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) |
Rewrite to better address the names (displacing "septimal major semitone", not attested anywhere) |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = septimal diatonic semitone, | | Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic semitone | ||
| Color name = ry1, ruyo unison | | Color name = ry1, ruyo unison | ||
| Sound = jid_15_14_pluck_adu_dr220.mp3 | | Sound = jid_15_14_pluck_adu_dr220.mp3 | ||
}} | }} | ||
{{Wikipedia|Septimal diatonic semitone}} | {{Wikipedia|Septimal diatonic semitone}} | ||
'''15/14''' is | '''15/14''' is an interval in [[7-limit]] [[just intonation]] measuring about 119.4 [[cent]]s, traditionally known as the '''septimal diatonic semitone''' for its proximity (and conflation in [[marvel]] tunings such as septimal [[meantone]]) with the classical diatonic semitone [[16/15]]. However, it functions as a ''[[chromatic semitone]]'', as is supported by [[Sagittal notation]], [[Helmholtz–Ellis notation]] and the [[Functional Just System]], viewed as the [[Pythagorean apotome]] altered by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic chromatic semitone''', or as [[Marc Sabat]] has taken to call it, the '''major chromatic 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 the difference between: | ||
* [[14/13]] and [[15/13]] | * [[14/13]] and [[15/13]] | ||
* [[14/11]] and [[15/11]] | * [[14/11]] and [[15/11]] | ||
| Line 21: | Line 27: | ||
* [[26/15]] and [[13/7]] | * [[26/15]] and [[13/7]] | ||
Finally, since it is a [[superparticular ratio]] with a numerator which is the fifth [[triangular number]], it is a [[triangle-particular]] ratio with factorization ([[25/24]])⋅([[36/35]]). | |||
== Approximation == | == Approximation == | ||
15/14 is very accurately approximated by [[10edo]] (1\10) and all [[linus]] temperaments. The [[linus comma]], 5. | 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). | 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 40: | ||
* [[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 10:46, 1 August 2026
| Interval information |
aberschismic chromatic semitone,
major chromatic semitone
reduced
[sound info]
15/14 is an interval in 7-limit just intonation measuring about 119.4 cents, traditionally known as the septimal diatonic semitone for its proximity (and conflation in marvel tunings such as septimal meantone) with the classical diatonic semitone 16/15. However, it functions as a chromatic semitone, as is supported by Sagittal notation, Helmholtz–Ellis notation and the Functional Just System, viewed as the Pythagorean apotome altered by an aberschisma. This gives rise to the more precise name aberschismic chromatic semitone, or as Marc Sabat has taken to call it, the major chromatic 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 the difference between:
Finally, since it is a superparticular ratio with a numerator which is the fifth triangular number, it is a triangle-particular ratio with factorization (25/24)⋅(36/35).
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.