Monzo - Revision history

FloraC: /* Examples */ Adopt template

2025-12-03T17:52:31Z

Examples: Adopt template

← Older revision		Revision as of 17:52, 3 December 2025
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	~~:'''Practical hint:'''~~ On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo\|read more…]]).		{{Tip\| On the wiki, the monzo template helps you getting correct brackets ([[Template: Monzo\|read more…]]). }}

	== Relationship with vals ==		== Relationship with vals ==

FloraC: Notational pet peeve (a, b, c, .. can't represent infinite variables)

2025-11-25T12:13:39Z

Notational pet peeve (a, b, c, .. can't represent infinite variables)

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	}}		}}
	{{Beginner\|Monzos and interval space}}		{{Beginner\|Monzos and interval space}}
	A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any "composite" interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo\| a ~~b c d e f~~ … }}, where ~~the columns~~ represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[~~Harmonic Limit~~\|prime limit]].		A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any composite interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, contribute to the interval's prime factorization, up to some [[harmonic limit\|prime limit]].

	Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries.		Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries.

FloraC: #Subgroup monzos looks good enough. See no point to revive the old text

2025-11-25T12:07:21Z

#Subgroup monzos looks good enough. See no point to revive the old text

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 = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n
 </math>
 == Generalizations ==
 === Subgroup monzos ===

FloraC: Cleanup +1

2025-11-25T12:06:21Z

Cleanup +1

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	== History and terminology ==		== History and terminology ==
	Monzos are named in honor of [[Joseph Monzo]], given by [[Gene Ward Smith]] in July 2003. These were also previously called ''factorads'' by [[John Chalmers]] in ''[[Xenharmonikôn]] 1'', although the basic idea goes back at least as far as [[Adriaan Fokker]] and probably further back, so that the entire naming situation can be viewed as an example of [[Wikipedia: Stigler%27s law of eponymy\|Stigler's law]] many times over. More descriptive but longer terms include '''prime-count vector'''<ref>Used by [[Douglas Blumeyer]] and [[Dave Keenan]] on this wiki, notably in [[Dave Keenan & Douglas Blumeyer's guide to RTT]]</ref>, '''prime-exponent vector'''<ref>[http://tonalsoft.com/enc/m/monzo.aspx Tonalsoft \| ''Monzo'']</ref>, and in the context of just intonation, '''harmonic space coordinates'''<ref>[https://www.plainsound.org/HEJI/ Plainsound Music Edition \| ''Plainsound Harmonic Space Calculator'']</ref>.		Monzos are named in honor of [[Joseph Monzo]], given by [[Gene Ward Smith]] in July 2003. These were also previously called ''factorads'' by [[John Chalmers]] in ''[[Xenharmonikôn]] 1'', although the basic idea goes back at least as far as [[Adriaan Fokker]] and probably further back, so that the entire naming situation can be viewed as an example of [[Wikipedia: Stigler%27s law of eponymy\|Stigler's law]] many times over. More descriptive but longer terms include '''prime-count vector'''<ref group="note">Used by [[Douglas Blumeyer]] and [[Dave Keenan]] on this wiki, notably in [[Dave Keenan & Douglas Blumeyer's guide to RTT]]. </ref>, '''prime-exponent vector'''<ref>[http://tonalsoft.com/enc/m/monzo.aspx Tonalsoft \| ''Monzo'']</ref>, and in the context of just intonation, '''harmonic space coordinates'''<ref>[https://www.plainsound.org/HEJI/ Plainsound Music Edition \| ''Plainsound Harmonic Space Calculator'']</ref>.

	== Examples ==		== Examples ==
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	-->		-->
	== Generalizations ==		== Generalizations ==
			=== Subgroup monzos ===
			{{Main\| Subgroup monzos and vals }}

	~~=== 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 {{monzo\| 1 -2 1 }} for 14/9 or 2.3.13/5 {{monzo\| 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 {{monzo\| 1 -2 … 1 }} for 14/9.
	~~{{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 ===		=== Tempered monzos ===
	{{Main\|Tempered monzos and vals}}		{{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).
			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 tempered monzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 {{monzo\| -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.		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.
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	=== Fractional monzos ===		=== Fractional monzos ===
	{{Main\|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.
			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 and stacks thereof. For example, {{monzo\| -1/2 1/2 }} is a monzo representing a neutral third equal to half of a perfect fifth, and {{monzo\| 1/12 }} is a monzo representing a 12edo semitone. {{monzo\| 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 ==		== See also ==
	* [[Extended ~~bra-ket~~ notation]]		* [[Extended bra–ket notation]]

