Constrained tuning: Difference between revisions

Line 338:

=== 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 [[Yahoo Groups tuning lists]], 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.

This sort of thing was important ~~historically~~ when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,

This sort of thing was important to early [[regular temperament theory]] pioneers when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,

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, many use the POTE tuning, 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 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.

Line 352:

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 ===

==== 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.

Line 364:

Another way to think of it is that as POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus can be thought of as sacrificing multiplicative (typically very large) ratios for divisive (typically very small) ratios, whereas CTE sticks to the original design book of TE-optimality without worrying about that.

~~Historically, there was also an observation~~ that ~~the~~ POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.

Observe that POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.

=== Using the Weil norm or Kees expressibility ===

@@ Line 338: / Line 338: @@
 === 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 [[Yahoo Groups tuning lists]], 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.
-This sort of thing was important historically when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,
+This sort of thing was important to early [[regular temperament theory]] pioneers when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,
-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, many use the POTE tuning, 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 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.
@@ Line 352: / Line 352: @@
 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 ===
+==== 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.
@@ Line 364: / Line 364: @@
 Another way to think of it is that as POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus can be thought of as sacrificing multiplicative (typically very large) ratios for divisive (typically very small) ratios, whereas CTE sticks to the original design book of TE-optimality without worrying about that.
-Historically, there was also an observation that the POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.
+Observe that POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.
 === Using the Weil norm or Kees expressibility ===