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 |
|---|---|---|---|
| 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, harmonic seventh | 12 | 960 | 7/4, 16/9 |
| Dominant 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.)