Constrained tuning: Difference between revisions
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→CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning: I'll try to find out the thread that suits a reference for the claim. For now I'm labeling it unsourced; style |
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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 == | ||
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. 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. | ||
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 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, | ||
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As a result of this, as a result, historically, the POTE tuning was used instead, which tunes it to the much less lopsided (and 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, as a result, historically, the POTE tuning was used instead, which tunes it to the much less lopsided (and 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 | === The problem === | ||
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 | 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. | ||
One criticism that has sometimes been brought up is to note the interval 13/1 is huge: it's four octaves large. We may not really care about 13/1 more than 13/8, or 15/1 more than 15/8, and so on. Instead, we often care most about intervals which are maybe within an octave or two at most in span. This can be viewed as a criticism of Tenney-weighting in general, perhaps, but it is has often been noted that the situation makes little difference there, | One criticism that has sometimes been brought up is to note the interval 13/1 is huge: it's four octaves large. We may not really care about 13/1 more than 13/8, or 15/1 more than 15/8, and so on. Instead, we often care most about intervals which are maybe within an octave or two at most in span. This can be viewed as a criticism of Tenney-weighting in general, perhaps, but it is has often been noted that the situation makes little difference there, | ||
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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. | 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. | ||
=== Using the Weil | === Using the Weil norm or Kees expressibility === | ||
Another way to solve this problem is to actually go back to the original objection that we perhaps don't care about 13/1 as much as 13/8 - or at least, that we don't care about it that much if we have to assign it to the entire equivalence class. So, we can take this objection seriously and use a different norm to begin with. | Another way to solve this problem is to actually go back to the original objection that we perhaps don't care about 13/1 as much as 13/8 - or at least, that we don't care about it that much if we have to assign it to the entire equivalence class. So, we can take this objection seriously and use a different norm to begin with. | ||