User:VectorGraphics/Vector's introduction to 15edo/Intervals

15edo contains 15 notes per octave, and thus 15 intervals to use in your music.

In order to classify these intervals, we will be using a "diatonic scale" with two different types of whole tones. Further justification for this scale can be found at User:VectorGraphics/Vector's introduction to 15edo/Scales.

A primer: Just intonation

In equal tunings, composers often think of harmony either in "just intonation", or some level of abstraction away from it. Just intonation is the tuning where all intervals are pure integer ratios, which is easy to tune and provides a distinct kind of mathematical structure. In this introduction, we will be providing just ratios that each interval corresponds to, but usually, you don't want to be working directly with these correspondences, partially because the JI ratios aren't actually all that close to 15edo's intervals. (Think about how 12edo composers almost never reference concepts from just intonation.)

Usually, just intonation intervals are generated by "stacking" (multiplying) primes. This looks like addition on a keyboard, but that's because pitch is logarithmic. While 12edo is a good representation of many ratios involving 2, 3, and 5, it falls flat at 7 (well, sharp, actually), and fails completely at 11, so 12edo can be described as a 5-limit system. However, 15edo is an 11-limit system, only failing at 13.

Interval categories

Here is a table of 15edo's intervals:

Name	Degree	Cents	Approximate Ratios	Note
Unison	0	0	1/1
Semitone	1	80	25/24, 16/15
Minor tone	2	160	10/9
Major tone, wolf third	3	240	8/7, 9/8
Minor third	4	320	6/5
Major third	5	400	5/4
Perfect fourth	6	480	4/3, 21/16
Small tritone, diminished fifth, wolf fourth	7	560	11/8, 7/5
Large tritone, augmented fourth, wolf fifth	8	640	16/11, 10/7
Perfect fifth	9	720	3/2, 32/21
Minor sixth	10	800	8/5
Major sixth	11	880	5/3
Wolf sixth, narrow minor seventh	12	960	7/4, 16/9
Wide minor seventh	13	1040	9/5
Major seventh	14	1120	48/25, 15/8
Octave	15	1200	2/1

Let's take a look at the zarlino diatonic scale (more elaboration on the structure of this scale in the page linked at the top). Zarlino is an MV3 scale, meaning that there are at most 3 sizes of any given interval.

So, let's lay out all the modes of zarlino, and see where our scale degrees fall:

1..2.34..5.6..71 // Ionian
1.2..34..5.6..71 // Ionian
1.23..4.5..67..1 // Dorian
1..23..4.5..67.1 // Dorian
12..3.4..56..7.1 // Phrygian
12..3.4..56.7..1 // Phrygian
1..2.3..45..6.71 // Lydian
1..2.3..45.6..71 // Lydian
1.2..34..5.67..1 // Mixolydian
1.2..34.5..67..1 // Mixolydian
1..23..4.56..7.1 // Aeolian
1..23.4..56..7.1 // Aeolian
12..3.45..6.7..1 // Locrian
12.3..45..6.7..1 // Locrian
. 

As you can see, except for seconds, the main diatonic major/minor dichotomies remain intact, albeit with the occasional exception. Specifically, for thirds, fourths, fifths, and sixths, you find "wolf intervals" on two of the fourteen modes for each, which for fifths are between perfect and diminished, and for thirds, they are below minor. Each of these types of wolf intervals is special because they correspond to the prime harmonics we have access to in 15edo, but not in 12edo: the wolf sixth, for example, represents the harmonic seventh ratio of 7/4 (and the wolf third represents its complement 8/7) (but note that these are also our largest seconds and smallest sevenths), and the wolf fourth represents the undecimal tritone, 11/8 (and of course, the wolf fifth represents its complement 16/11.)

The torus

In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as contorted in 2.3.7 and 2.5. So, the only solution is to use a two-dimensional "circle", which forms a torus-shaped "harmonic table":

0c	400c	800c	0c	400c	800c	0c
480c	880c	80c	480c	880c	80c	480c
960c	160c	560c	960c	160c	560c	960c
240c	640c	1040c	240c	640c	1040c	240c
720c	1120c	320c	720c	1120c	320c	720c
0c	400c	800c	0c	400c	800c	0c
480c	880c	80c	480c	880c	80c	480c
960c	160c	560c	960c	160c	560c	960c
240c	640c	1040c	240c	640c	1040c	240c
720c	1120c	320c	720c	1120c	320c	720c
0c	400c	800c	0c	400c	800c	0c

We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

• fifths are found by stepping 1 step up the Y-axis.
• harmonic sevenths are found by stepping 2 steps down the Y-axis.
• major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

• the diatonic semitone, or limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
• the chromatic semitone, or chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
• the ptolemaic semitone, or syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
• the compound semitone, or diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this "compound semitone" is actually the complement of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

Note Names

Type	Name	Cents	Note on C
1sn	Unison	0	C
1sn	Chroma	80	C#
2nd	Diminished second	0	Dbb
2nd	Minor second	80	Db
2nd	Minor tone	160	D
2nd	Major tone	240	D#
2nd	Augmented second	320	D##
3rd	Diminished third	160	Ebbb
3rd	Wolf third	240	Ebb
3rd	Minor third	320	Eb
3rd	Major third	400	E
3rd	Augmented third	480	E#
4th	Diminished fourth	400	Fb
4th	Perfect fourth	480	F
4th	Wolf fourth	560	F#
4th	Augmented fourth	640	F##
5th	Diminished fifth	560	Gbb
5th	Wolf fifth	640	Gb
5th	Perfect fifth	720	G
5th	Augmented fifth	800	G#
6th	Diminished sixth	720	Abb
6th	Minor sixth	800	Ab
6th	Major sixth	880	A
6th	Wolf sixth	960	A#
6th	Augmented sixth	1040	A##
7th	Diminished seventh	880	Bbbb
7th	Narrow minor seventh	960	Bbb
7th	Wide minor seventh	1040	Bb
7th	Major seventh	1120	B
7th	Augmented seventh	1200	B#
8ve	Diminished octave	1120	Cb
8ve	Octave	1200	C