Monzo: Difference between revisions

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For example, the interval [[15/8]] can be thought of as having <math>5 \cdot 3</math> in the numerator, and <math>2 \cdot 2 \cdot 2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.

~~:'''Practical hint:''' the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).~~

Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them:

Here are some common [[5-limit]] monzos, ~~for your reference~~:

{| class="wikitable center-1"

|-

! Ratio

! Factors

! Monzo

|-

| [[3/2]]

| <math>2^{-1} \cdot 3</math>

| {{monzo| -1 1 0 }}

|-

| [[5/4]]

| <math>2^{-2} \cdot 5</math>

| {{monzo| -2 0 1 }}

|-

| [[9/8]]

| <math>2^{-3} \cdot 3^2</math>

| {{monzo| -3 2 0 }}

|-

| [[81/80]]

| <math>2^{-4} \cdot 3^4 \cdot 5^{-1}</math>

| {{monzo| -4 4 -1 }}

|}

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|-

! Ratio

! Factors

! Monzo

|-

| [[7/4]]

| <math>2^{-2} \cdot 7</math>

| {{monzo| -2 0 0 1 }}

|-

| [[7/6]]

| <math>2^{-1} \cdot 3^{-1} \cdot 7</math>

| {{monzo| -1 -1 0 1 }}

|-

| [[7/5]]

| <math>5^{-1} \cdot 7</math>

| {{monzo| 0 0 -1 1 }}

|}

:'''Practical hint:''' On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).

== Relationship with vals ==

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Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.

-->

== Generalizations ==

=== Subgroup monzos ===

A subgroup monzo is like a standard monzo, except that it is in a just intonation [[subgroup]] that is not a prime limit. It is usually provided along with the subgroup it is in, such as 2.3.7 [1 -2 1⟩ for 14/9 or 2.3.13/5 [1 -1 1⟩ for 26/15. For subgroups with primes as basis elements (such as 2.3.7), a subgroup monzo can be seen as a standard monzo abbreviated to hide zeroes. To make this more clear, the abbreviated sections may be replaced with ellipses, such as 2.3.7 [1 -2 ... 1⟩ for 14/9.

=== Tempered monzos ===

A tempered monzo represents a tempered interval, or an interval in a regular temperament. It functions similarly to a subgroup monzo, except each entry represents not a JI interval but a generator for that regular temperament. For example, the tmonzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 [-6 4⟩. (Note that we write the generators with tildes to indicate that they are tempered intervals). This means that to find the major third, you go up four perfect twelfths (~3/1) and down six octaves (~2/1).

More generally, tempered monzos are applicable to any regular tuning, regardless of JI mapping, so corresponding intervals in two different regular temperaments that are tuned the same way have the same tempered monzo.

=== Fractional monzos ===

Any of the previous categories of monzo can also be a "fractional monzo", allowing entries to be fractions or non-integer rational numbers as opposed to just integers. This allows monzos to express equal divisions of just intervals (or stacks thereof). For example, [-1/2 1/2⟩ is a monzo representing a neutral third equal to half of a perfect fifth, and [1/12⟩ is a monzo representing a 12edo semitone. [1/12 1/13⟩ is a monzo representing 1\12edo stacked with 1\13edt. (Numerically, this is the 156th root of 2<sup>13</sup>*3<sup>12</sup>.) Note that we write the fractional monzo entries with forward slashes (as they represent fractions), despite writing edosteps with backslashes.

== See also ==

* [[Extended bra-ket notation]]

* [[Smonzos and svals]] — subgroup monzos and vals

* [[Tmonzos and tvals]] — tempered monzos and vals

== External links ==

@@ Line 16: / Line 16: @@
 For example, the interval [[15/8]] can be thought of as having <math>5 \cdot 3</math> in the numerator, and <math>2 \cdot 2 \cdot 2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.
-:'''Practical hint:''' the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).
+Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them:
-Here are some common [[5-limit]] monzos, for your reference:
 {| class="wikitable center-1"
 |-
 ! Ratio
+! Factors
 ! Monzo
 |-
 | [[3/2]]
+| <math>2^{-1} \cdot 3</math>
 | {{monzo| -1 1 0 }}
 |-
 | [[5/4]]
+| <math>2^{-2} \cdot 5</math>
 | {{monzo| -2 0 1 }}
 |-
 | [[9/8]]
+| <math>2^{-3} \cdot 3^2</math>
 | {{monzo| -3 2 0 }}
 |-
 | [[81/80]]
+| <math>2^{-4} \cdot 3^4 \cdot 5^{-1}</math>
 | {{monzo| -4 4 -1 }}
 |}
@@ Line 43: / Line 46: @@
 |-
 ! Ratio
+! Factors
 ! Monzo
 |-
 | [[7/4]]
+| <math>2^{-2} \cdot 7</math>
 | {{monzo| -2 0 0 1 }}
 |-
 | [[7/6]]
+| <math>2^{-1} \cdot 3^{-1} \cdot 7</math>
 | {{monzo| -1 -1 0 1 }}
 |-
 | [[7/5]]
+| <math>5^{-1} \cdot 7</math>
 | {{monzo| 0 0 -1 1 }}
 |}
+:'''Practical hint:''' On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).
 == Relationship with vals ==
@@ Line 89: / Line 98: @@
 Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.
 -->
+== Generalizations ==
+=== Subgroup monzos ===
+{{Main|Subgroup monzos and vals}}
+A subgroup monzo is like a standard monzo, except that it is in a just intonation [[subgroup]] that is not a prime limit. It is usually provided along with the subgroup it is in, such as 2.3.7 [1 -2 1⟩ for 14/9 or 2.3.13/5 [1 -1 1⟩ for 26/15. For subgroups with primes as basis elements (such as 2.3.7), a subgroup monzo can be seen as a standard monzo abbreviated to hide zeroes. To make this more clear, the abbreviated sections may be replaced with ellipses, such as 2.3.7 [1 -2 ... 1⟩ for 14/9.
+=== Tempered monzos ===
+{{Main|Tempered monzos and vals}}
+A tempered monzo represents a tempered interval, or an interval in a regular temperament. It functions similarly to a subgroup monzo, except each entry represents not a JI interval but a generator for that regular temperament. For example, the tmonzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 [-6 4⟩. (Note that we write the generators with tildes to indicate that they are tempered intervals). This means that to find the major third, you go up four perfect twelfths (~3/1) and down six octaves (~2/1).
+More generally, tempered monzos are applicable to any regular tuning, regardless of JI mapping, so corresponding intervals in two different regular temperaments that are tuned the same way have the same tempered monzo.
+=== Fractional monzos ===
+{{Main|Fractional monzos}}
+Any of the previous categories of monzo can also be a "fractional monzo", allowing entries to be fractions or non-integer rational numbers as opposed to just integers. This allows monzos to express equal divisions of just intervals (or stacks thereof). For example, [-1/2 1/2⟩ is a monzo representing a neutral third equal to half of a perfect fifth, and [1/12⟩ is a monzo representing a 12edo semitone. [1/12 1/13⟩ is a monzo representing 1\12edo stacked with 1\13edt. (Numerically, this is the 156th root of 2<sup>13</sup>*3<sup>12</sup>.) Note that we write the fractional monzo entries with forward slashes (as they represent fractions), despite writing edosteps with backslashes.
 == See also ==
 * [[Extended bra-ket notation]]
-* [[Smonzos and svals]] — subgroup monzos and vals
-* [[Tmonzos and tvals]] — tempered monzos and vals
 == External links ==