Intervals

[Dan]

The Music page of the DSGB website recommends referring to intervals by the number of semitones between the two notes, and thereby to ignore the distinction between, e.g., augmented fourths and diminished fifths. This works on an equally-tempered instrument, but not necessarily in general.

It is however possible to get the precision of traditional interval names without the complications of inclusive counting and remembering which ones are "perfect" and which ones have "major" and "minor" varieties. An interval can be specified in terms of the physical interval, the number of semitones (modulo 12), and the scale interval, the number of scale degrees (modulo 7), using half-inclusive counting.

An interval can be notated by a (physical interval, scale interval). The intervals within the major scale are:

0,0 = unison
1,1 = minor second
2,1 = major second
3,2 = minor third
4,2 = major third
5,3 = perfect fourth
6,3 = augmented fourth
6,4 = diminished fifth
7,4 = perfect fifth
8,5 = minor sixth
9,5 = major sixth
Χ,6 = minor seventh
Ɛ,6 = major seventh

Futhermore, a note can be denoted as root+interval. For example:

D=C+(2,1)
E=C+(4,2)
F=C+(5,3)
G=C+(7,4)
A=C+(9,5)
B=C+(Ɛ,6)

To be continued...

[Listerine]

I think a chromatic interval could be specified along with a diatonic scale degree. The scale degrees would be counted inclusively, but cyclically (first, second, third, fourth, fifth, sixth, sevth, first, second...).

Then you'd have this (chromatic interval on left, scale degree on right)...


0th-1st / unison (P1)
1st-2nd (m2)
2nd-2nd (M2)
3rd-3rd (m3)
4th-3rd (M3)
5th-4th (P4)
6th-4th (A4)
6th-5th (d5)
7th-5th (P5)
8th-6th (m6)
9th-6th (M6)
th-7th (m7)
th-7th (M7)
10th-1st (P8)
11st-2nd (m9)
and so on...

A diminished third would be a 2nd-3rd, a diminished seventh would be a 9th-7th, etc...

[Listerine]

I've settled on this; label the degrees of a scale A B C D E F G, which are movable. Then just attach the physical span in dozenths of an octave (A10)

A0 = unison
B1 = minor second
B2 = major second
C3 = minor third
C4 = major third
D5 = perfect fourth
D6 = augmented fourth
E6 = diminished fifth
E7 = perfect fifth
F8 = minor sixth
F9 = major sixth
Gλ = minor seventh
GƐ = major seventh
A10 = octave
B11 = minor ninth
B12 = major ninth
C13 = minor tenth
C14 = major tenth
etc...

And of course a B0 would be a diminished second and all that.

Every triad is A C E (be it A0 C4 E7, A0 C3 E7, A0 C3 E6, A0 C4 E8). Every seventh chord (we don't refer to them as such, though) is A C E G. Then ya have A C E G B, A C E G B D and so on.

Use a degree symbol to refer to a specific scale degree (it doesn't mean diminished). In a major scale, °F is B2 above °E. In a natural minor scale, °F is B1 above °E.

To show a diatonic triad, give the degree of the root and a superscript 2 to show it's tertiated twice over through the scale. °A² is a triad starting from the root, the tonic chord. A superscript 3 shows a G ("seventh") chord. °E³ would be a dominant seventh chord in a major key. And you can go higher, using superscript 4, 5 and 6. Other intervals can be added to further modify them. + and - serve as accidentals.

With this system, we don't really rely on the major scale for interval names, and it serves tonal music just the same as modal.

If you're using someone other than a heptatonic scale, letters may be added or taken away. ABCDE for pentatonic and ABCDEFGH for octatonic, for example.