User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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# Is compositeness heard? | # Is compositeness heard? | ||
# Are divisive ratios more important than multiplicative ratios?<ref>Prior to this material, the two problems are often said in the other order, but this essay inverts them since weight is usually considered before | # Are divisive ratios more important than multiplicative ratios?<ref>Prior to this material, the two problems are often said in the other order, but this essay inverts them since weight is usually considered before skew in tuning optimization. </ref> | ||
In fact, they can be modeled in terms of parameters of the norm used in optimization. The first problem is about the weight, and the second about the skew. The order of the norm is the third parameter. Although not versed into a "hard problem" rhetoric since it is a little bit abstract, we must still consider it along with the first two. Collectively, they are parameters of the norm. Being independent of specific temperaments, they are genuine metaproblems of tuning optimization, and well worth a dive. | In fact, they can be modeled in terms of parameters of the norm used in optimization. The first problem is about the weight, and the second about the skew. The order of the norm is the third parameter. Although not versed into a "hard problem" rhetoric since it is a little bit abstract, we must still consider it along with the first two. Collectively, they are parameters of the norm. Being independent of specific temperaments, they are genuine metaproblems of tuning optimization, and well worth a dive. | ||
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== Chapter I. Harmonic Rootedness == | == Chapter I. Harmonic Rootedness == | ||
There are two main categories of rootedness: chordal rootedness and tonal rootedness. | There are two main categories of rootedness: chordal rootedness and tonal rootedness. | ||
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== Chapter II. Divisive and Multiplicative Ratios == | == Chapter II. Divisive and Multiplicative Ratios == | ||
Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a divisive | Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a divisive ratio; 15/1 is a multiplicative ratio. The question is, thus, if ratios of the form ''n''/''d'' are more important than those of the form ''nd''. | ||
The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed | The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed in the last chapter, whereas tuning optimization has to do with percept formation and excitation, and to the better end, minimization of mistuned beating. These are fundamentally different effects – this essay takes the liberty of being the first to treat them separately. | ||
The odd-limit tonality diamond fully favors divisive ratios to multiplicative ones, as the odd limit of a ratio is equal to the exponentiation of the Kees height, a norm in a lattice skewed towards divisive ratios by 1/12 turn. It is useful in just chord construction. Consider the just major triad again. While 5/1 and 3/1 are the only ratios used to build the chord, the interval between them – 5/3 – is a real, played interval, unlike the multiplicative ratio 15/1, which is not played, only present in the harmonics. Likewise, using any harmonics as components of a just chord causes all the ratios between them to be played, and thus to be emergent. Unless we stick to bare dyads, it could not be more appropriate than adopting a metric that favors divisive ratios, especially the tonality diamonds. | The odd-limit tonality diamond fully favors divisive ratios to multiplicative ones, as the odd limit of a ratio is equal to the exponentiation of the Kees height, a norm in a lattice skewed towards divisive ratios by 1/12 turn. It is useful in just chord construction. Consider the just major triad again. While 5/1 and 3/1 are the only ratios used to build the chord, the interval between them – 5/3 – is a real, played interval, unlike the multiplicative ratio 15/1, which is not played, only present in the harmonics. Likewise, using any harmonics as components of a just chord causes all the ratios between them to be played, and thus to be emergent. Unless we stick to bare dyads, it could not be more appropriate than adopting a metric that favors divisive ratios, especially the tonality diamonds. | ||
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The same cannot be assumed for tuning optimization, since that is a vastly different scenario. In a just major triad, the 15th harmonic exists in three ways: as the harmonic of the root, of the 3rd harmonic, and of the 5th harmonic. Figure 1 is the frequency spectrum of the triad played in the semisine waveform, which has been proposed as the standard ear-training waveform in [[User:FloraC/Proposed standard ear-training waveform|''Proposed Standard Ear-Training Waveform'']]. | The same cannot be assumed for tuning optimization, since that is a vastly different scenario. In a just major triad, the 15th harmonic exists in three ways: as the harmonic of the root, of the 3rd harmonic, and of the 5th harmonic. Figure 1 is the frequency spectrum of the triad played in the semisine waveform, which has been proposed as the standard ear-training waveform in [[User:FloraC/Proposed standard ear-training waveform|''Proposed Standard Ear-Training Waveform'']]. | ||
If we play such a triad in a tempered tuning profile, the quality of the chord is determined by how the three components said above line up. In a tuning profile characterized by the | If we play such a triad in a tempered tuning profile, the quality of the chord is determined by how the three components said above line up. In a tuning profile characterized by the error map | ||
