Monzo: Difference between revisions

(One intermediate revision by one other user not shown)

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 92:

Line 101:

=== 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).

@@ 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 92: / Line 101: @@
 === Subgroup monzos ===
-{{Main|Smonzos and svals}}
+{{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|Tmonzos and tvals}}
+{{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).