Negative interval

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 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 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)³, a minor 6th minus three minor 3rds.

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