Monzos and interval space - Revision history

Sintel: -legacy

2025-03-29T18:09:53Z

-legacy

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	{{Expert\|Monzo}}		{{Expert\|Monzo}}
	~~{{Legacy}}~~
	This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.		This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.

Lériendil at 16:52, 17 February 2025

2025-02-17T16:52:59Z

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	{{Expert\|Monzo}}		{{Expert\|Monzo}}
	{{Legacy~~\|Monzos_and_interval_space~~}}		{{Legacy}}
	This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.		This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.

Lériendil at 16:06, 17 February 2025

2025-02-17T16:06:44Z

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	{{Expert\|Monzo}}		{{Expert\|Monzo}}
	{{Legacy\|Monzos_and_interval_space}}		{{Legacy\|Monzos_and_interval_space}}

	This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.		This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.

Lériendil: deploying new cat

2025-02-17T16:06:34Z

deploying new cat

← Older revision		Revision as of 16:06, 17 February 2025
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	{{Expert\|Monzo}}		{{Expert\|Monzo}}
			{{Legacy\|Monzos_and_interval_space}}

	This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.		This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.

Lériendil: Changed protection level for "Monzos and interval space" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

2025-02-17T16:00:13Z

Changed protection level for "Monzos and interval space" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:00, 17 February 2025
(No difference)

FloraC: +interwiki; mark as expert page and rework the intro correspondingly

2024-04-20T08:48:43Z

+interwiki; mark as expert page and rework the intro correspondingly

← Older revision		Revision as of 08:48, 20 April 2024
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	~~<span style~~=~~"display: block; text-align: right;">[[de:~~Intervallraum~~]]</span>~~		{{Interwiki
			\| de = Intervallraum
			\| en = Monzos and interval space
			\| ja = モンゾと音程空間
			}}
			{{Expert\|Monzo}}

	This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''~~. For a simpler explanation with examples, visit the [[monzos]] page~~.		This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.

	== Definition ==		== Definition ==

FloraC: -todo

2021-12-10T16:01:29Z

-todo

← Older revision		Revision as of 16:01, 10 December 2021
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Monzo]]		[[Category:Monzo]]

	~~{{todo\| cleanup }}~~

FloraC: Styling; internalize wikipedia links; improve categories

2021-12-10T16:00:48Z

Styling; internalize wikipedia links; improve categories

← Older revision		Revision as of 16:00, 10 December 2021
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	== Definition ==		== Definition ==

	A [[~~Harmonic_Limit~~\|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving		A [[Harmonic limit\|''p''-limit]] rational number ''q'' can by definition be factored into primes of size less than or equal to ''p'', giving

	<math>q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}</math>		<math>q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}</math>

	where the exponents are integers (positive, negative, or zero.) This is often written in [http://mathworld.wolfram.com/Ket.html ket vector] ([~~http~~:~~//en.wikipedia.org/wiki/~~Bra-~~ket_notation wp~~]) notation as		where the exponents are integers (positive, negative, or zero.) This is often written in [http://mathworld.wolfram.com/Ket.html ket vector] (→ [[Wikipedia: Bra-ket notation]]) notation as

	<math>\|e_2 \, e_3 \, e_5 \dotso e_p\rangle</math>		<math>\|e_2 \, e_3 \, e_5 \dotso e_p\rangle</math>
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	<math>\\| \|e_2 \, e_3 \dotso e_p \rangle \\| = \|e_2\| + \|e_3\| \log_2 3 + \dotsb + \|e_p\| \log_2 p</math>		<math>\\| \|e_2 \, e_3 \dotso e_p \rangle \\| = \|e_2\| + \|e_3\| \log_2 3 + \dotsb + \|e_p\| \log_2 p</math>