	== External links ==		== External links ==
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	== Notes ==		== Notes ==
			<references group="note"/>

			== References ==
	<references />		<references />

FloraC: Add back Kite's conversion formula (moved from Ratio page) and cleanup

2025-11-25T11:59:30Z

Add back Kite's conversion formula (moved from Ratio page) and cleanup

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	== Examples ==		== Examples ==
	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 }}.		To find the monzo of an interval in [[ratio]] form, factor the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. If any primes smaller than the largest prime do not appear, include them using a zero exponent. Enter the exponents into the monzo.

			For example, the interval [[15/8]] can be thought of as having 5 × 3 in the numerator, and 2 × 2 × 2 in the denominator. This can be compactly represented by the expression 2<sup>-3</sup> × 3<sup>1</sup> × 5<sup>1</sup>, 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 }}.

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

VectorGraphics at 04:22, 4 November 2025

2025-11-04T04:22:36Z

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	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.		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.

			A tempered monzo may be called a ''generator-count vector'', and conversely a standard monzo can be called a ''prime-count vector''.

	=== Fractional monzos ===		=== Fractional monzos ===

VectorGraphics at 05:05, 16 May 2025

2025-05-16T05:05:35Z

← Older revision		Revision as of 05:05, 16 May 2025
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	=== Subgroup monzos ===		=== 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.		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 ===		=== 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).		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).

Sintel: /* Examples */ Add factors, to be very explicit

2025-05-15T18:18:54Z

Examples: Add factors, to be very explicit

← Older revision		Revision as of 18:18, 15 May 2025
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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 }}.		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"		{\| class="wikitable center-1"
	\|-		\|-
	! Ratio		! Ratio
			! Factors
	! Monzo		! Monzo
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
			\| <math>2^{-1} \cdot 3</math>
	\| {{monzo\| -1 1 0 }}		\| {{monzo\| -1 1 0 }}
	\|-		\|-
	\| [[5/4]]		\| [[5/4]]
			\| <math>2^{-2} \cdot 5</math>
	\| {{monzo\| -2 0 1 }}		\| {{monzo\| -2 0 1 }}
	\|-		\|-
	\| [[9/8]]		\| [[9/8]]
			\| <math>2^{-3} \cdot 3^2</math>
	\| {{monzo\| -3 2 0 }}		\| {{monzo\| -3 2 0 }}
	\|-		\|-
	\| [[81/80]]		\| [[81/80]]
			\| <math>2^{-4} \cdot 3^4 \cdot 5^{-1}</math>
	\| {{monzo\| -4 4 -1 }}		\| {{monzo\| -4 4 -1 }}
	\|}		\|}
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	\|-		\|-
	! Ratio		! Ratio
			! Factors
	! Monzo		! Monzo
	\|-		\|-
	\| [[7/4]]		\| [[7/4]]
			\| <math>2^{-2} \cdot 7</math>
	\| {{monzo\| -2 0 0 1 }}		\| {{monzo\| -2 0 0 1 }}
	\|-		\|-
	\| [[7/6]]		\| [[7/6]]
			\| <math>2^{-1} \cdot 3^{-1} \cdot 7</math>
	\| {{monzo\| -1 -1 0 1 }}		\| {{monzo\| -1 -1 0 1 }}
	\|-		\|-
	\| [[7/5]]		\| [[7/5]]
			\| <math>5^{-1} \cdot 7</math>
	\| {{monzo\| 0 0 -1 1 }}		\| {{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 ==		== Relationship with vals ==

VectorGraphics at 01:43, 15 May 2025

2025-05-15T01:43:20Z

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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}}.		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}}.
	-->		-->
	== Subgroup monzos ==		== Generalizations ==

			=== Subgroup monzos ===
	{{Main\|Smonzos and svals}}		{{Main\|Smonzos and svals}}
	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.		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 ==		=== Tempered monzos ===
	{{Main\|Tmonzos and tvals}}		{{Main\|Tmonzos and tvals}}
	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).		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).
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	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.		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 ==		=== Fractional monzos ===
	{{Main\|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.		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.

VectorGraphics at 01:42, 15 May 2025

2025-05-15T01:42:54Z

@@ Line 89: / Line 89: @@
 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}}.
 -->
 == See also ==
 * [[Extended bra-ket notation]]
 == External links ==