15/14: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox Interval
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name = septimal diatonic semitone, septimal major semitone
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 15:17:20 UTC</tt>.<br>
| Color name = ry1, ruyo unison
: The original revision id was <tt>245079967</tt>.<br>
| Sound = jid_15_14_pluck_adu_dr220.mp3
: The revision comment was: <tt></tt><br>
}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
{{Wikipedia|Septimal diatonic semitone}}
<h4>Original Wikitext content:</h4>
'''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:
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The septimal diatonic semitone, 15/14, is a [[superparticular]] number with a numerator which is the fifth triangular number, and is the interval between 16/15 and 8/7, 14/13 and 15/13, 7/6 and 5/4, 6/5 and 9/7, 14/11 and 15/11, 4/3 and 10/7, 7/5 and 3/2, 22/15 and 11/7, 14/9 and 5/3, 8/5 and 12/7, 26/15 and 13/7 and 7/4 and 15/8.
* [[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]]


[[http://en.wikipedia.org/wiki/Septimal_diatonic_semitone]]</pre></div>
It also arises in higher limits as:
<h4>Original HTML content:</h4>
* [[14/13]] and [[15/13]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;15_14&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The septimal diatonic semitone, 15/14, is a &lt;a class="wiki_link" href="/superparticular"&gt;superparticular&lt;/a&gt; number with a numerator which is the fifth triangular number, and is the interval between 16/15 and 8/7, 14/13 and 15/13, 7/6 and 5/4, 6/5 and 9/7, 14/11 and 15/11, 4/3 and 10/7, 7/5 and 3/2, 22/15 and 11/7, 14/9 and 5/3, 8/5 and 12/7, 26/15 and 13/7 and 7/4 and 15/8.&lt;br /&gt;
* [[14/11]] and [[15/11]]
&lt;br /&gt;
* [[22/15]] and [[11/7]]
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diatonic_semitone" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Septimal_diatonic_semitone&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[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
Ratio 15/14
Factorization 2-1 × 3 × 5 × 7-1
Monzo [-1 1 1 -1⟩
Size in cents 119.4428¢
Names septimal diatonic semitone,
septimal major semitone
Color name ry1, ruyo unison
FJS name [math]\displaystyle{ \text{A1}^{5}_{7} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 7.71425
Weil norm (log2 max(n, d)) 7.81378
Wilson norm (sopfr(nd)) 17

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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:

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:

Several 10th-octave temperaments treat ~15/14 as the period, including decoid and linus.

See also

References

  1. ↑ Marc Sabat. Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space. Plainsound Music Edition, 2008.
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