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Old 27th March 2013, 07:16 AM   #95
calebprime
Penultimate Amazing
 
Join Date: Jul 2006
Posts: 13,001
Thanks, I feel much better.

In fact, mapping a 16 or 12 or 10-element series to a 7-note scale works ok. It just means that there are moments when intervals other than octaves are produced in the extreme registers. Mostly this is ok. Everything is a trade-off. A 16-element series mapped to a 7+7 diatonic scale occasionally creates an interval of 2 octaves and a third when you exceed the confines of one modulus. That's a fairly euphonious interval, so all is well.


Originally Posted by calebprime View Post
B..D..C..Eb.Bb.F#.C#.E..Ab.G..F..A.
Ab.F#.A..E..C..G..Bb.D..C#.B..Eb.F.
Ab.C..A..Bb.F#.G..E..D..B..C#.F..Eb
C..Eb.Bb.F#.C#.E..D..F..G..B..A..Ab



...
Having chosen my foreground patterns (flesh) and this background chorale (bones), there's a stage that involves some simple analysis.

The analysis involves figuring out which scales will work.

Each chord might go with any of basically 6 types of scales: Whole tone, hexatonic (1+3 semitones), major, melodic minor, harmonic major, harmonic minor, or octatonic (1/2 step + whole step, 1+2 semitones). Harmonic major and harmonic minor are the same "type". All the modes of these scale-types are included, so major includes Dorian, Lydian, Locrian, etc.

I'm only including scales with no 2 adjacent minor seconds, for purposes of sanity, euphony, and best mapping.

For now, I'm calling an overtone series scale Melodic Minor type.

Even with these simplifying assumptions, things get slightly more complicated.

Each chord can be thought of as:

1) By itself
2) With an A pedal
3) With the next chord
4) With the previous chord.

In each of these four cases, different scales are possible.

"Possible" is defined as: No 012, or two adjacent semitones (too dissonant and not a scale).

Each chord in each of the four scenarios above has a specific "space" or set of possible scales.

A is my overall tonality, but I might want to move away from it. I might want a smooth connection with the next chord (3), or a more abrupt one (1).

This is not taking into account possible staggering of voices -- where one or another comes in early or late -- which reduces the possible number of scales, usually. It changes the game.

The first chord, C,Ab,Ab,B can be: Ab Hexatonic, A Harmonic Minor, E Harmonic Major, Ab/B Octotonic, or A Melodic Minor, when considered purely by itself.

The exact intonation of a given scale has a number of possibilities even within the basic material of my tuning system, which is an 87EDO approximation of these ratios (36 pitches per octave) 1/1, 16/15, 13/12, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 32/27, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 15/11, 11/8, 7/5, 10/7, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 15/8.

This 36-note set can be thought of modally, also, meaning that there are a number of options for tuning the chorale, by moving the tuning base or anchor -- the 1/1 -- to different frequencies.

Self-similar patterns tend to create every possible combination of pitches within a scale, so this tends to make your choice of tunings more conservative, more 3-limit, if you will. The first priority is to avoid the bad approximations of 5ths (3/2 ratios) that inevitably seem to occur. In ye olden days, these were called "wolf" 5ths.

Perhaps a meantone tuning or 12-tone equal temperament would be better for a piece generated by self-similar series, for the above reason. But I'm committed to my 36-note tuning system, and have practiced it so long that I can play it in my sleep, so it's fairly "easy".

The piece might have passages where the self-similar series are not being used, where the strengths of the 36-note 13-limit tuning can really be heard.


The seductive thing about the self-similar idea is that you can generate complex textures from a little string of numbers as easily as falling off a log. It's a little music-making machine, and I find machines of this kind fascinating. Polyrhythms are a natural thing to explore in this world, because they're already there, embedded in the series.

Last edited by calebprime; 27th March 2013 at 07:27 AM.
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