Radical interval: Difference between revisions

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A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''2 ''e''3 … ''e''p }} is a fractional monzo, then it represents 2''e''2 3''e''3 … ''p''''e''''p'' just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 21/13 3-1/13 57/26. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)1/26. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.

A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''2 ''e''3 … ''e''p }} is a fractional monzo, then it represents 2''e''2 3''e''3 … ''p''''e''''p'' just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 21/13 3-1/13 57/26. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)1/26. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.

Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.

~~Starting with the next section~~, ~~we discuss projection matrices in~~ the ~~formal mathematical language~~. For ~~a beginner-level introduction~~, ~~see~~ [[~~Projection]]~~.

== Tunings in terms of fractional monzos ==

Fractional monzos can be used to notate any number that can be expressed as a root, so they can be used to express the degrees of [[Equal-step tuning|equal tunings]]. For example, 12edo's fifth can be expressed as [7/12⟩, and the Bohlen-Pierce supermajor third may be expressed as [0 3/13⟩.

~~== Fractional projection matrices ==~~

What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by [7/12 -3/13⟩. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like [[slendric]], except that while those temperaments follow a 2-step process of 1) equally dividing a just interval and 2) assigning the divisions to another just interval, fractional monzos provide a framework for skipping the second step (if you deem it unnecessary). In fact, slendric can be described as equating [3 0 0 -1⟩ and [-1/3 1/3⟩.

~~A square matrix P~~ is ~~a [[Wikipedia: Projection (linear algebra)|projection]] if P2 = P~~. ~~A nontrivial projection~~, ~~meaning~~ one ~~which is neither~~ the ~~zero matrix nor the identity matrix~~, ~~has~~ [~~[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues~~. If the ~~rows~~ of ~~P represent a tuning~~ of ~~a regular temperament as vectors in either weighted or unweighted~~ [[~~Monzos and interval space|interval space~~]], ~~then~~ a ~~comma c~~ of ~~the temperament (in the appropriate coordinates~~) ~~times P from~~ the ~~left~~, ~~cP, will be~~ the ~~zero vector~~. ~~A val of the temperament v~~, ~~times P on the right, Pv, will satisfy Pv = v~~.

~~In particular, this is true of matrices~~ with ~~rows consisting of fractional monzos. This is of interest since several of~~ the ~~most important tunings~~, in ~~particular [[Minimax tuning|minimax]] and [[Least squares tuning|least squares]], have tuning values which can be expressed as fractional monzos~~. For ~~example, the fractional monzo we have used as an example is the tuning for~~ a ~~fifth in the 7/26~~-~~comma Woolhouse meantone. Indeed~~, ~~any meantone whose tuning is expressed as a fraction of a comma has an associated 3×3 projection matrix defining the tuning~~.

Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]].

== ~~Tunings in terms of fractional monzos~~ ==

== Mathematical interpretation ==

If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.

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If the target set is a ''q''-odd-limit diamond, eigenmonzos in the range 1 < ''x'' < sqrt(2) may be found by simply finding all of the sets of ''r'' - 1 ''q''-odd-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.

== Algebraic considerations ==

=== Fractional projection matrices ===

A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and interval space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.

In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular [[Minimax tuning|minimax]] and [[Least squares tuning|least squares]], have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3×3 projection matrix defining the tuning.

=== Algebraic considerations ===

For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[Wikipedia: Free abelian group|free abelian group]] (or equivalently, Z-module) of rank ''n'' equal to the number of primes less than or equal to ''p'' for the ''p''-limit in question. Fractional monzos do not define a free group but rather a [[Wikipedia:Divisible group|divisible group]], meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a [[Wikipedia:Vector space|vector space]] (of dimension ''n'') over the rational numbers. They are also torsion-free (equivalently, [[Wikipedia:Flat module|flat]]) abelian groups, and are the [[Wikipedia:Injective hull|injective hulls]] of the corresponding monzos.

@@ Line 5: / Line 5: @@
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 }}{{Legacy}}
-A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.
+A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.
 Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.
-Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]].
+== Tunings in terms of fractional monzos ==
+Fractional monzos can be used to notate any number that can be expressed as a root, so they can be used to express the degrees of [[Equal-step tuning|equal tunings]]. For example, 12edo's fifth can be expressed as [7/12⟩, and the Bohlen-Pierce supermajor third may be expressed as [0 3/13⟩.
-== Fractional projection matrices ==
+What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by [7/12 -3/13⟩. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like [[slendric]], except that while those temperaments follow a 2-step process of 1) equally dividing a just interval and 2) assigning the divisions to another just interval, fractional monzos provide a framework for skipping the second step (if you deem it unnecessary). In fact, slendric can be described as equating [3 0 0 -1⟩ and [-1/3 1/3⟩.
-A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and interval space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.
-In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular [[Minimax tuning|minimax]] and [[Least squares tuning|least squares]], have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3×3 projection matrix defining the tuning.
+Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]].
-== Tunings in terms of fractional monzos ==
+== Mathematical interpretation ==
 If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' projection matrix by means of a list of ''n'' - ''r'' commas c and ''r'' '''eigenmonzos''' e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1.
@@ Line 31: / Line 31: @@
 If the target set is a ''q''-odd-limit diamond, eigenmonzos in the range 1 &lt; ''x'' &lt; sqrt(2) may be found by simply finding all of the sets of ''r'' - 1 ''q''-odd-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.
-== Algebraic considerations ==
+=== Fractional projection matrices ===
+A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and interval space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.
+In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular [[Minimax tuning|minimax]] and [[Least squares tuning|least squares]], have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3×3 projection matrix defining the tuning.
+=== Algebraic considerations ===
 For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[Wikipedia: Free abelian group|free abelian group]] (or equivalently, Z-module) of rank ''n'' equal to the number of primes less than or equal to ''p'' for the ''p''-limit in question. Fractional monzos do not define a free group but rather a [[Wikipedia:Divisible group|divisible group]], meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a [[Wikipedia:Vector space|vector space]] (of dimension ''n'') over the rational numbers. They are also torsion-free (equivalently, [[Wikipedia:Flat module|flat]]) abelian groups, and are the [[Wikipedia:Injective hull|injective hulls]] of the corresponding monzos.