Wyschnegradsky and Blackwood

: Intro

Ivan Wyschnegradsky (1893 - 1979) and Easley Blackwood Jr. (1933 - 2033) were two noted microtonal composers who did a good amount of work in 24-EDO.

I'd like to learn more about their theories. 

: Wyschnegradsky

Wyschnegradsky laid out his theoriest in his "Manual of Quarter-Tone Harmony", but I haven't read it. I'll start with things I've heard about it.

Wyschnegradsky used quartertones in dissonant ornaments between consonances - passing tones, neighbor tones, maybe note cambiate - as well as using quartertones in unprepared grace notes, appoggiaturas.

He also looked at triads and tetrads which were altered from traditional tonal ones by a qurter tone in one note.

He also had a special 13-note scale. He probably had lots of scales, but this is more species. It's called "the quasi-diatonic" or "the diatonized chromatic scale" or "the chromatic scale diatonicized to 13 tones". It has 13 notes.

Suppose we start at zero steps of 24-EDO, and add on 11 steps repeatedly, subtracting 24 if we exceed 24. Then we get this sequence:

    [0, 11, 22, 9, 20, 7, 18, 5, 16, 3, 14, 1, 12, ...] -> 

Which, if we cut it short at 13 notes, can be ordered to give this sequence:

    [0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22] 

which has these relative intervals, plus 2 steps to reach the octave:

    [1, 2, 2, 2, 2, 2]  [1, 2, 2, 2, 2, 2] + [2]

Wyschnegradsky's quasi-diatonics scale is a rotation and rebracketing of this. Here it is in absolute steps:

[0, 2, 4, 6, 8, 10, 11, 13, 15, 17, 19, 21, 23, 24]

And relative steps:

     [2, 2, 2, 2, 2, 1] [2] [2, 2, 2, 2, 2, 1]

and pitches, not quite in his notation but hopefully close enough:

[C, C#, D, D#, E, F, Ft, Gd, Gt, Ad, At, Bd, Bt, C]

This scale has two identical subscales of 7 notes spanning 11\24, which are joined by a 2\24 minor second. The same way you can modulate from C major to F major or G readily by major moving around a circle of 5ths, in the process retaining all the notes of one tetrachord, Wyschnegradsky would readily modulate from this scale to one rooted on Ft or Gd, along the circle of 11\24 steps, and retain notes of the upper or lower tetrachord.

Wyschnegradsky is basically giving up on using the perfect fifth with this scale, but has to option to use the octave reduced 11th harmonic, aka the ascendant fourth, justly tuned to 11/8 and tuned by 24-EDO to 11\24 all over the place.

I think he would commonly form chords within this scale by going up three scale degrees. Here are tetrads constructed in that way on each scale degree:

[C, D#, Ft, Ad]

[C#, E, Gb, At]

[D, F, Gt, Bd]

[D#, Ft, Ad, Bt]

[E, Gd, At, C]

[F, Gt, Bd, C#]

[Ft, Ad, Bt, D]

[Gd, At, C, D#]

[Gt, Bd, C#, E]

[Ad, Bt, D, F]

[At, C, D#, Ft]

[Bd, C#, E, Gd]

[Bt, D, F, Gt]

If I wanted to sound a little like Wyschnegradsky, I'd play through the skip-3 triads and tetrads like these, see which ones chain together well, and start making progressions.

That's about all that I know about Wyschnegradsky. I actually learned all of this by mistake: I got Wyschnegradsky confused with Blackwood and looked up the wrong microtonalist. But some of it was interesting.

:: Blackwood

Blackwood was a great innovator of microtonal counterpoint. I don't know that his theories are necessary to make microtonal counterpoint, but boy does he have a productive theory.

For comparison, the theory of microtonal counterpoint that I've been coding up takes  normal rules of counterpoint and expands the set of intervals that are considered melodically fluid and harmonically consonant. Do that, and you get microtonal counterpoint for free.

Blackwood, in contrast, creates a new harmonic category and treats it differently: for him there are consonancea, dissonances, and even harsher discords.

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