Constrained tuning: Difference between revisions
m →CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning: Moved the introduction which was opinionated into the criticisms section and retitled the criticism and defense of CTE sections for clarity. |
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== CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning == | == CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning == | ||
=== Criticism of CTE === | |||
People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0-386-720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle. | People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0-386-720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle. | ||
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As a result of this, historically, the POTE tuning was used instead, which tunes it to a result that is approximately [[delta-rational]] 0-400-720 cents. People have also suggested using the Kees-Euclidean or KE tuning, also known as the constrained-Weil-Euclidean or CWE tuning. Here is a summary of the math involved and the historical reasoning behind this. | As a result of this, historically, the POTE tuning was used instead, which tunes it to a result that is approximately [[delta-rational]] 0-400-720 cents. People have also suggested using the Kees-Euclidean or KE tuning, also known as the constrained-Weil-Euclidean or CWE tuning. Here is a summary of the math involved and the historical reasoning behind this. | ||
The CTE tuning can be thought of as a modified TE tuning in which the weighting (in monzo space) on the 2/1 coordinate has been changed to 0, making it a kind of seminorm rather than a norm. As a result, all elements in the same octave-equivalence class are weighted identically: they are all given complexity equal to the ''representative'' in each equivalence class in which all factors of 2 have been removed. Thus 5/4 is given the same complexity as 5/1, 13/8 as 13/1, and so on. | The CTE tuning can be thought of as a modified TE tuning in which the weighting (in monzo space) on the 2/1 coordinate has been changed to 0, making it a kind of seminorm rather than a norm. As a result, all elements in the same octave-equivalence class are weighted identically: they are all given complexity equal to the ''representative'' in each equivalence class in which all factors of 2 have been removed. Thus 5/4 is given the same complexity as 5/1, 13/8 as 13/1, and so on. | ||
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This is one of the reasons why the tuning of 1-5/4-3/2 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 1-5/4-3/2 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 compact 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, instead of 1-5/4-3/2. | This is one of the reasons why the tuning of 1-5/4-3/2 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 1-5/4-3/2 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 compact 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, instead of 1-5/4-3/2. | ||
=== | === Defense of CTE === | ||
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. | 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. | ||