Tempered monzos and vals: Difference between revisions

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A regular temperament is a 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 refers to an equivalence class of functions separated by unimodular transformations (which are functionally 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 '''tval'''.

Tmonzos and tvals are like regular monzos and vals, except they work in a space of tempered intervals (i.e. the intervals found in meantone) rather than in just intonation. A tval specifies a tuning or further temperament of the generators of a temperament (for example, the 31edo tval for meantone is {31 49}), and a tmonzo specifies a particular tempered interval in terms of stacking the tenperament's generators (for example, the tmonzo form of the major third in meantone is {-6 4}). Taking the [[Mathematical guide/Matrix operations|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 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 refers to an equivalence class of functions separated by unimodular transformations (which are functionally 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 '''tval'''.

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're 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 Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists #Normal val list|normal val list]] form.

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-A regular temperament is a 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 refers to an equivalence class of functions separated by unimodular transformations (which are functionally 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 '''tval'''.
+Tmonzos and tvals are like regular monzos and vals, except they work in a space of tempered intervals (i.e. the intervals found in meantone) rather than in just intonation. A tval specifies a tuning or further temperament of the generators of a temperament (for example, the 31edo tval for meantone is {31 49}), and a tmonzo specifies a particular tempered interval in terms of stacking the tenperament's generators (for example, the tmonzo form of the major third in meantone is {-6 4}). Taking the [[Mathematical guide/Matrix operations|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 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 refers to an equivalence class of functions separated by unimodular transformations (which are functionally 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 '''tval'''.
 Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're 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 Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists #Normal val list|normal val list]] form.