Constrained tuning: Difference between revisions
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'''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo|unit eigenmonzos, or unchanged-intervals]]). | '''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo|unit eigenmonzos, or unchanged-intervals]]). | ||
'''CTE tuning''', ('''constrained | '''CTE tuning''', ('''constrained Tenney–Euclidean tuning'''), is the most basic version of this idea. It has a more sophisticated variant, '''CWE tuning''' ('''constrained Weil–Euclidean tuning'''), a.k.a. '''KE tuning''' ('''Kees–Euclidean tuning'''), created to address some subtle problems perceived by the tuning-math community with CTE. Both of these are special cases of the '''CTWE tuning''' ('''constrained Tenney–Weil–Euclidean tuning'''). | ||
These tunings will be the focus of this article. Otherwise normed tunings can be defined and computed analogously. | These tunings will be the focus of this article. Otherwise normed tunings can be defined and computed analogously. | ||
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All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a {{w|least squares|least squares problem}}, CTE tuning can be viewed as an equality-constrained least squares problem. | All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a {{w|least squares|least squares problem}}, CTE tuning can be viewed as an equality-constrained least squares problem. | ||
The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 (pure-octave pure-fifth) constrained tuning. For a rank-''r'' temperament, specifying a rank-''m'' constraint list will yield {{nowrap|''r'' | The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 (pure-octave pure-fifth) constrained tuning. For a rank-''r'' temperament, specifying a rank-''m'' constraint list will yield {{nowrap|''r'' − ''m''}} {{w|degrees of freedom}} to be optimized. | ||
== Definition == | == Definition == | ||
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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 === | === 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 | 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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=== 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 | 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 often 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 | Music making, that is when we are not abstractly naming the chords, is often 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 | 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 === | === Using destretch === | ||
The POTE tuning, which simply "de-stretches" the TE tuning and tunes the | The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 1–5/4–3/2 to 0–400–720, translates the near-delta-rational 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). | ||
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. | 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. | ||
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=== Using the Weil norm or Kees expressibility === | === 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/ | 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. | ||
The Weil norm of <math>\max(n,d)</math> can be thought of as the average of Tenney Height and the interval's span, and so inherently does this: 5/1 and 5/4 are weighted equally, so that the psychoacoustic importance of the former and small manageable size of the latter balance out. We can then do a constrained optimization using the Weil norm, and if we are using the Weil-Euclidean norm, we get the '''constrained Weil-Euclidean''' or '''CWE tuning'''. The term '''Kees-Euclidean''' has also sometimes been used for this (although the term has occasionally been used to refer to a de-stretched Weil-Euclidean tuning instead). | The Weil norm of <math>\max(n,d)</math> can be thought of as the average of Tenney Height and the interval's span, and so inherently does this: 5/1 and 5/4 are weighted equally, so that the psychoacoustic importance of the former and small manageable size of the latter balance out. We can then do a constrained optimization using the Weil norm, and if we are using the Weil-Euclidean norm, we get the '''constrained Weil-Euclidean''' or '''CWE tuning'''. The term '''Kees-Euclidean''' has also sometimes been used for this (although the term has occasionally been used to refer to a de-stretched Weil-Euclidean tuning instead). | ||
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As mentioned above, if we constrain the equave to be pure, and look for the tuning map that is closest to the JIP using the WE norm, we get the CWE tuning, a.k.a. KE tuning. | As mentioned above, if we constrain the equave to be pure, and look for the tuning map that is closest to the JIP using the WE norm, we get the CWE tuning, a.k.a. KE tuning. | ||
It has sometimes been noted that the Weil norm can give less-than-perfect results in other | It has sometimes been noted that the Weil norm can give less-than-perfect results in other ways—for instance, it weights 13/8, 13/9, 13/10, 13/11, and 13/12 all equally. This doesn't seem to cause quite as much of a problem with the WE or KE tunings, or even the minimax Kees tuning, as it does with the minimax Weil tuning. But, this could sometimes be an issue. | ||
So, one simple solution is to interpolate between the two, giving the ''' | So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap|k {{=}} 0}} is the TE norm, {{nowrap|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE. | ||
=== Comparison === | === Comparison === | ||
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<math>\val{1200.000 & 1920.000 & 2786.314}</math> | <math>\val{1200.000 & 1920.000 & 2786.314}</math> | ||
Since prime 5 is not involved in the comma to begin with, it is understandable that it is tuned pure as in 5-limit JI. This, as mentioned above, leads to very lopsided behavior for compact chords like | Since prime 5 is not involved in the comma to begin with, it is understandable that it is tuned pure as in 5-limit JI. This, as mentioned above, leads to very lopsided behavior for compact chords like 1–5/4–3/2. Note that the tunings for KE and POTE distribute the error between 5/4 and 6/5 relatively evenly; both are very close to the delta-rational 0–397–720. The CTE tuning, on the other hand, has that chord tuned to 0–386–720, so that all of the error is on the 6/5 at about 18 cents sharp. | ||
== Special constraint == | == Special constraint == | ||
The special eigenmonzo ''X'''''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning. | The special eigenmonzo ''X'''''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning. | ||
It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained | It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained Tenney–Euclidean tuning'''). | ||
=== For equal temperaments === | === For equal temperaments === | ||
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<math>\displaystyle n = 1/g = \operatorname{mean} (V_X)</math> | <math>\displaystyle n = 1/g = \operatorname{mean} (V_X)</math> | ||
Unlike TE or TOP, the optimal edo number space in TOC is linear with respect to ''V''. That is, if ''V'' = ''αV''<sub>1</sub> + ''βV''<sub>2</sub>, then | Unlike TE or TOP, the optimal edo number space in TOC is linear with respect to ''V''. That is, if {{nowrap|''V'' {{=}} ''αV''<sub>1</sub> + ''βV''<sub>2</sub>}}, then | ||
<math>\displaystyle | <math>\displaystyle | ||