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

VectorGraphics at 20:46, 8 April 2025

2025-04-08T20:46:03Z

Lériendil at 01:56, 23 March 2025

2025-03-23T01:56:23Z

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

Lériendil at 01:55, 23 March 2025

2025-03-23T01:55:37Z

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

VectorGraphics at 23:35, 22 March 2025

2025-03-22T23:35:30Z

← Older revision		Revision as of 23:35, 22 March 2025
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	* 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 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 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.		* 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.

VectorGraphics at 22:51, 22 March 2025

2025-03-22T22:51:09Z

← Older revision		Revision as of 22:51, 22 March 2025
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	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.)		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":
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			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.

VectorGraphics: Created page with "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..."

2025-03-22T04:12:47Z

Created page with "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..."

New page

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:
{| class="wikitable"
|+
!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.)