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the ~15/1 will be a combination of harmonics with pitch errors of | the ~15/1 will be a combination of harmonics with pitch errors of −''ε'', 0, and +''ε''. In addition, the harmonic itself can be played as a dyad and its pitch error is 0. | ||
Now consider the contrasting profile | Now consider the contrasting profile | ||
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$$ | $$ | ||
the ~15/1 will be a combination of harmonics with pitch errors of | the ~15/1 will be a combination of harmonics with pitch errors of −''ε'', −''ε'', and 0, but the played harmonic is at −2''ε''. So we see this harmonic will get pretty off the track whenever played. | ||
Regarding 5/3, it is the opposite situation. Ɛ<sub>2</sub> comes out superior to Ɛ<sub>1</sub> as it perfectly hits 5/3 whereas Ɛ<sub>1</sub>'s ~5/3 is off by +2''ε''. | Regarding 5/3, it is the opposite situation. ''Ɛ''<sub>2</sub> comes out superior to ''Ɛ''<sub>1</sub> as it perfectly hits 5/3 whereas ''Ɛ''<sub>1</sub>'s ~5/3 is off by +2''ε''. | ||
However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point ( | However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point (harmonic series) for it to beat against, so it lacks relevance in tuning optimization. The only scenario to account for its accuracy is where it is played on the chord's formal root, in which case its 3rd harmonic beats against the formal root's 5th harmonic, for example. That is still not a good argument for its relative importance since we would have manipulated the chord structure just in order to obtain this result. A chord with ~15/1 played on the formal root would call for an accurate ~15/1 and then neutralize the demand for an accurate ~5/3 as previously posed. For example, the demands posed by the just major sixth chord 1–5/4–3/2–5/3 and by the just major seventh chord 1–5/4–3/2–15/8 cancel each other out up to octave equivalence. More generally, for any chord featuring a divisive ratio on the formal root, there is a counterpart featuring a multiplicative ratio alike. | ||
We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in positive harmony. | We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in positive harmony. | ||
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== Chapter III. Power in Proportion == | == Chapter III. Power in Proportion == | ||
The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map P as | The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map P as | ||
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Let us denote the just tuning map in cents by T<sub>J</sub>, the | Let us denote the just tuning map in cents by ''T''<sub>''J''</sub>, the error map ''Ɛ'' is | ||
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That is the 1/5-comma tuning, in which harmonics 3 and 5 have an equal magnitude and an opposite sign of error. | That is the 1/5-comma tuning, in which harmonics 3 and 5 have an equal magnitude and an opposite sign of error. | ||
TOP tuning works principally the same, except that harmonic 2 is no longer constrained to pure and that the allowed error of ''q'' is log<sub>2</sub> (''q'') times that of prime 2. The TOP | TOP tuning works principally the same, except that harmonic 2 is no longer constrained to pure and that the allowed error of ''q'' is log<sub>2</sub> (''q'') times that of prime 2. The TOP error map of 5-limit meantone is | ||
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The | The error map is | ||
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The | The error map is | ||
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== Chapter IV. Art of Compromise == | == Chapter IV. Art of Compromise == | ||
Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' (''c''. 122 BC) – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave!<ref>"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", ''Ethnomusicology''. Fritz A. Kuttner. </ref> | Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' (''c''. 122 BC) – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave!<ref>"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", ''Ethnomusicology''. Fritz A. Kuttner. </ref> So what about this essay? Most likely, it will be no end of a debate, but inviting more. It is high time we confront the last hard problem: compositeness of the harmonics. | ||
If we play the interval of 15/1, does it somehow suggest 5/1 and 3/1? It seems even if we do not hear 15/1 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less. | If we play the interval of 15/1, does it somehow suggest 5/1 and 3/1? It seems even if we do not hear 15/1 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less. | ||
On one hand, we want the majority of chords to be in tune, so obviously the most common intervals should get the best care. The question is then what probability distribution is followed without knowing what kind of harmony will be used in | On one hand, we want the majority of chords to be in tune, so obviously the most common intervals should get the best care. The question is then what probability distribution is followed without knowing what kind of harmony will be used in advance. A chi distribution would certainly make sense if we were to talk about randomly generated "tonal" music with no regards of psychoacoustics – since each voice's number of generator steps from the tonic was supposed to follow a normal distribution. In a world with human beings and with harmonic clarity rather than the abstract number of generator steps playing the predominant role of forming tonality, the right assumption for commonness is definitely not that but to be inversely related to complexity. The metric can be taken as the inner product of a uniform distribution and the inverse complexity, and if the uniform distribution is replaced with something that favors structurally tonal music such as a chi distribution, we obtain a commonness curve that biases heavily towards simplicity more than many would expect. | ||
