15/14: Difference between revisions
Add temperaments (generated by this interval) section, mention marvel in terminology section (not obvious to people not quite familiar with septimal meantone, so it clarifies the context) |
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{{Wikipedia|Septimal diatonic semitone}} | {{Wikipedia|Septimal diatonic semitone}} | ||
'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It | '''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It 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<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>. | ||
Because it contains exactly one of each prime up to 7, it appears as the interval between many simple [[7-limit]] ratios. In particular, it is the difference between certain [[interval qualities]] of seconds, thirds, sixths, and sevenths: between classical minor and supermajor, and between subminor and classical major. These are the pairs of intervals separated by 15/14: | |||
* [[28/27]] and [[10/9]] | |||
* [[16/15]] and [[8/7]] | * [[16/15]] and [[8/7]] | ||
* [[7/6]] and [[5/4]] | * [[7/6]] and [[5/4]] | ||
* [[6/5]] and [[9/7]] | * [[6/5]] and [[9/7]] | ||
* [[14/9]] and [[5/3]] | * [[14/9]] and [[5/3]] | ||
* [[8/5]] and [[12/7]] | * [[8/5]] and [[12/7]] | ||
* [[7/4]] and [[15/8]] | * [[7/4]] and [[15/8]] | ||
* [[9/5]] and [[27/14]] | |||
In addition, it separates the perfect fourth from the larger septimal tritone, and the perfect fifth from the smaller septimal tritone: | |||
* [[4/3]] and [[10/7]] | |||
* [[7/5]] and [[3/2]] | |||
It also arises in higher limits as: | It also arises in higher limits as: | ||
| Line 21: | Line 27: | ||
* [[26/15]] and [[13/7]] | * [[26/15]] and [[13/7]] | ||
== | == Approximation == | ||
15/14 is | 15/14 is very accurately approximated by [[10edo]] (1\10) and all [[linus]] temperaments. The [[linus comma]], 5.6{{c}}, 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)<sup>5</sup> × (15/14)<sup>2</sup> = 2 / (mercurial comma). | |||
{{Interval edo approximation|max edo=131|15/14}} | |||
== Temperaments == | == Temperaments == | ||
| Line 33: | Line 38: | ||
* [[Septidiasemi]] | * [[Septidiasemi]] | ||
* [[Subsedia]] | * [[Subsedia]] | ||
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]]. | |||
{{todo|complete list}} | {{todo|complete list}} | ||
== See also == | == See also == | ||
* [[28/15]] – its [[octave complement]] | * [[28/15]] – its [[octave complement]] | ||
* [[7/5]] – its [[fifth complement]] | * [[7/5]] – its [[fifth complement]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
Latest revision as of 12:05, 1 May 2026
| Interval information |
septimal major semitone
reduced
[sound info]
15/14 is a superparticular ratio with a numerator which is the fifth triangular number. It 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].
Because it contains exactly one of each prime up to 7, it appears as the interval between many simple 7-limit ratios. In particular, it is the difference between certain interval qualities of seconds, thirds, sixths, and sevenths: between classical minor and supermajor, and between subminor and classical major. These are the pairs of intervals separated by 15/14:
- 28/27 and 10/9
- 16/15 and 8/7
- 7/6 and 5/4
- 6/5 and 9/7
- 14/9 and 5/3
- 8/5 and 12/7
- 7/4 and 15/8
- 9/5 and 27/14
In addition, it separates the perfect fourth from the larger septimal tritone, and the perfect fifth from the smaller septimal tritone:
It also arises in higher limits as:
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 |
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.