Kite's thoughts on negative intervals: Difference between revisions

(4 intermediate revisions by 3 users not shown)

Line 1:

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 "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 ==

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 ==

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 ==

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.

Line 22:

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.

== See also ==

* [[Undirected value]]

~~TO DO~~: ~~convert wikipedia links to footnotes.~~

== Notes ==

[[Category:Interval]]

[[Category:Terms]]

@@ Line 1: / Line 1: @@
-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&#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.
+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. (See [[wikipedia:Interval_(music)|en.wikipedia.org/wiki/Interval_(music)]].)
+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>.
 "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.
-"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 ==
-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 ==
 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&#x266F;. 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&#x266F;&#x266F;&#x266F;.
+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&#x266F;. 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&#x266F;&#x266F;&#x266F;.
 == 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.
@@ Line 22: / Line 22: @@
 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.
+== See also ==
+* [[Undirected value]]
-TO DO: convert wikipedia links to footnotes.
+== Notes ==
+<references group="note"/>
+[[Category:Interval]]
+[[Category:Terms]]