User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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== Chapter III. Power in Proportion == | == Chapter III. Power in Proportion == | ||
The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map P as | The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map ''P'' as | ||
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The ''n'' is a positive integer determining the highest relevant harmonic. Specifically, the weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit, so different values of ''n'' can alter the relative weights of the primes. | The ''n'' is a positive integer determining the highest relevant harmonic. Specifically, the weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit, so different values of ''n'' can alter the relative weights of the primes. | ||
To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have | To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have unit weight since they can only be stacked once in the integer limit. 11 and beyond have zero weight because they cannot be stacked at all. If optimization is to be carried out for a 13-limit temperament then we have the weights 3, 2, 1, 1, 0, 0 for primes 2 to 13. The weights are different if ''n'' = 7, or 7-integer-limit, for example. The weight of 2 is 2, of 3, 5 and 7 is unity, and of 11 and 13 zero, giving 2, 1, 1, 1, 0, 0 for primes 2 to 13. | ||
The Hahn[''n''] weight matrix is given as | The Hahn[''n''] weight matrix is given as | ||