5-odd-limit - Revision history

Overthink: start section

2026-07-31T16:58:31Z

start section

← Older revision		Revision as of 16:58, 31 July 2026
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	This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.		This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.

			== Intervals separating adjacent 5-odd-limit ratios ==
			{{WIP}}
			=== 9/8 ===
			{{See also\|9-odd-limit #9/8\|Pythagorean tuning #Major second}}
			9/8 is the difference between 3/2 and 4/3, and is known as the Pythagorean major 2nd, or (major) whole tone. It is traditionally considered an imperfect dissonance, being less dissonant than the semitones and the tritone, but less consonant than the 5-odd-limit consonances.

			=== 16/15 ===
			=== 25/24 ===

Overthink: specify

2026-06-21T18:19:53Z

specify

← Older revision		Revision as of 18:19, 21 June 2026
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	{{Odd-limit navigation}}		{{Odd-limit navigation}}
	The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are ~~considered equivalent~~. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.		The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are the unison, octave, and its multiples. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

	The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino, constructed from the numbers 1, 2, 3, 4, 5, 6, and 8. (Note the absence of 7.)		The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino, constructed from the numbers 1, 2, 3, 4, 5, 6, and 8. (Note the absence of 7.)

Kili: /* Perfect fifth (3/2) */ lijnkao

2026-06-01T18:57:18Z

Perfect fifth (3/2): lijnkao

← Older revision		Revision as of 18:57, 1 June 2026
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	''Main article: [[Perfect fifth]]''		''Main article: [[Perfect fifth]]''

	The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the [[Pythagorean]] system of tuning.		The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the [[Pythagorean tuning\|Pythagorean]] system of tuning.

	In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.		In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.

Vector at 02:18, 16 April 2026

2026-04-16T02:18:18Z

← Older revision		Revision as of 02:18, 16 April 2026
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	The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are considered equivalent. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.		The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are considered equivalent. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

	The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.		The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino, constructed from the numbers 1, 2, 3, 4, 5, 6, and 8. (Note the absence of 7.)

	== Table of 5-odd-limit intervals ==		== Table of 5-odd-limit intervals ==

Vector at 02:17, 16 April 2026

2026-04-16T02:17:34Z

← Older revision		Revision as of 02:17, 16 April 2026
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	{{Odd-limit navigation}}		{{Odd-limit navigation}}
	The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.		The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are considered equivalent. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

	The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.		The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.

Overthink: add navigation

2026-04-14T01:11:01Z

add navigation

← Older revision		Revision as of 01:11, 14 April 2026
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			{{Odd-limit navigation}}
	The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.		The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

Inthar: /* Approximation by edos */

2026-04-13T03:31:37Z

Approximation by edos

← Older revision		Revision as of 03:31, 13 April 2026
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	The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.		The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

	The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as five major thirds stacking to a fifth. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.		The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as [[Magic\|five major thirds stacking to a fifth]]. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

	== Intervals of the 5-odd-limit ==		== Intervals of the 5-odd-limit ==

Inthar: /* Approximation by edos */

2026-04-13T03:28:33Z

Approximation by edos

← Older revision		Revision as of 03:28, 13 April 2026
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	== Approximation by edos ==		== Approximation by edos ==
	The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[~~dicot~~]], does not lead to categorical conflicts the same way 3edo's [[~~father~~]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.		The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

	The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. [[19edo]] is a ~~Menatone~~ tuning like 12edo ~~but~~ more accurate~~, but~~ the arithmetic of 5-limit intervals may lead to ~~different~~ results, such as five major thirds stacking to a fifth. [[22edo]] has the opposite tuning tendencies to 12edo.		The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as five major thirds stacking to a fifth. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

	== Intervals of the 5-odd-limit ==		== Intervals of the 5-odd-limit ==

Inthar: /* Approximation by edos */

2026-04-13T03:27:17Z

Approximation by edos

← Older revision		Revision as of 03:27, 13 April 2026
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	The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[dicot]], does not lead to categorical conflicts the same way 3edo's [[father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.		The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[dicot]], does not lead to categorical conflicts the same way 3edo's [[father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

	The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo~~. This~~ is one factor that led to 12edo's worldwide standardization. 22edo has the opposite tuning tendencies to 12edo.		The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. [[19edo]] is a Menatone tuning like 12edo but more accurate, but the arithmetic of 5-limit intervals may lead to different results, such as five major thirds stacking to a fifth. [[22edo]] has the opposite tuning tendencies to 12edo.

	== Intervals of the 5-odd-limit ==		== Intervals of the 5-odd-limit ==

Inthar at 03:25, 13 April 2026

2026-04-13T03:25:16Z

← Older revision		Revision as of 03:25, 13 April 2026
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	The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as		The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance\|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

	The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.		The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.