 | 2\21 | 114.29 | -5.16 | -9.02 |
| 30 | 3\30 | 120.00 | +0.56 | +1.39 |
| 31 | 3\31 | 116.13 | -3.31 | -8.56 |
| 40 | 4\40 | 120.00 | +0.56 | +1.86 |
| 41 | 4\41 | 117.07 | -2.37 | -8.10 |
| 50 | 5\50 | 120.00 | +0.56 | +2.32 |
| 51 | 5\51 | 117.65 | -1.80 | -7.63 |
| 60 | 6\60 | 120.00 | +0.56 | +2.79 |
| 61 | 6\61 | 118.03 | -1.41 | -7.17 |
| 70 | 7\70 | 120.00 | +0.56 | +3.25 |
| 71 | 7\71 | 118.31 | -1.13 | -6.70 |
| 80 | 8\80 | 120.00 | +0.56 | +3.71 |
| 81 | 8\81 | 118.52 | -0.92 | -6.24 |
| 90 | 9\90 | 120.00 | +0.56 | +4.18 |
| 91 | 9\91 | 118.68 | -0.76 | -5.77 |
| 100 | 10\100 | 120.00 | +0.56 | +4.64 |
| 101 | 10\101 | 118.81 | -0.63 | -5.31 |
| 110 | 11\110 | 120.00 | +0.56 | +5.11 |
| 111 | 11\111 | 118.92 | -0.52 | -4.85 |
| 120 | 12\120 | 120.00 | +0.56 | +5.57 |
| 121 | 12\121 | 119.01 | -0.43 | -4.38 |
| 130 | 13\130 | 120.00 | +0.56 | +6.04 |
| 131 | 13\131 | 119.08 | -0.36 | -3.92 |
Temperaments
The following linear temperaments are generated by a ~15/14:
In addition, this fractional-octave temperament is generated by a ~15/14:
- Tertiosec (1\3)
Several 10th-octave temperaments treat ~15/14 as the period, including decoid and linus.
See also
- 28/15 – its octave complement
- 7/5 – its fifth complement
- List of superparticular intervals
- Gallery of just intervals
References
- ↑ Marc Sabat. Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space. Plainsound Music Edition, 2008.