Tempered monzos and vals - Revision history

FloraC: - breadcrumbs (this isn't a subpage).

2025-05-17T10:31:40Z

- breadcrumbs (this isn't a subpage).

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	~~{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}~~'''Tempered monzos''' and '''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 tempered val (for short, "'''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 tempered monzo (for short, "'''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.		'''Tempered monzos''' and '''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 tempered val (for short, "'''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 tempered monzo (for short, "'''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.

	Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).		Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

VectorGraphics at 05:04, 16 May 2025

2025-05-16T05:04:07Z

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	{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}'''Tempered monzos''' and '''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 tempered monzo (for short, "'''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.		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}'''Tempered monzos''' and '''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 tempered val (for short, "'''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 tempered monzo (for short, "'''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.

	Tmonzos are rather straightforward, and ~~tempered vals (for short, "'''~~tvals~~'''")~~ act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).		Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

	Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.		Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.

VectorGraphics at 05:03, 16 May 2025

2025-05-16T05:03:38Z

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	{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}'''~~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.		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}'''Tempered monzos''' and '''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 tempered monzo (for short, "'''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.

	Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).		Tmonzos are rather straightforward, and tempered vals (for short, "'''tvals'''") act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

	Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.		Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.

VectorGraphics at 05:02, 16 May 2025

2025-05-16T05:02:40Z

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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.		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}'''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.

	Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).		Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

VectorGraphics: VectorGraphics moved page Tmonzos and tvals to Tempered monzos and vals

2025-05-16T04:37:35Z

VectorGraphics moved page Tmonzos and tvals to Tempered monzos and vals

← Older revision	Revision as of 04:37, 16 May 2025
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VectorGraphics at 18:32, 3 May 2025

2025-05-03T18:32:12Z

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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.		'''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.

	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.		Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

			Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.

	== Example ==		== Example ==
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	This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations#Multiply matrix by vector\|apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.		This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations#Multiply matrix by vector\|apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

			Now, let's use a tval {{val\| 31 18 }} to figure out what the tuning of the tempered 5/4 is in 31edo. By applying this tval to the tmonzo {{monzo\| -2 4 }}, we get -62 + 72 = 10 edosteps.

	== See also ==		== See also ==

VectorGraphics: removed regtemp definition as per talk page

2025-04-23T16:33:49Z

removed regtemp definition as per talk page

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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.		'''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.		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.

VectorGraphics: Fix link

2025-04-15T08:48:43Z

Fix link

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	$$		$$

	This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations\|apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.		This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations#Multiply matrix by vector\|apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

	== See also ==		== See also ==

VectorGraphics: Re-add apply link

2025-04-15T08:48:18Z

Re-add apply link

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	$$		$$

	This mapping represents meantone temperament. If we apply this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.		This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations\|apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

	== See also ==		== See also ==

FloraC: The generators need to be specified. Mapping is a clipping of temperament mapping matrix. Cleanup. +some links to Wikipedia

2025-04-15T08:44:44Z

The generators need to be specified. *Mapping* is a clipping of *temperament mapping matrix*. Cleanup. +some links to Wikipedia

← Older revision		Revision as of 08:44, 15 April 2025
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	Tmonzos and tvals are like regular ~~monzos~~ and ~~vals~~, 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 ~~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.		'''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''.

	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 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.

	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.

	== Example ==		== Example ==
	As an example, consider the mapping		As an example, consider the mapping matrix

	~~[{{val\| 1 1 0 }}]~~

	[{~~{val\|~~ 0 1 4 }}]		$$
			\begin{bmatrix}
			1 & 1 & 0 \\
			0 & 1 & 4
			\end{bmatrix}
			$$

	This mapping~~, which is referred to in mathematical terms as a "matrix",~~ represents meantone temperament. If we ~~[[Mathematical guide/Matrix multiplication\|~~apply]] this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.		This mapping represents meantone temperament. If we apply this mapping to the monzo {{monzo\| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo\| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo\| -1 1 0 }}, we get the tmonzo {{monzo\| 0 1 }} (one tempered 3/2). That 2/1 and 3/2 map to {{monzo\| 1 0 }} and {{monzo\| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo {{monzo\| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo\| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

	== See also ==		== See also ==

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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.		'''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.		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.