Tempered monzos and vals: Difference between revisions

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'''Tmonzos''' (for '''tempered monzos''') and '''tvals''' (for '''tempered vals''') are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tval specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tmonzo specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.

Mathematically, a regular temperament is a {{w|homomorphism}} (a kind of function) from the space of just intervals to the space of tempered intervals generated by that temperament, where both these spaces are [[stacking|abelian groups]]. Technically, a regular temperament is an equivalence class of functions separated by {{w|unimodular}} transformations, which represent the same temperament. An element of the space of tempered intervals is called a ''tempered monzo'', or for short ''tmonzo'', and represents the number of steps of each generator required to reach a tempered interval, and an element of the dual module ''K''* is called a ''tempered val'', or for short ''tval''.

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they are {{w|linear functional|linear functionals}} which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[temperament mapping matrix|mapping matrix]] for the temperament which is in [[normal lists #Normal val list|normal val list]] form.

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 '''Tmonzos''' (for '''tempered monzos''') and '''tvals''' (for '''tempered vals''') are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tval specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tmonzo specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.
-Mathematically, a regular temperament is a {{w|homomorphism}} (a kind of function) from the space of just intervals to the space of tempered intervals generated by that temperament, where both these spaces are [[stacking|abelian groups]]. Technically, a regular temperament is an equivalence class of functions separated by {{w|unimodular}} transformations, which represent the same temperament. An element of the space of tempered intervals is called a ''tempered monzo'', or for short ''tmonzo'', and represents the number of steps of each generator required to reach a tempered interval, and an element of the dual module ''K''* is called a ''tempered val'', or for short ''tval''.
 Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they are {{w|linear functional|linear functionals}} which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[temperament mapping matrix|mapping matrix]] for the temperament which is in [[normal lists #Normal val list|normal val list]] form.