	which is a [~~http~~:~~//en.wikipedia.org/wiki/Normed_vector_space~~ vector space norm]; hence we may [~~http~~:~~//en.wikipedia.org/wiki/~~Embedding embed] the p-limit monzos into a normed vector I space of dimension n = π(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [~~http~~:~~//en.wikipedia.org/wiki/Lattice_~~%28group%29 lattice], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by ~~log2~~(k), then the norm becomes the standard [http://mathworld.wolfram.com/L1-Norm.html L1 norm]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while {{monzo\|1 0}} represents 2, so does {{monzo\|0 ~~log3~~(2)}}.		which is a [[Wikipedia: Normed vector space\|vector space norm]]; hence we may [[Wikipedia: Embedding\|embed]] the ''p''-limit monzos into a normed vector I space of dimension ''n'' = π (''p'') via a map M:monzos ⟶ I. The monzos under this embedding now define a [[Wikipedia:Lattice %28group%29\|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime ''k'' by log<sub>2</sub> (''k''), then the norm becomes the standard [http://mathworld.wolfram.com/L1-Norm.html L1 norm]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while {{monzo\| 1 0 }} represents 2, so does {{monzo\| 0 log<sub>3</sub> (2)}}.

	Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-~~Euclidean_metrics~~\|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo {{monzo\|~~e2 e3 ... ep~~}} then the Tenney-Euclidean norm, or TE norm, of it is		Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics\|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo {{monzo\| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>''p''</sub> }} then the Tenney-Euclidean norm, or TE norm, of it is

	<math>\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</math>		<math>\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</math>
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	and if the coordinates are the weighted interval space coordinates, then the TE norm is the [http://mathworld.wolfram.com/L2-Norm.html standard Euclidean, or L2, norm].		and if the coordinates are the weighted interval space coordinates, then the TE norm is the [http://mathworld.wolfram.com/L2-Norm.html standard Euclidean, or L2, norm].

	== Alternate ~~Definition~~ ==		== Alternate definition ==
			Given a rational number ''q'', we can rewrite it in monzo form by the following definition:
	Given a rational number q, we can rewrite it in monzo form by the following definition:

	<math>q = \|v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle</math>		<math>q = \|v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle</math>

	The [[~~Tenney_Height\|~~Tenney height]] of this monzo is given by		The [[Tenney height]] of this monzo is given by

	<math>\\| \|v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \\| = \|v_2 (q)\| + \|v_3 (q)\| \log_2 3 + \dotsb + \|v_p (q)\| \log_2 p</math>		<math>\\| \|v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \\| = \|v_2 (q)\| + \|v_3 (q)\| \log_2 3 + \dotsb + \|v_p (q)\| \log_2 p</math>

	Where vp(q) is the [~~http~~:~~//en.wikipedia.org/wiki/~~P-~~adic_order~~ p-adic valuation] of q.		Where ''v''<sub>''p''</sub> (''q'') is the [[Wikipedia: P-adic order\|''p''-adic valuation]] of ''q''.

	== Example ==		== Example ==
			The 5-limit interval 16/15 factors as 2<sup>4</sup> 3<sup>-1</sup> 5<sup>-1</sup>, so it has a monzo representation of {{monzo\| 4 -1 -1 }}. In weighted coordinates, that becomes {{monzo\| 4 -log<sub>2</sub> (3) -log<sub>2</sub> (5) }}, approximately {{monzo\| 4 -1.585 -2.322 }}.
	The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of {{monzo\|4 -1 -1}}. In weighted coordinates, that becomes {{monzo\|4 -~~log2~~(3) -~~log2~~(5)}}, approximately {{monzo\|4 -1.585 -2.322}}.

	The TE norm is therefore		The TE norm is therefore
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	== See also ==		== See also ==

	* [[Fractional monzos]]		* [[Fractional monzos]]
	* [[Vals and tuning space]]		* [[Vals and tuning space]]

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Interval measure]]~~
	[[Category:Interval space]]		[[Category:Interval space]]
	[[Category:Math]]		[[Category:Math]]

Mike Battaglia: Protected "Monzos and interval space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:01:42Z

Protected "Monzos and interval space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:01, 15 November 2021
(No difference)

FloraC: Improve categories

2021-04-03T09:41:14Z

Improve categories

← Older revision		Revision as of 09:41, 3 April 2021
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	* [[Vals and tuning space]]		* [[Vals and tuning space]]

			[[Category:Regular temperament theory]]
	[[Category:Interval measure]]		[[Category:Interval measure]]
	[[Category:Interval space]]		[[Category:Interval space]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Monzo]]		[[Category:Monzo]]
	~~[[Category:Theory]]~~
			{{todo\| cleanup }}