Constrained tuning: Difference between revisions

Line 243:

The problem becomes much more severe when we essentially assign these weightings to entire equivalence classes. In this situation, all of these small, intra-octave intervals essentially have their weightings scrambled relative to one another. Small intervals which used to be similarly-calibrated are now totally different: 16/13 is weighted much more strongly than 6/5; 15/8 and 6/5 are given equal weight; 5/4 is now much more important than 6/5, and so on. This is because some of these intervals happen to be octave-equivalent to prime numbers 3-4 octaves up, and others aren't. As a result, when we build simple triads out of these manageably-sized intervals, they tend to be tuned in a very lopsided fashion.

This is one of the reasons why the tuning of 4:5:6 is so skewed in CTE Blackwood~~: 5/4 is prioritized more strongly than 6/5~~. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE Blackwood has 4:5:6 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes important triads has not quite translated to CTE. ~~What has basically happened~~ is that 1:3:5 is tuned as nicely as possible, at the cost of 4:5:6.

This is one of the reasons why the tuning of 4:5:6 is so skewed in CTE Blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE Blackwood has 4:5:6 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes important triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1:3:5 is tuned as nicely as possible, at the cost of 4:5:6.

The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 4:5:6 to 0-400-720, does much better in translating this useful property of TE to a pure-octave tuning: the relative sizes of all intervals are preserved. Thus, we retain these nice properties for small intervals, as well as small triads, etc. In fact, one notes that if one were to actually measure the tuning error on all triads, tetrads, etc, as well as dyads, we may very well get something closer to the POTE tuning than the CTE tuning. One also notes that "de-stretching" the POTE tuning is, to first order, approximately the same as stretching all chords in it "linearly," so that "isoharmonic," "proportional," "delta-rational" chords remain so after the transformation (approximately).

@@ Line 243: / Line 243: @@
 The problem becomes much more severe when we essentially assign these weightings to entire equivalence classes. In this situation, all of these small, intra-octave intervals essentially have their weightings scrambled relative to one another. Small intervals which used to be similarly-calibrated are now totally different: 16/13 is weighted much more strongly than 6/5; 15/8 and 6/5 are given equal weight; 5/4 is now much more important than 6/5, and so on. This is because some of these intervals happen to be octave-equivalent to prime numbers 3-4 octaves up, and others aren't. As a result, when we build simple triads out of these manageably-sized intervals, they tend to be tuned in a very lopsided fashion.
-This is one of the reasons why the tuning of 4:5:6 is so skewed in CTE Blackwood: 5/4 is prioritized more strongly than 6/5. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE Blackwood has 4:5:6 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes important triads has not quite translated to CTE. What has basically happened is that 1:3:5 is tuned as nicely as possible, at the cost of 4:5:6.
+This is one of the reasons why the tuning of 4:5:6 is so skewed in CTE Blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE Blackwood has 4:5:6 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes important triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1:3:5 is tuned as nicely as possible, at the cost of 4:5:6.
 The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 4:5:6 to 0-400-720, does much better in translating this useful property of TE to a pure-octave tuning: the relative sizes of all intervals are preserved. Thus, we retain these nice properties for small intervals, as well as small triads, etc. In fact, one notes that if one were to actually measure the tuning error on all triads, tetrads, etc, as well as dyads, we may very well get something closer to the POTE tuning than the CTE tuning. One also notes that "de-stretching" the POTE tuning is, to first order, approximately the same as stretching all chords in it "linearly," so that "isoharmonic," "proportional," "delta-rational" chords remain so after the transformation (approximately).