15edo - Revision history

Vector at 14:55, 4 April 2026

2026-04-04T14:55:06Z

← Older revision		Revision as of 14:55, 4 April 2026
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	==== Chords ====		==== Chords ====
	15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).		15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

			=== Derivation ===
			15edo may be derived in a number of ways

			* by tripling 5edo (perhaps to collapse [[Blackwood\|blackwood[10]]] to an edo)
			** as the intersection of blackwood and augmented
			*** to attain a near-DR 4:5:6
			* as the simplest tuning of Zarlino diatonic
			* as the simplest reasonable tuning of Porcupine

	== Intervals ==		== Intervals ==

Vector at 08:48, 24 March 2026

2026-03-24T08:48:32Z

← Older revision		Revision as of 08:48, 24 March 2026
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	The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones (Note, however, that in Greek music theory onyx is a limiting case of a diatonic scale, much like 15edo's zarlino). In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.		The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones (Note, however, that in Greek music theory onyx is a limiting case of a diatonic scale, much like 15edo's zarlino). In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

	The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third ~~span the same number of scale steps~~; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.		The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third are found on separate degrees; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

	The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).		The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

Vector at 08:46, 24 March 2026

2026-03-24T08:46:35Z

← Older revision		Revision as of 08:46, 24 March 2026
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	=== Non-diatonic scales ===		=== Non-diatonic scales ===
	The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.		The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones (Note, however, that in Greek music theory onyx is a limiting case of a diatonic scale, much like 15edo's zarlino). In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

	The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.		The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

Inthar: /* Notation */

2026-02-23T04:58:58Z

Notation

← Older revision		Revision as of 04:58, 23 February 2026
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	Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation\|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation\|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.		Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation\|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation\|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

			{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Inthar: Reverted edit by Inthar (talk) to last revision by Fitzgerald Lee

2026-02-19T16:13:13Z

Reverted edit by Inthar (talk) to last revision by Fitzgerald Lee

← Older revision		Revision as of 16:13, 19 February 2026
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	==== Edostep interpretations ====		==== Edostep interpretations ====
	15edo's edostep has the following interpretations in the ~~2...~~11 ~~subgroup~~:		15edo's edostep has the following interpretations in the [[11-limit]]:

	* 16/15 (the difference between 5/4 and 4/3)		* 16/15 (the difference between 5/4 and 4/3)
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	* 36/35 (the difference between 5/4 and 9/7)		* 36/35 (the difference between 5/4 and 9/7)
	* 22/21 (the difference between 7/6 and 11/9)		* 22/21 (the difference between 7/6 and 11/9)
			* 21/20 (the difference between 4/3 and 7/5)

	==== JI approximation ====		==== JI approximation ====

Inthar: Reverted edit by Fitzgerald Lee (talk) to last revision by Vector

2026-02-19T16:12:59Z

Reverted edit by Fitzgerald Lee (talk) to last revision by Vector

← Older revision		Revision as of 16:12, 19 February 2026
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	==== Edostep interpretations ====		==== Edostep interpretations ====
	15edo's edostep has the following interpretations in the [[11~~-limit]]~~:		15edo's edostep has the following interpretations in the 2...11 subgroup:

	* 16/15 (the difference between 5/4 and 4/3)		* 16/15 (the difference between 5/4 and 4/3)
Line 12:		Line 12:
	* 36/35 (the difference between 5/4 and 9/7)		* 36/35 (the difference between 5/4 and 9/7)
	* 22/21 (the difference between 7/6 and 11/9)		* 22/21 (the difference between 7/6 and 11/9)
	* 21/20 (the difference between 4/3 and 7/5)

	==== JI approximation ====		==== JI approximation ====

Fitzgerald Lee: /* Edostep interpretations */

2026-02-19T10:54:01Z

Edostep interpretations

← Older revision		Revision as of 10:54, 19 February 2026
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	==== Edostep interpretations ====		==== Edostep interpretations ====
	15edo's edostep has the following interpretations in the ~~2...~~11 ~~subgroup~~:		15edo's edostep has the following interpretations in the [[11-limit]]:

	* 16/15 (the difference between 5/4 and 4/3)		* 16/15 (the difference between 5/4 and 4/3)
Line 12:		Line 12:
	* 36/35 (the difference between 5/4 and 9/7)		* 36/35 (the difference between 5/4 and 9/7)
	* 22/21 (the difference between 7/6 and 11/9)		* 22/21 (the difference between 7/6 and 11/9)
			* 21/20 (the difference between 4/3 and 7/5)

	==== JI approximation ====		==== JI approximation ====

Vector: fixed the "15" bug hopefully

2026-02-08T21:13:28Z

fixed the "15" bug hopefully

← Older revision		Revision as of 21:13, 8 February 2026
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	The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).		The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

	After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.		After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

	==== Blackwood ====		==== Blackwood ====
	Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler\|15\|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler\|15\|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)		Blackwood is based on our "fifth-generated diatonic" {{Interval ruler\|15\|0, 240, 480, 720, 960, 1200}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{Interval ruler\|15\|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

	==== Augmented ====		==== Augmented ====

Vector at 01:24, 8 February 2026

2026-02-08T01:24:59Z

← Older revision		Revision as of 01:24, 8 February 2026
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	There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.		There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.
			[[File:Valentine.png\|thumb\|A comparison of 15edo and 16edo, showing their similarities (see [[Carlos Alpha]]) and 15edo's "Onyx" scale.]]
	And the classical chroma x limma scale:		And the classical chroma x limma scale:

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	=== Regular temperaments ===		=== Regular temperaments ===
	15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.		15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo. It is also a tuning of [[Carlos Alpha]], like [[16edo]].

	== Notation ==		== Notation ==

Hkm: /* Non-diatonic scales */

2026-01-26T22:11:47Z

Non-diatonic scales

← Older revision		Revision as of 22:11, 26 January 2026
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	The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.		The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

	The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), ~~peltonic~~ ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).		The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

	After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.		After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

← Older revision		Revision as of 08:48, 24 March 2026
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	The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones (Note, however, that in Greek music theory onyx is a limiting case of a diatonic scale, much like 15edo's zarlino). In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.		The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler\|15\|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler\|15\|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones (Note, however, that in Greek music theory onyx is a limiting case of a diatonic scale, much like 15edo's zarlino). In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

	The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third ~~span the same number of scale steps~~; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.		The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler\|15\|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler\|15\|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third are found on separate degrees; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

	The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).		The scales generated by stacking either tritone are called pentic ({{Interval ruler\|15\|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler\|15\|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler\|15\|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler\|15\|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).