Delta-rational chord: Difference between revisions

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=== Limitations ===

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, 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 inaccurate. This is because DR simplicity is only preserved by inversion and revoicing for simple JI chords.

For example, take 0.0570 (the LSD 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 LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. 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, all three inversions are barely acceptable to unacceptable: 4:6:7 has LSD error 0.0574, 6:7:8 has LSD error 0.0652, and 7:8:12 has LSD error 0.0570. Of course, this depends on what error tolerance you prefer.

== Finding approximate DR chords in edos ==

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+=== Limitations ===
+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, 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 inaccurate. This is because DR simplicity is only preserved by inversion and revoicing for simple JI chords.
+For example, take 0.0570 (the LSD 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 LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. 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, all three inversions are barely acceptable to unacceptable: 4:6:7 has LSD error 0.0574, 6:7:8 has LSD error 0.0652, and 7:8:12 has LSD error 0.0570. Of course, this depends on what error tolerance you prefer.
 == Finding approximate DR chords in edos ==