15/14 - Revision history

FloraC: - misinformation (dyads are neither otonal nor utonal). - duplicate information. Re-organize

2026-05-01T12:05:50Z

- misinformation (dyads are neither otonal nor utonal). - duplicate information. Re-organize

← Older revision		Revision as of 12:05, 1 May 2026
Line 5:		Line 5:
	}}		}}
	{{Wikipedia\|Septimal diatonic semitone}}		{{Wikipedia\|Septimal diatonic semitone}}
	'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. Because it contains exactly one of each prime up to 7, it appears as the interval between many simple [[7-limit]] ratios, ~~including~~:		'''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]]
	* [[4/3]] and [[10/7]]
	* [[7/5]] and [[3/2]]
	* [[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 20:		Line 26:
	* [[22/15]] and [[11/7]]		* [[22/15]] and [[11/7]]
	* [[26/15]] and [[13/7]]		* [[26/15]] and [[13/7]]

	~~== Theory ==~~
	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>.

	15/14 is the difference between certain [[interval qualities]] of thirds and sixths: between (classic) minor and supermajor, and between subminor and (classic) major. These are the pairs of intervals separated by 15/14:
	* 7/6 and 5/4 (otonal thirds)
	* 6/5 and 9/7 (utonal thirds)
	* 14/9 and 5/3 (otonal sixths)
	* 8/5 and 12/7 (utonal sixths)
	~~[[36/35]] separates the otonal pairs from the utonal pairs; 4/3 separates the thirds from the sixths.~~

	~~They can also be organized into pairs of intervals differing by 36/35:~~
	* 7/6 and 6/5 (minor thirds)
	* 5/4 and 9/7 (major thirds)
	* 14/9 and 8/5 (minor sixths)
	* 5/3 and 12/7 (major sixths)
	~~15/14 separates the minor pairs from the major pairs; 4/3 separates the thirds from the sixths.~~

	~~36/35 separates 15/14 from [[25/24]] on one side and [[54/49]] on the other.~~

	== Approximation ==		== 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.		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}}
			{{Interval edo approximation\|max edo=131\|15/14}}

	== Temperaments ==		== Temperaments ==

Squib: /* Theory */as relationship between thirds

2026-02-08T19:43:18Z

Theory: as relationship between thirds

@@ Line 21: / Line 21: @@
 * [[26/15]] and [[13/7]]
-== Terminology ==
+== Theory ==
 /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 ==

Squib at 18:40, 8 February 2026

2026-02-08T18:40:15Z

← Older revision		Revision as of 18:40, 8 February 2026
Line 5:		Line 5:
	}}		}}
	{{Wikipedia\|Septimal diatonic semitone}}		{{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:		'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. Because it contains exactly one of each prime up to 7, it appears as the interval between many simple [[7-limit]] ratios, including:
	* [[16/15]] and [[8/7]]		* [[16/15]] and [[8/7]]
	* [[7/6]] and [[5/4]]		* [[7/6]] and [[5/4]]

Lucius Chiaraviglio: /* Approximation */ Highest approximation of golden meantone was 31edo — bumped this up to 131edo

2025-12-30T12:41:06Z

Approximation: Highest approximation of golden meantone was 31edo — bumped this up to 131edo

← Older revision		Revision as of 12:41, 30 December 2025
Line 28:		Line 28:

	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 ==

FloraC: Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

2025-11-03T13:13:04Z

Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

← Older revision		Revision as of 13:13, 3 November 2025
Line 28:		Line 28:

	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\|15/14}}

	== Temperaments ==		== Temperaments ==

Pailiaq: /* Approximation */

2025-11-03T06:44:11Z

Approximation

← Older revision		Revision as of 06:44, 3 November 2025
Line 27:		Line 27:
	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.		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).		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}}

	== Temperaments ==		== Temperaments ==

BudjarnLambeth: /* See also */

2025-08-10T10:47:55Z

Fredg999: /* Temperaments */ Oops, most likely not all 10th-octave temperaments treat ~15/14 as the period

2025-08-05T17:56:54Z

Temperaments: Oops, most likely not all 10th-octave temperaments treat ~15/14 as the period

← Older revision		Revision as of 17:56, 5 August 2025
Line 37:		Line 37:
	* [[Tertiosec]] (1\3)		* [[Tertiosec]] (1\3)

	[[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].		Several [[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].
	{{todo\|complete list}}		{{todo\|complete list}}

Fredg999: /* Temperaments */ Specify period of tertiosec

2025-08-05T17:55:38Z

Temperaments: Specify period of tertiosec

← Older revision		Revision as of 17:55, 5 August 2025
Line 35:		Line 35:

	In addition, this [[fractional-octave temperament]] is generated by a ~15/14:		In addition, this [[fractional-octave temperament]] is generated by a ~15/14:
	* [[Tertiosec]]		* [[Tertiosec]] (1\3)

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

Fredg999: /* Temperaments */ Add tertiosec

2025-08-05T17:53:45Z

Temperaments: Add tertiosec

← Older revision		Revision as of 17:53, 5 August 2025
Line 33:		Line 33:
	* [[Septidiasemi]]		* [[Septidiasemi]]
	* [[Subsedia]]		* [[Subsedia]]

			In addition, this [[fractional-octave temperament]] is generated by a ~15/14:
			* [[Tertiosec]]

			[[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].
	{{todo\|complete list}}		{{todo\|complete list}}

	~~In addition, several [[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].~~

	== See also ==		== See also ==

← Older revision		Revision as of 10:47, 10 August 2025
Line 43:		Line 43:
	* [[28/15]] – its [[octave complement]]		* [[28/15]] – its [[octave complement]]
	* [[7/5]] – its [[fifth complement]]		* [[7/5]] – its [[fifth complement]]
	* [[1ed15/14]] - its [[ambitonal sequence]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]