15/14: Difference between revisions
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{{Infobox Interval | |||
| Name = septimal diatonic semitone, septimal major semitone | |||
| Color name = ry1, ruyo unison | |||
| Sound = jid_15_14_pluck_adu_dr220.mp3 | |||
}} | |||
{{Wikipedia|Septimal diatonic semitone}} | |||
'''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: | |||
* [[14/13]] and [[15/13]] | |||
* [[14/11]] and [[15/11]] | |||
* [[22/15]] and [[11/7]] | |||
* [[26/15]] and [[13/7]] | |||
== 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 ''[[Wikipedia:chromatic semitone|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>. | |||
[[ | == 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)<sup>5</sup> × (15/14)<sup>2</sup> = 2 / (mercurial comma). | |||
== Temperaments == | |||
The following [[linear temperament]]s are [[generate]]d by a [[~]]15/14: | |||
* [[Septidiasemi]] | |||
* [[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}} | |||
== See also == | |||
* [[28/15]] – its [[octave complement]] | |||
* [[7/5]] – its [[fifth complement]] | |||
* [[List of superparticular intervals]] | |||
* [[Gallery of just intervals]] | |||
== References == | |||
<references/> | |||
[[Category:Semitone]] | |||
[[Category:Chroma]] | |||
[[Category:Mercurial]] | |||
Latest revision as of 10:47, 10 August 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).
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.