Kite's thoughts on negative intervals - Revision history

TallKite at 21:23, 14 February 2026

2026-02-14T21:23:28Z

← Older revision		Revision as of 21:23, 14 February 2026
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	A '''negative interval''' is an [[interval]] that goes down the [[scale]] but up in [[pitch]], and vice versa. For example, in [[just intonation]] the [[Pythagorean comma]] is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to [[3-limit]] JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''negative'' 2nd.		A '''negative interval''' is an [[interval]] that goes down the [[scale]] but up in [[pitch]], and vice versa. For example, in [[just intonation]] the [[Pythagorean comma]] is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to [[3-limit]] JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''negative'' 2nd.

	The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd. We say "equivalently" because the interval ''between'' two notes is a vertical or harmonic interval, whereas the interval ''from'' one note ''to'' another is a horizontal or melodic interval<ref group="note">See [[Wikipedia: Interval (music)]].</ref>.		The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd, written -d2. We say "equivalently" because the interval ''between'' two notes is a vertical or harmonic interval, whereas the interval ''from'' one note ''to'' another is a horizontal or melodic interval<ref group="note">See [[Wikipedia: Interval (music)]].</ref>.

	"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.		"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.
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	Within a single piece of music, it's quite rare to find two notes a Pythagorean comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the [[schisma]] = [-15 8 1⟩ = 2¢, also rare.		Within a single piece of music, it's quite rare to find two notes a Pythagorean comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the [[schisma]] = [-15 8 1⟩ = 2¢, also rare.

	But in other ~~tunings~~ negative 2nds are commonplace. For example, in 7-limit JI, the interval from [[7/5]] (a diminished 5th) up to [[10/7]] (an augmented 4th) is [[50/49]] = 35¢, a negative diminished 2nd. Furthermore, the interval from [[16/15]] (a minor 2nd) up to [[15/14]] (an augmented unison) is [[225/224]] = 8¢, another negative diminished 2nd.		But in other prime limits negative 2nds are commonplace. For example, in 7-limit JI, the interval from [[7/5]] (a diminished 5th) up to [[10/7]] (an augmented 4th) is [[50/49]] = 35¢, a negative diminished 2nd. Furthermore, the interval from [[16/15]] (a minor 2nd) up to [[15/14]] (an augmented unison) is [[225/224]] = 8¢, another negative diminished 2nd.

	Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.		Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.

TallKite: moved the reentrant scale stuff back to the scale page where it belongs better

2025-08-20T03:16:12Z

moved the reentrant scale stuff back to the scale page where it belongs better

← Older revision		Revision as of 03:16, 20 August 2025
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	Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.		Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.

	~~== Reentrant scales ==~~
	A '''reentrant scale''' has at least one negative interval, going backwards relative to the general direction of the scale. Although a reentrant scale is not strictly ascending or descending, its ascending and descending forms are determined by its general direction.

	Reentrant scales are mostly relevant when applying extreme tunings to abstract scales, causing some steps to have a negative size in order to preserve the abstract scale's usual structure. For example, if you try to generate a [[MOS scale]] with a [[generator]] whose size falls outside of the generator range of all possible MOS patterns with the same given number of notes, you will obtain a MOS scale with a negative [[step ratio]]. A 7-tone scale with a 295{{cent}} generator is just outside of the range for [[4L 3s]], and can be interpreted as a 4L 3s scale with 315{{cent}} large steps and -20{{cent}} (negative) small steps, whereas considering the scale's pitches in ascending order leads to a [[ternary]], [[Maximum variety\|MV4]] scale interpretation.

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

TallKite: TallKite moved page Negative interval to Kite's thoughts on negative intervals: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T01:04:00Z

TallKite moved page Negative interval to Kite's thoughts on negative intervals: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 01:04, 20 August 2025
(No difference)

Fredg999: Move "reentrant scale" from Scale, replace 1 Wikipedia link with a footnote, replaced other Wikipedia link with internal link, add 1 link to see also, misc. edits

2025-07-30T03:00:43Z

Move "reentrant scale" from Scale, replace 1 Wikipedia link with a footnote, replaced other Wikipedia link with internal link, add 1 link to see also, misc. edits

← Older revision		Revision as of 03:00, 30 July 2025
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	A negative interval is an interval that goes down the scale but up in pitch. For example, in just intonation the [[~~pythagorean~~ comma]] is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to 3-limit JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''negative'' 2nd.		A '''negative interval''' is an [[interval]] that goes down the [[scale]] but up in [[pitch]], and vice versa. For example, in [[just intonation]] the [[Pythagorean comma]] is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to [[3-limit]] JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''negative'' 2nd.