Try thinking of it this way: one could spend their life making music of only plain octaves and fifths without being bored at all. That was what happened in many cultures around the world and no one seemed to have a problem. It is actually our expression of harmonic feelings in intricate multidigit ratios that is the more peculiar endeavor. | Try thinking of it this way: one could spend their life making music of only plain octaves and fifths without being bored at all. That was what happened in many cultures around the world and no one seemed to have a problem. It is actually our expression of harmonic feelings in intricate multidigit ratios that is the more peculiar endeavor. | ||
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On the other hand, it is argued that complex intervals need relatively more care since it is harder to capture their identities. It is believed that complex intervals have a smaller range of tolerance in which their identities will be revealed, which is fairly easy to understand. | On the other hand, it is argued that complex intervals need relatively more care since it is harder to capture their identities. It is believed that complex intervals have a smaller range of tolerance in which their identities will be revealed, which is fairly easy to understand. | ||
The Tenney weight is the weight that <s>turns a deaf ear to</s> strikes a perfect balance on those considerations. In fact, it is the only weight in which tunings on composite subgroups coincide with tunings on prime subgroups, meaning that optimizing a temperament on 2.3.5 or 2.9.5 will render the same result for all the intervals they share. The reason is each prime ''q'' in the prime list Q has an importance rating of 1/log<sub>2</sub> (''q''), represented by the matrix | The Tenney weight is the weight that <s>turns a deaf ear to</s> strikes a perfect balance on those considerations. In fact, it is the only weight in which tunings on composite subgroups coincide with tunings on prime subgroups, meaning that optimizing a temperament on 2.3.5 or 2.9.5 will render the same result for all the intervals they share. The reason is each prime ''q'' in the prime list ''Q'' has an importance rating of 1/log<sub>2</sub> (''q''), represented by the matrix | ||
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That is pretty wrong if gazed from the universe of Tenney weight, as it makes harmonic 8 three times distant with three times the error of 13. One can immediately see the bumps in the complexity curve of integer harmonics. Nonetheless, it | That is pretty wrong if gazed from the universe of Tenney weight, as it makes harmonic 8 three times distant with three times the error of 13. One can immediately see the bumps in the complexity curve of integer harmonics. Nonetheless, it reasonably holds itself as it demands the same absolute tolerance for all primes. It only highlights higher primes in a mild manner if the standard is, as they argued, a diminishing tolerance. | ||
The Wilson weight does the opposite as it puts 1/''q'' importance rating to the prime ''q'', represented by the matrix | The Wilson weight does the opposite to the equilateral weight, as it puts 1/''q'' importance rating to the prime ''q'', represented by the matrix | ||
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which indicates that the prime ''q'' in Q has the weight equal to floor (log<sub>''q''</sub> (''n'')). | which indicates that the prime ''q'' in ''Q'' has the weight equal to floor (log<sub>''q''</sub> (''n'')). | ||
The Tenney weight is a special case of the Hahn[''n''] weight, where ''n'' → infinity. The only thing that sets Hahn[''n''] apart from Tenney is the floor function (since log<sub>Q</sub> (''n'') = log<sub>2</sub> (''n'')/log<sub>2</sub> (Q) and log<sub>2</sub> (''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other Hahn[''n''] weights are algebraic. | The Tenney weight is a special case of the Hahn[''n''] weight, where ''n'' → infinity. The only thing that sets Hahn[''n''] apart from Tenney is the floor function (since log<sub>''Q''</sub> (''n'') = log<sub>2</sub> (''n'')/log<sub>2</sub> (''Q'') and log<sub>2</sub> (''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other Hahn[''n''] weights are algebraic. | ||
That defines the H<sub>''n''</sub>C, H<sub>''n''</sub>E, and H<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CH<sub>''n''</sub>E tuning. For example, CH<sub>13</sub>E, CH<sub>14</sub>E, CH<sub>15</sub>E, and CH<sub>16</sub>E are all equivalent and one should always write CH<sub>16</sub>E. | That defines the H<sub>''n''</sub>C, H<sub>''n''</sub>E, and H<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CH<sub>''n''</sub>E tuning. For example, CH<sub>13</sub>E, CH<sub>14</sub>E, CH<sub>15</sub>E, and CH<sub>16</sub>E are all equivalent and one should always write CH<sub>16</sub>E. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Temperament !! | ! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E) | ||
|- | |- | ||
| 5-limit meantone || {{val| 0 -4.7407 +2.5436 }} || {{val| 0 -4.3013 +4.3013 }} | | 5-limit meantone || {{val| 0 -4.7407 +2.5436 }} || {{val| 0 -4.3013 +4.3013 }} | ||
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© 2023 Flora Canou | © 2023 Flora Canou | ||
Version Stable | Version Stable 2 | ||
This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | ||