Delta-rational chord: Difference between revisions

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Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them less accurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI; that all inversions of a chord preserve the JI chord's delta signature with low enough error is thus a sensible criterion for a "good" temperament.

For example, take 0.00807 (the ~~naive~~ least-squares error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with ~~naive~~ least-squares error '''0.0118'''; the 7:8:12 has an even higher error of '''0.0178'''. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, the 4:6:7, 6:7:8, and 7:8:12 have errors '''0.0108''', '''0.0106''', and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, '''0.0112''', and '''0.0126'''.

For example, take 0.00807 (the direct least-squares error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with direct least-squares error '''0.0118'''; the 7:8:12 has an even higher error of '''0.0178'''. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, the 4:6:7, 6:7:8, and 7:8:12 have errors '''0.0108''', '''0.0106''', and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, '''0.0112''', and '''0.0126'''.

== Higher-order differences of frequency ==

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 Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them less accurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI; that all inversions of a chord preserve the JI chord's delta signature with low enough error is thus a sensible criterion for a "good" temperament.
-For example, take 0.00807 (the naive least-squares error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with naive least-squares error '''0.0118'''; the 7:8:12 has an even higher error of '''0.0178'''. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, the 4:6:7, 6:7:8, and 7:8:12 have errors '''0.0108''', '''0.0106''', and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, '''0.0112''', and '''0.0126'''.
+For example, take 0.00807 (the direct least-squares error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with direct least-squares error '''0.0118'''; the 7:8:12 has an even higher error of '''0.0178'''. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, the 4:6:7, 6:7:8, and 7:8:12 have errors '''0.0108''', '''0.0106''', and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, '''0.0112''', and '''0.0126'''.
 == Higher-order differences of frequency ==