	The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd. We say "equivalently" because the interval ''between'' two notes is a vertical or harmonic interval, whereas the interval ''from'' one note ''to'' another is a horizontal or melodic interval~~. (~~See [[~~wikipedia~~:~~Interval_~~(music)~~\|en~~.~~wikipedia.org/wiki~~/~~Interval_(music)]]~~.)		The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd. We say "equivalently" because the interval ''between'' two notes is a vertical or harmonic interval, whereas the interval ''from'' one note ''to'' another is a horizontal or melodic interval<ref group="note">See [[Wikipedia: Interval (music)]].</ref>.

	"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.		"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.

	"Negative" does not mean "inverted". The inversion of a diminished 2nd is an augmented 7th ~~(see [[wikipedia:Inversion_(music)#Intervals\|en.wikipedia.org/wiki/Inversion_(music)#Intervals]])~~. The inversion of a negative diminished 2nd is a diminished 9th.		"Negative" does not mean "inverted". The [[inversion]] of a diminished 2nd is an augmented 7th. The inversion of a negative diminished 2nd is a diminished 9th.

	== Temperaments ==		== Temperaments ==
	In certain temperaments such as [[meantone]], the fifth is flattened sufficiently such that the ~~pythagorean~~ comma becomes descending. It's no longer negative, and is simply a descending diminished 2nd. However, negative 2nds do occur in meantone. (In fact, multiple negative 2nds, 3rds, etc. inevitably occur in every tuning of rank-2 or higher. We can simply repeatedly diminish a 2nd or a 3rd until it becomes descending, then flip it to make it ascending.) In the case of meantone, the kleisma ([[fifthspan]] of +19) is a negative 2nd.		In certain temperaments such as [[meantone]], the fifth is flattened sufficiently such that the Pythagorean comma becomes descending. It's no longer negative, and is simply a descending diminished 2nd. However, negative 2nds do occur in meantone. (In fact, multiple negative 2nds, 3rds, etc. inevitably occur in every tuning of rank-2 or higher. We can simply repeatedly diminish a 2nd or a 3rd until it becomes descending, then flip it to make it ascending.) In the case of meantone, the kleisma ([[fifthspan]] of +19) is a negative 2nd.

	== Interval arithmetic ==		== Interval arithmetic ==
	Adding or subtracting a negative interval is the same as subtracting or adding the corresponding positive interval.		Adding or subtracting a negative interval is the same as subtracting or adding the corresponding positive interval.

	For example, what is an octave plus a ~~pythagorean~~ comma? We must subtract a diminished 2nd from an octave. We know that P8 - m2 = M7. If we diminish what we're subtracting (m2), we will augment the result. Thus P8 - d2 = A7, an augmented 7th, e.g. C-B♯. Likewise a major 3rd minus a ~~pythagorean~~ comma is a diminished 4th, e.g. C-F♭. An extreme example: the sum of two ~~pythagorean~~ commas is a negative triply-diminished 3rd, e.g. C-A♯♯♯.		For example, what is an octave plus a Pythagorean comma? We must subtract a diminished 2nd from an octave. We know that P8 - m2 = M7. If we diminish what we're subtracting (m2), we will augment the result. Thus P8 - d2 = A7, an augmented 7th, e.g. C-B♯. Likewise a major 3rd minus a Pythagorean comma is a diminished 4th, e.g. C-F♭. An extreme example: the sum of two Pythagorean commas is a negative triply-diminished 3rd, e.g. C-A♯♯♯.

	== Prevalence in just intonation ==		== Prevalence in just intonation ==
	Within a single piece of music, it's quite rare to find two notes a ~~pythagorean~~ comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the [[schisma]] = [-15 8 1⟩ = 2¢, also rare.		Within a single piece of music, it's quite rare to find two notes a Pythagorean comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the [[schisma]] = [-15 8 1⟩ = 2¢, also rare.

	But in other tunings negative 2nds are commonplace. For example, in 7-limit JI, the interval from [[7/5]] (a diminished 5th) up to [[10/7]] (an augmented 4th) is [[50/49]] = 35¢, a negative diminished 2nd. Furthermore, the interval from [[16/15]] (a minor 2nd) up to [[15/14]] (an augmented unison) is [[225/224]] = 8¢, another negative diminished 2nd.		But in other tunings negative 2nds are commonplace. For example, in 7-limit JI, the interval from [[7/5]] (a diminished 5th) up to [[10/7]] (an augmented 4th) is [[50/49]] = 35¢, a negative diminished 2nd. Furthermore, the interval from [[16/15]] (a minor 2nd) up to [[15/14]] (an augmented unison) is [[225/224]] = 8¢, another negative diminished 2nd.
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	Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.		Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.

			== Reentrant scales ==
			A '''reentrant scale''' has at least one negative interval, going backwards relative to the general direction of the scale. Although a reentrant scale is not strictly ascending or descending, its ascending and descending forms are determined by its general direction.

