Golden sequences and tuning: Difference between revisions

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An approach to solving this problem is to use the properties of the golden ratio (in this case, logarithmic phi). The golden ratio is as far as possible from any simple rational number. Thus, by successively stacking the golden ratio, one avoids having intervals coincide. By setting the ratio of step sizes in any MOS to the golden ratio, the generator can then be characterized by an expression in terms of logarithmic phi. By continuing to stack generators that are tuned this way, one never runs into overly small commas.

If we tune diatonic this way, the result is golden meantone, with a generator around 696.2 cents. However, we can, in any case, define an "original" scale that is the first place that particular golden tuning is encountered. For example, the golden tuning for diatonic is also the golden tuning for pentic, but not the golden tuning for trial (2L 1s), which is simply logarithmic phi. Notably, in the general case, any MOS always shares its golden generator with its last oligolarge ancestor. For example, the golden tuning of 7L 2s is the same as that of 2L 5s, but not the same as that of ~~3L 2s~~.

If we tune diatonic this way, the result is golden meantone, with a generator around 696.2 cents. However, we can, in any case, define an "original" scale that is the first place that particular golden tuning is encountered. For example, the golden tuning for diatonic is also the golden tuning for pentic, but not the golden tuning for trial (2L 1s), which is simply logarithmic phi. Notably, in the general case, any MOS always shares its golden generator with its last oligolarge ancestor. For example, the golden tuning of 7L 2s is the same as that of 2L 5s, but not the same as that of 2L 3s.

Interestingly, if we take the step sizes of any oligolarge MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that oligolarge MOS! (Note that this also happens to be the list of EDOs that approximate the golden tuning for this series of MOSes, by the definition of a golden sequence.)

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 An approach to solving this problem is to use the properties of the golden ratio (in this case, logarithmic phi). The golden ratio is as far as possible from any simple rational number. Thus, by successively stacking the golden ratio, one avoids having intervals coincide. By setting the ratio of step sizes in any MOS to the golden ratio, the generator can then be characterized by an expression in terms of logarithmic phi. By continuing to stack generators that are tuned this way, one never runs into overly small commas.
-If we tune diatonic this way, the result is golden meantone, with a generator around 696.2 cents. However, we can, in any case, define an "original" scale that is the first place that particular golden tuning is encountered. For example, the golden tuning for diatonic is also the golden tuning for pentic, but not the golden tuning for trial (2L&nbsp;1s), which is simply logarithmic phi. Notably, in the general case, any MOS always shares its golden generator with its last oligolarge ancestor. For example, the golden tuning of 7L&nbsp;2s is the same as that of 2L&nbsp;5s, but not the same as that of 3L&nbsp;2s.
+If we tune diatonic this way, the result is golden meantone, with a generator around 696.2 cents. However, we can, in any case, define an "original" scale that is the first place that particular golden tuning is encountered. For example, the golden tuning for diatonic is also the golden tuning for pentic, but not the golden tuning for trial (2L&nbsp;1s), which is simply logarithmic phi. Notably, in the general case, any MOS always shares its golden generator with its last oligolarge ancestor. For example, the golden tuning of 7L&nbsp;2s is the same as that of 2L&nbsp;5s, but not the same as that of 2L 3s.
 Interestingly, if we take the step sizes of any oligolarge MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that oligolarge MOS! (Note that this also happens to be the list of EDOs that approximate the golden tuning for this series of MOSes, by the definition of a golden sequence.)