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> | 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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== Chapter IV. Art of Compromise == | == Chapter IV. Art of Compromise == | ||
Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave! This essay will be no end of a debate, but inviting more. It is high time we confront the last hard problem: compositeness of the harmonics. | Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' (''c''. 122 BC) – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave!<ref>"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", ''Ethnomusicology''. Fritz A. Kuttner. </ref> This essay will be no end of a debate, but inviting more. It is high time we confront the last hard problem: compositeness of the harmonics. | ||
If we play the interval of 15/1, does it somehow suggest 5/1 and 3/1? It seems even if we do not hear 15/1 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less. | If we play the interval of 15/1, does it somehow suggest 5/1 and 3/1? It seems even if we do not hear 15/1 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less. | ||
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with ''w''<sub>1</sub> being free since the octave is constrained. This weight always yields rational projection maps for some reasons. It can be used to tune all 5-limit temperaments alike, and the weight ratio between 5 and 3 is sqrt (2), very close to log<sub> | with ''w''<sub>1</sub> being free since the octave is constrained. This weight always yields rational projection maps for some reasons. It can be used to tune all 5-limit temperaments alike, and the weight ratio between 5 and 3 is 1/sqrt (2), very close to log<sub>5</sub> (3) in Tenney. In general, the weight ratio between ''q'' and 3 should be close to log<sub>''q''</sub> (3) and the exact values are left to readers to experiment with. | ||
== Chapter V. Towards an Optimization Strategy == | == Chapter V. Towards an Optimization Strategy == | ||