Talk:Marvel: Difference between revisions

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: [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:27, 16 January 2025 (UTC)

:: In response to the choice of odd-limits/tonality diamonds, the answer is very simple: we assume pure octaves so that all octave-equivalents and octave-inversions of every interval in the tonality diamond is included implicitly with equal weighting. In other words, part of the rationale for using pure octaves is that it makes the use of odd-limits perfectly natural because there is no bias to specific octave-equivalents or octave-inversions, so that we can truly just think about the approximations of pairs of odd numbers which vastly simplifies analysis (the usefulness of which should not be underestimated). This is also important because any nonzero tempering of the octave causes an infinite number of intervals of any odd-limit to become arbitrarily inconsistent, though in practice this usually doesn't matter. In other words, it's a far clearer direction to optimize for an explicit set of harmonies and then consider octave-tempering afterwards than first allowing octave-tempering and then being unnecessarily paralysed by choice, which IMO causes things like TE to look artificially natural as solutions because they ~~accept~~ the fact that the choice becomes very arbitrary if you allow tempered octaves to justify a disembodied method of optimization that is purely in terms of the basis. That's not to say such a scheme of optimization has no value of course, but it's important to acknowledged the limitations and especially the assumptions of any optimization scheme.

:: In response to the choice of odd-limits/tonality diamonds, the answer is very simple: we assume pure octaves so that all octave-equivalents and octave-inversions of every interval in the tonality diamond is included implicitly with equal weighting. In other words, part of the rationale for using pure octaves is that it makes the use of odd-limits perfectly natural because there is no bias to specific octave-equivalents or octave-inversions, so that we can truly just think about the approximations of pairs of odd numbers which vastly simplifies analysis (the usefulness of which should not be underestimated). This is also important because any nonzero tempering of the octave causes an infinite number of intervals of any odd-limit to become arbitrarily inconsistent, though in practice this usually doesn't matter. In other words, it's a far clearer direction to optimize for an explicit set of harmonies and then consider octave-tempering afterwards than first allowing octave-tempering and then being unnecessarily paralysed by choice, which IMO causes things like TE to look artificially natural as solutions because they use the fact that the choice becomes very arbitrary if you allow tempered octaves to justify a disembodied method of optimization that is purely in terms of the basis. That's not to say such a scheme of optimization has no value of course, but it's important to acknowledged the limitations and especially the assumptions of any optimization scheme.

:: In response to "all complexity weighting faces the paradox that a growing weight suddenly plunges to zero at the edge of the limit" - this is not a paradox, this is completely expected, and this is exactly why I showed a large range of different odd-limits you could consider, to show that generally the same systems appear for basically all the 7-limited odd-limit targets one might want to consider practically.

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 : [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:27, 16 January 2025 (UTC)
-:: In response to the choice of odd-limits/tonality diamonds, the answer is very simple: we assume pure octaves so that all octave-equivalents and octave-inversions of every interval in the tonality diamond is included implicitly with equal weighting. In other words, part of the rationale for using pure octaves is that it makes the use of odd-limits perfectly natural because there is no bias to specific octave-equivalents or octave-inversions, so that we can truly just think about the approximations of pairs of odd numbers which vastly simplifies analysis (the usefulness of which should not be underestimated). This is also important because any nonzero tempering of the octave causes an infinite number of intervals of any odd-limit to become arbitrarily inconsistent, though in practice this usually doesn't matter. In other words, it's a far clearer direction to optimize for an explicit set of harmonies and then consider octave-tempering afterwards than first allowing octave-tempering and then being unnecessarily paralysed by choice, which IMO causes things like TE to look artificially natural as solutions because they accept the fact that the choice becomes very arbitrary if you allow tempered octaves to justify a disembodied method of optimization that is purely in terms of the basis. That's not to say such a scheme of optimization has no value of course, but it's important to acknowledged the limitations and especially the assumptions of any optimization scheme.
+:: In response to the choice of odd-limits/tonality diamonds, the answer is very simple: we assume pure octaves so that all octave-equivalents and octave-inversions of every interval in the tonality diamond is included implicitly with equal weighting. In other words, part of the rationale for using pure octaves is that it makes the use of odd-limits perfectly natural because there is no bias to specific octave-equivalents or octave-inversions, so that we can truly just think about the approximations of pairs of odd numbers which vastly simplifies analysis (the usefulness of which should not be underestimated). This is also important because any nonzero tempering of the octave causes an infinite number of intervals of any odd-limit to become arbitrarily inconsistent, though in practice this usually doesn't matter. In other words, it's a far clearer direction to optimize for an explicit set of harmonies and then consider octave-tempering afterwards than first allowing octave-tempering and then being unnecessarily paralysed by choice, which IMO causes things like TE to look artificially natural as solutions because they use the fact that the choice becomes very arbitrary if you allow tempered octaves to justify a disembodied method of optimization that is purely in terms of the basis. That's not to say such a scheme of optimization has no value of course, but it's important to acknowledged the limitations and especially the assumptions of any optimization scheme.
 :: In response to "all complexity weighting faces the paradox that a growing weight suddenly plunges to zero at the edge of the limit" - this is not a paradox, this is completely expected, and this is exactly why I showed a large range of different odd-limits you could consider, to show that generally the same systems appear for basically all the 7-limited odd-limit targets one might want to consider practically.