Domain basis: Difference between revisions
Cmloegcmluin (talk | contribs) →Intersecting: attempt to explain how it works |
Cmloegcmluin (talk | contribs) note about non-JI |
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Note for comparison that a comma basis is also a type of basis. In the same way that an interval basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament — to be precise, the subspace of all commas that are tempered out.<ref>In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of an interval basis, only the basis vectors are formal primes. The spanned vectors in general are not formal primes. This is why we do not call an interval basis a "formal prime basis".</ref> | Note for comparison that a comma basis is also a type of basis. In the same way that an interval basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament — to be precise, the subspace of all commas that are tempered out.<ref>In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of an interval basis, only the basis vectors are formal primes. The spanned vectors in general are not formal primes. This is why we do not call an interval basis a "formal prime basis".</ref> | ||
= JI interval subspaces = | = Nonstandard JI interval subspaces = | ||
For some advanced mathematical information about non-standard just intonation (JI) interval subspaces, as well as a gateway to browse temperaments within popular interval subspaces of this sort, see [[Just intonation subgroup|this page]]. | For some advanced mathematical information about non-standard just intonation (JI) interval subspaces, as well as a gateway to browse temperaments within popular interval subspaces of this sort, see [[Just intonation subgroup|this page]]. | ||
= Non-JI interval subspaces = | |||
The behavior of these are not yet well-described. For, for instance, is best to represent them as matrices? If a formal prime is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language. | |||
= Interval subspaces as subspaces of other interval subspaces = | = Interval subspaces as subspaces of other interval subspaces = | ||