Radical interval - Revision history

VectorGraphics at 05:05, 16 May 2025

2025-05-16T05:05:06Z

← Older revision		Revision as of 05:05, 16 May 2025
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	=== Fractional monzos ===		=== Fractional monzos ===
	For the sake of clarity, monzos representing radical intervals are called '''fractional monzos~~''' or '''fmonzos~~'''. Mathematically, ~~fmonzos~~ behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9.		For the sake of clarity, monzos representing radical intervals are called '''fractional monzos'''. Mathematically, fractional monzos behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.
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	A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup.		A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup.

	== ~~Fmonzos~~ in projection matrices ==		== Fractional monzos in projection matrices ==
	{{Main\| Projection matrices }}		{{Main\| Projection matrices }}

VectorGraphics at 05:04, 16 May 2025

2025-05-16T05:04:46Z

← Older revision		Revision as of 05:04, 16 May 2025
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	\| ja =		\| ja =
	}}		}}
	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals.

			=== Fractional monzos ===
			For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.

FloraC: Neither exponent nor coefficient is right here. -superfluous whitespace. Style

2025-04-03T17:47:35Z

Neither exponent nor coefficient is right here. -superfluous whitespace. Style

← Older revision		Revision as of 17:47, 3 April 2025
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			A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9.
	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo~~\|EDO~~]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that ~~exponents~~ 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.
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	Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.		Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.

	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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.		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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.

	=== Radical subgroups ===		=== Radical subgroups ===

Sintel: -legacy

2025-03-29T17:59:35Z

-legacy

← Older revision		Revision as of 17:59, 29 March 2025
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	~~}}{{Legacy\|Fractional monzo~~}}		}}

	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that exponents 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that exponents 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>, which may also be written as 1\26ed312500/9.

VectorGraphics: changed "coefficients" to "exponents" because fancy math people overuse "coefficient" too much; what they are is exponents

2025-03-07T18:18:01Z

changed "coefficients" to "exponents" because fancy math people overuse "coefficient" too much; what they are is exponents

← Older revision		Revision as of 18:18, 7 March 2025
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	}}{{Legacy\|Fractional monzo}}		}}{{Legacy\|Fractional monzo}}

	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that exponents 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.

VectorGraphics: Capitalization of EDO

2025-03-06T17:16:17Z

Capitalization of EDO

← Older revision		Revision as of 17:16, 6 March 2025
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	}}{{Legacy\|Fractional monzo}}		}}{{Legacy\|Fractional monzo}}

	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.
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	Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.		Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.

	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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.		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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.

	=== Radical subgroups ===		=== Radical subgroups ===

FloraC: Fix a link

2025-03-06T08:56:47Z

Fix a link

← Older revision		Revision as of 08:56, 6 March 2025
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	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(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 ''n''-th root of a positive rational number which corresponds to it.		Vectors in [[monzos and interval space\|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 ''n''-th root of a positive rational number which corresponds to it.

	== Tunings in terms of radical intervals ==		== Tunings in terms of radical intervals ==

FloraC: Equal divisions implying an equave is a common misconception. Correct wording. Clarify a few things. style and formatting improvements

2025-03-06T08:55:08Z

Equal divisions implying an equave is a common misconception. Correct wording. Clarify a few things. style and formatting improvements

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	{{interwiki		{{interwiki
	\| de = ~~Nichtganzzahlige_Intervallvektoren~~		\| de = Nichtganzzahlige Intervallvektoren
	\| en = ~~Fractional monzo~~		\| en = Radical interval
	\| es =		\| es =
	\| ja =		\| ja =
	}}{{Legacy\|~~Fractional_monzo~~}}		}}{{Legacy\|Fractional monzo}}

	A '''radical interval''' is an interval whose ratio can be expressed in terms of roots of integers (i.e. sqrt(2)), as opposed to [[~~Just intonation\|~~just ~~intervals~~]] which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[~~Equal~~ tuning~~\|equal tunings~~]] such as [[~~EDO\|EDOs~~]], and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s ~~and [[Majestazic scale-building]]~~. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as 2^(1/2) * 3^(-1/13)). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos.''' Mathematically, fmonzos behave the same as ordinary [[~~Monzo\|monzos~~]], 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>~~. (This~~ may be ~~notated~~ 1\26ed312500/9~~, but that generally implies that 312500/9 is being considered the equave~~.)		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the cent value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with {{val\| ~~cents~~ (2) ~~cents~~ (3) … ~~cents~~ (p) }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example (1/13)~~×1200.0~~ - (1/13)~~×cents~~ (3) + (7/26)~~×cents~~ (5) = 696.1648 cents.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(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.		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 ''n''-th root of a positive rational number which corresponds to it.

	== Tunings in terms of radical intervals ==		== Tunings in terms of radical intervals ==
	Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of ~~[[Equal-step tuning\|~~equal tunings]]. For example, 12edo's fifth can be expressed as [7/~~12⟩~~ or 2^(7/12), and the ~~Bohlen-Pierce~~ supermajor third may be expressed as [0 3/~~13⟩~~ or 3^(3/13).		Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.

	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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, 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, radical intervals 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⟩.		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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.

	=== Radical subgroups ===		=== Radical subgroups ===
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	== Fmonzos in projection matrices ==		== Fmonzos in projection matrices ==
	''Main ~~article: [[~~Projection matrices~~]]''~~		{{Main\| Projection matrices }}

	~~== See also ==~~

	* [[EDO]]
	* [[Equal tuning]]

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Tuning]]		[[Category:Tuning]]

VectorGraphics at 22:14, 5 March 2025

2025-03-05T22:14:43Z

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	=== Radical subgroups ===		=== Radical subgroups ===
	A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be in the 2.4ed5 subgroup.		A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup.

	== Fmonzos in projection matrices ==		== Fmonzos in projection matrices ==

VectorGraphics at 22:14, 5 March 2025

2025-03-05T22:14:29Z

← Older revision		Revision as of 22:14, 5 March 2025
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	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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, 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, radical intervals 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⟩.		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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, 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, radical intervals 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⟩.

			=== Radical subgroups ===
			A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be in the 2.4ed5 subgroup.

	== Fmonzos in projection matrices ==		== Fmonzos in projection matrices ==

← Older revision		Revision as of 18:18, 7 March 2025
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	}}{{Legacy\|Fractional monzo}}		}}{{Legacy\|Fractional monzo}}

	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, except that exponents 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.

← Older revision		Revision as of 17:16, 6 March 2025
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	}}{{Legacy\|Fractional monzo}}		}}{{Legacy\|Fractional monzo}}

	A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.		A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo\|EDO]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap\| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos''' or '''fmonzos'''. Mathematically, fmonzos behave the same as ordinary [[monzo]]s, 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>, which may also be written as 1\26ed312500/9.

	By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.		By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val\| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap\| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.
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	Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.		Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo\| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo\| 0 3/13 }} or 3<sup>3/13</sup>.

	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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.		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 {{monzo\| 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]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo\| 3 0 0 -1 }} and {{monzo\| -1/3 1/3 }}.

	=== Radical subgroups ===		=== Radical subgroups ===