"Tonality" revisited, Part II

> > A short interlude on the definition of the word "Tonality"
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> > In the previous article concerning this, Tonality was used improperly. Properly, it means a system of organizing music according to specific rules of consonance, dissonance, and chord progression. That this system is superior to others is unprovable.
> > The improper way I used it, it meant the ordering of pitches according to a specific method that is much broader than and includes tonality, yet does not include all pitched music, such as some, but not all, serial and set-class pieces.
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> > On the mathematics
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> > Two vibrations at the unison have a ratio of 1:1
> > Two vibrations at the octave have a ratio of 1:2
> > Two vibrations at the octave+perfect fifth have a ratio of 1:3
> > Two vibrations at the perfect fifth have a ratio of 2:3
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> > Assuming that two notes an octave apart are equivalent notes, which seems a self-evident assumption, though it might not be, the next notes in the series are
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> > Two vibrations at the major second have a ratio of 8:9
> > Two vibrations at the major sixth have a ratio of 16:27
> > Two vibrations at the major third have a ratio of 64:81
> > Two vibrations at the major seventh have a ratio of 128:243
> > Two vibrations at the tritone have a ratio of...you figure it out.
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> > Invert using octave equivalence, and you now have all the possible relationships between notes in the order of simplest ratio to most complex ratio. According to me and many other theorists who I follow, the music ought to incorporate these into its composition as per above. The closer the relathionship
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> > On the answering of objections
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> > Obj. Saying that if two objects have one mathematical ratio as the relationship between them and two other objects have a different one, then one set is more closely related than the other is arbitrary or determined by biological or cultural factors, for mathematics does not decree one relationship as better than another.
> > Ans. It is true that many things are related merely arbitrarily, culturally, or because of biology, but non-arbitrary relationships can exist. For example, the relationship between me and my parents is necessarily a closer relationship than the relationship between me and my cousins, because if I would have had different parents, I would be a different person, but if I had a different cousin, I would remain the same person; thus the relationship of parent is essential, while the relationship of cousin is not.
> > Into what category do ratios fall? It seems that it is self evident, independent of biology or culture, that the closer the relationship approaches 1 (or the closer the notes approach to a unison), the closer (and thus, more relational) the relationship between them; a yardstick is more closely related to a meterstick than it is to a 2.5-meterstick because it is more close to it in length.
> > The other sort of self-evident, culturally and biologically independent relationship is a bit more complicated. Consider the intervals of perfect fifth and major sixth, with the ratios of 2/3 and 16/27 respectively. In regards to choosing between them, the composer can say "I choose ____" or "I choose ____ because of ____" If they are trying to be completely objective and independent from biology and culture and they choose to have the second option, what is left to them is to choose on the basis of the ratios. And which makes more sense: to say "I choose the fifth because it has the relationship of the ratio of 2/3, not 16/27," or "I choose the sixth because it has the relationship of the ratio of 16/27, not 2/3?" The second statement is hilarious and ludicrous and unbelieveable and seems to be self-evidently less valid than the first. One could, of course, simply say "I choose ____" or even "I aleotorate ___," but then, they are choosing, and, considered in the abstract, their choice has no basis. It would be better to choose with an abstract basis in the way outlined above.
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