Normal forms: Difference between revisions
Cmloegcmluin (talk | contribs) references and footnotes |
Cmloegcmluin (talk | contribs) consistent hyphenation of "prime-count vector" |
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=== Positive ratio form === | === Positive ratio form === | ||
Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the defactored Hermite form ensures that the first nonzero entry of each comma's prime count vector is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''. | Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the defactored Hermite form ensures that the first nonzero entry of each comma's [[prime-count vector]] is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''. | ||
For example, the defactored Hermite form of meantone's comma basis is {{bra|{{vector| 4 -4 1 }}}}. HNF has ensured that the first number is positive. But the vector {{vector| 4 -4 1 }} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch. | For example, the defactored Hermite form of meantone's comma basis is {{bra|{{vector| 4 -4 1 }}}}. HNF has ensured that the first number is positive. But the vector {{vector| 4 -4 1 }} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch. | ||