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

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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..."
 
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!Cents
!Cents
!Approximate Ratios
!Approximate Ratios
!Note
|-
|-
|Unison
|Unison
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|0
|0
|1/1
|1/1
|
|-
|-
|Semitone
|Semitone
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|80
|80
|'''25/24,''' 16/15
|'''25/24,''' 16/15
|
|-
|-
|Minor tone
|Minor tone
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|160
|160
|'''10/9'''
|'''10/9'''
|
|-
|-
|Major tone, wolf third
|Major tone, wolf third
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|240
|240
|'''8/7''', 9/8
|'''8/7''', 9/8
|
|-
|-
|Minor third
|Minor third
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|320
|320
|'''6/5'''
|'''6/5'''
|
|-
|-
|Major third
|Major third
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|400
|400
|'''5/4'''
|'''5/4'''
|
|-
|-
|Perfect fourth
|Perfect fourth
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|480
|480
|'''4/3''', 21/16
|'''4/3''', 21/16
|
|-
|-
|Small tritone, diminished fifth, wolf fourth
|Small tritone, diminished fifth, wolf fourth
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|560
|560
|'''11/8''', 7/5
|'''11/8''', 7/5
|
|-
|-
|Large tritone, augmented fourth, wolf fifth
|Large tritone, augmented fourth, wolf fifth
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|640
|640
|'''16/11''', 10/7
|'''16/11''', 10/7
|
|-
|-
|Perfect fifth
|Perfect fifth
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|720
|720
|'''3/2''', 32/21
|'''3/2''', 32/21
|
|-
|-
|Minor sixth
|Minor sixth
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|800
|800
|'''8/5'''
|'''8/5'''
|
|-
|-
|Major sixth
|Major sixth
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|880
|880
|'''5/3'''
|'''5/3'''
|
|-
|-
|Wolf sixth, harmonic seventh
|Wolf sixth, narrow minor seventh
|12
|12
|960
|960
|'''7/4''', 16/9
|'''7/4''', 16/9
|
|-
|-
|Dominant seventh
|Wide minor seventh
|13
|13
|1040
|1040
|'''9/5'''
|'''9/5'''
|
|-
|-
|Major seventh
|Major seventh
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|1120
|1120
|'''48/25,''' 15/8
|'''48/25,''' 15/8
|
|-
|-
|Octave
|Octave
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|1200
|1200
|2/1
|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.
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.
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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":
{| class="wikitable"
|+
|'''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 ==
{| class="wikitable"
!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
|}
Retrieved from "https://en.xen.wiki/w/User:VectorGraphics/Vector%27s_introduction_to_15edo/Intervals"