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Modelling Tonality (1) Diatonic Transposition (some intuitions)

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I'm thinking about the formal conditions of the diatonic transposition.

It may seem trivial, but actually is much more difficult than it appears.

I think that modelling tonality and diatonic stuff is far more difficult

 than  the 12-tone operations.

One of the things I think is that in diatonic transposition you need to

 inform more to the machine, a kind of axis point or map, because the transposition

 of each degree is going to be different.

In the case of the diatonic set, the MyHill property assures that each diatonic distance

 will be in exactly 2 sizes. Seconds: major and minor; Thirds: major and minor;

Fourths: perfect and augmented; Fifths: perfect and diminished and so on...


The Morris pitch spaces are also part of the problem...


Do you have any hint in relation to this intuitions to share ?

Maybe the way that OM make this operations ?





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Here's my way for diatonic transposition.

It is very simple but do exactly what i want when composing


I use this system extensively in all my compositions now, not always in diatonic context but also with synthetic modes, row, algorithmic pitch material etc...

I love the concept of degree and transpositions inside a scale and use that technique very often.

(setf motiv '((q c4 e4 g4)(q c4 e4 g4)(q c4 e4 g4)(q c4 e4 g4)(q c4 e4 g4)))

(setf degree '(1 4 2 5 1))
(setf harmonic-path (harmonic-progression
                     '(c4 major)
                     :step 1
                     :size 7
                     :base 1
(setf p1 (tonality-map (mclist harmonic-path)  motiv))




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You already know how to translate a MIDI pitch number into a representation consisting of two pieces of information, a pitch class integer and an octave integer. The conversion can be applied in both directions.


Now, you can apply a similar conversion to a pitch class integers, converting them into a scale degree (integer) and an accidental (another integer), depending on a scale (a set of pitch classes). Again, this conversion can be done in both ways (there are multiple solutions if you allow for enharmonic equivalence).


Once you have such a representation, you can then do diatonic transpositions (depending on whatever scale) within the scale degree domain, and finally translate your  results back into pitch classes, or MIDI pitch numbers.


I have a draft of a paper discussing this formally. A preliminary version of this paper has been published at SMC, http://uobrep.openrepository.com/uobrep/handle/10547/622264 Don't get scared away by the constraint programming aspect. Such conversions can also be implemented as plain deterministic functions.


This can be seen as a kind of mapping, if you want, if you also see the relation between MIDI pitch numbers and pitch classes as a mapping :)





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Thank you, Torsten. You are digging a lot into the realm of modelling tonality. I think that this is really something more interesting than brownian motions, stochastic stuff,  because there are many constraints in the tonality and also idiomatisms. Models like the xenakian one are amazing but the are in the realm of math. I'd like to find something more idiomatic.



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