Constrained tuning: Difference between revisions

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=== The defense ===

Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the ~~original version~~ being ~~part of~~ the [[chord of nature]]. It is thus important to recognize that ~~1-3-5 is the true basis of harmony and that 1~~-~~5/4-3/2 is~~ but ~~a simplification~~ for our cognitive processes.

Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.

Music making, that is when we are not abstractly naming the chords, is all about various open voicings. The archaic {{w|Alberti bass}} is one of the few examples of close voicing, used as a bassline to accompany other materials. It should be noted that 13/1, dismissed as too wide in the section above, is still within the range of a full choir, not to mention a {{w|rock band}}, {{w|concert band}} or {{w|orchestra}}. {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Symphony No. 3 (Beethoven)|''Symphony No. 3''}} opens with 1-2-5/2-4-5-6-8-10-12-16. Such a chord will be much overtempered, its tuning profile unreasonably squeezed and strained, if we set 1-5/4-3/2 as our target.

CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure~~. And furthermore,~~ any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.

CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure; any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.

=== Using destretch ===

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 === The defense ===
-Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the original version being part of the [[chord of nature]]. It is thus important to recognize that 1-3-5 is the true basis of harmony and that 1-5/4-3/2 is but a simplification for our cognitive processes.
+Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.
 Music making, that is when we are not abstractly naming the chords, is all about various open voicings. The archaic {{w|Alberti bass}} is one of the few examples of close voicing, used as a bassline to accompany other materials. It should be noted that 13/1, dismissed as too wide in the section above, is still within the range of a full choir, not to mention a {{w|rock band}}, {{w|concert band}} or {{w|orchestra}}. {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Symphony No. 3 (Beethoven)|''Symphony No. 3''}} opens with 1-2-5/2-4-5-6-8-10-12-16. Such a chord will be much overtempered, its tuning profile unreasonably squeezed and strained, if we set 1-5/4-3/2 as our target.
-CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure. And furthermore, any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.
+CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure; any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.
 === Using destretch ===