Kite's thoughts on negative intervals: Difference between revisions

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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.

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.

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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.

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.

~~== 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 ==

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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&#x266F; 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&#x266F; and D♭ (or equivalently between D♭ and C&#x266F;) 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&#x266F; and D♭ (or equivalently between D♭ and C&#x266F;) 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&#x266F; 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&#x266F; 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.
-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.
-== 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 ==