Rank-3 scale: Difference between revisions
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Not all trivalent scales are PWF. Only a single scale - ''abcba'' - is trivalent and not PWF. | Not all trivalent scales are PWF. Only a single scale - ''abcba'' - is trivalent and not PWF. | ||
Only a single PWF scale is mirror-symmetric - abacaba. | Only a single PWF scale is mirror-symmetric - ''abacaba''. | ||
Apart from ''abacaba'', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period. | Apart from ''abacaba'', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period. | ||
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PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from ''abacaba''. | PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from ''abacaba''. | ||
The scale abacaba is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric trivalent scale. | The scale ''abacaba'' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric trivalent scale. | ||
The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales. | The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales. | ||
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'''Conjecture''': The only SN scale that is trivalent is ''abacaba''. | '''Conjecture''': The only SN scale that is trivalent is ''abacaba''. | ||
'''Conjecture''': The only SN scales that are MV3 are abacaba, and scales of the form ''a…ba…c''. | '''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''a…ba…c''. | ||
'''Conjecture''': The only SN scales that are MV3, and have mean variety < 3 are those of the form ''a…ba…c''. | '''Conjecture''': The only SN scales that are MV3, and have mean variety < 3 are those of the form ''a…ba…c''. | ||