	{{~~todo~~\|~~inline~~=~~1\|add links\|text~~=~~Convert wikipedia links to footnotes.}}~~		Reentrant scales are mostly relevant when applying extreme tunings to abstract scales, causing some steps to have a negative size in order to preserve the abstract scale's usual structure. For example, if you try to generate a [[MOS scale]] with a [[generator]] whose size falls outside of the generator range of all possible MOS patterns with the same given number of notes, you will obtain a MOS scale with a negative [[step ratio]]. A 7-tone scale with a 295{{cent}} generator is just outside of the range for [[4L 3s]], and can be interpreted as a 4L 3s scale with 315{{cent}} large steps and -20{{cent}} (negative) small steps, whereas considering the scale's pitches in ascending order leads to a [[ternary]], [[Maximum variety\|MV4]] scale interpretation.
	[[Category:Interval]][[Category:Terms]]
			== See also ==
			* [[Undirected value]]

			== Notes ==
			<references group="note"/>

			[[Category:Interval]]
			[[Category:Terms]]

BudjarnLambeth at 07:33, 30 April 2025

2025-04-30T07:33:49Z

← Older revision		Revision as of 07:33, 30 April 2025
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	{{todo\|inline=1\|text=Convert wikipedia links to footnotes.}}		{{todo\|inline=1\|add links\|text=Convert wikipedia links to footnotes.}}
	[[Category:Interval]][[Category:Terms]]		[[Category:Interval]][[Category:Terms]]

BudjarnLambeth: Modernise todo; categorise uncategorised Page

2025-04-30T07:33:29Z

Modernise todo; categorise uncategorised Page

← Older revision		Revision as of 07:33, 30 April 2025
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	~~TO DO: convert~~ wikipedia links to footnotes.		{{todo\|inline=1\|text=Convert wikipedia links to footnotes.}}
			[[Category:Interval]][[Category:Terms]]

TallKite: Created page with "A negative interval is an interval that goes down the scale but up in pitch. For example, in just intonation the pythagorean comma is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to 3-limit JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''..."

2025-04-30T05:38:33Z

Created page with "A negative interval is an interval that goes down the scale but up in pitch. For example, in just intonation the pythagorean comma is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to 3-limit JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''..."

New page

A negative interval is an interval that goes down the scale but up in pitch. For example, in just intonation the [[pythagorean comma]] is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to 3-limit JI.) But because this comma is an augmented unison ''minus'' a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a ''negative'' 2nd.

The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd. We say "equivalently" because the interval ''between'' two notes is a vertical or harmonic interval, whereas the interval ''from'' one note ''to'' another is a horizontal or melodic interval. (See [[wikipedia:Interval_(music)|en.wikipedia.org/wiki/Interval_(music)]].)

"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.

"Negative" does not mean "inverted". The inversion of a diminished 2nd is an augmented 7th (see [[wikipedia:Inversion_(music)#Intervals|en.wikipedia.org/wiki/Inversion_(music)#Intervals]]). The inversion of a negative diminished 2nd is a diminished 9th.

== Temperaments ==
In certain temperaments such as [[meantone]], the fifth is flattened sufficiently such that the pythagorean comma becomes descending. It's no longer negative, and is simply a descending diminished 2nd. However, negative 2nds do occur in meantone. (In fact, multiple negative 2nds, 3rds, etc. inevitably occur in every tuning of rank-2 or higher. We can simply repeatedly diminish a 2nd or a 3rd until it becomes descending, then flip it to make it ascending.) In the case of meantone, the kleisma ([[fifthspan]] of +19) is a negative 2nd.

== Interval arithmetic ==
Adding or subtracting a negative interval is the same as subtracting or adding the corresponding positive interval.

For example, what is an octave plus a pythagorean comma? We must subtract a diminished 2nd from an octave. We know that P8 - m2 = M7. If we diminish what we're subtracting (m2), we will augment the result. Thus P8 - d2 = A7, an augmented 7th, e.g. C-B♯. Likewise a major 3rd minus a pythagorean comma is a diminished 4th, e.g. C-F♭. An extreme example: the sum of two pythagorean commas is a negative triply-diminished 3rd, e.g. C-A♯♯♯.

== Prevalence in just intonation ==
Within a single piece of music, it's quite rare to find two notes a pythagorean comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the [[schisma]] = [-15 8 1⟩ = 2¢, also rare.

But in other tunings negative 2nds are commonplace. For example, in 7-limit JI, the interval from [[7/5]] (a diminished 5th) up to [[10/7]] (an augmented 4th) is [[50/49]] = 35¢, a negative diminished 2nd. Furthermore, the interval from [[16/15]] (a minor 2nd) up to [[15/14]] (an augmented unison) is [[225/224]] = 8¢, another negative diminished 2nd.

Negative minor 2nds are possible but rare. For example, [[1728/1715]] = [6 3 -1 -3⟩ = 13¢ is equal to ([[8/5]])/([[7/6]])<sup>3</sup>, a minor 6th minus three minor 3rds.

TO DO: convert wikipedia links to footnotes.