Subgroup monzos and vals - Revision history

FloraC: - breadcrumbs (this isn't a subpage).

2025-05-17T10:28:47Z

- breadcrumbs (this isn't a subpage).

← Older revision		Revision as of 10:28, 17 May 2025
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	~~{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}~~

	A '''subgroup monzo''' is a [[monzo]] whose elements refer to powers of the basis elements of a [[JI subgroup]] (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered [[prime interval\|primes]]. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo {{monzo\| 0 0 -1 0 0 1 }} (13/5) may be abbreviated to the subgroup monzo 5.13 {{monzo\| -1 1 }}. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 {{monzo\| 1 -1 1 }} for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.		A '''subgroup monzo''' is a [[monzo]] whose elements refer to powers of the basis elements of a [[JI subgroup]] (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered [[prime interval\|primes]]. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo {{monzo\| 0 0 -1 0 0 1 }} (13/5) may be abbreviated to the subgroup monzo 5.13 {{monzo\| -1 1 }}. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 {{monzo\| 1 -1 1 }} for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

FloraC: - abuse of quotation marks. Adopt template: monzo. Unhighlight elementary methods. It quickly gets complex when the numbers are large. Move it to the bottom. Also move the step-by-step tutorial to a collapsed box

2025-05-17T10:28:22Z

- abuse of quotation marks. Adopt template: monzo. Unhighlight elementary methods. It quickly gets complex when the numbers are large. Move it to the bottom. Also move the step-by-step tutorial to a collapsed box

← Older revision		Revision as of 10:28, 17 May 2025
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	{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}

	A '''subgroup monzo''' is a monzo whose elements refer to powers of the "basis elements" of a JI subgroup (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.		A '''subgroup monzo''' is a [[monzo]] whose elements refer to powers of the basis elements of a [[JI subgroup]] (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered [[prime interval\|primes]]. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo {{monzo\| 0 0 -1 0 0 1 }} (13/5) may be abbreviated to the subgroup monzo 5.13 {{monzo\| -1 1 }}. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 {{monzo\| 1 -1 1 }} for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

	A '''subgroup val''' is like a standard val, but the entries are the ~~mappings~~ of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.		A '''subgroup val''' is like a standard [[val]], but the entries are the [[mapping]]s of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.

	For example, the subgroup val 2.3.7 ⟨5 8 14] tells you that prime 2 is mapped to 5 steps, prime 3 is mapped to 8 steps, and prime 7 is mapped to 14 steps, without specifying a mapping for 5.		For example, the subgroup val 2.3.7 {{val\| 5 8 14 }} tells you that prime 2 is mapped to 5 steps, prime 3 is mapped to 8 steps, and prime 7 is mapped to 14 steps, without specifying a mapping for 5.

	For short, a subgroup monzo may be referred to as an '''smonzo''', and a subgroup val may be referred to as an '''sval'''.		For short, a subgroup monzo may be referred to as an '''smonzo''', and a subgroup val may be referred to as an '''sval'''.

	== Conversion ==		== Conversion ==
	~~The simplest way to convert between standard monzos~~ and ~~subgroup monzos~~ is ~~to evaluate the monzo as~~ a ~~rational number, then start factoring out either primes or~~ subgroup basis ~~elements. To convert 2.3.13/5 [1 -1 1⟩ to~~ a ~~standard monzo, we first look at~~ the ~~interval itself, which is 26/15. You can see that there~~'~~s 213 on the top~~, ~~and~~ then ~~35 on~~ the ~~bottom, and so the correct~~ monzo is ~~[1 -1 -1 0 0 1⟩.~~		If '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by

	To convert back, knowing the subgroup 2.3.13/5, we evaluate this monzo as a fraction (26/15). Recognizing the factor of 13/5 is a little tricky, but you can rearrange the expression as (26/5)/3 to make it more visible. Once you factor it out you have 13/5 * 2 on top, and 3 on the bottom, which can be used to write the subgroup monzo [1 -1 1⟩.		$$\vec m = S\vec m_G$$

	~~=== Without rationals ===~~		Conversely, if '''m''' is a monzo for an interval in the subgroup, then we can take the [[pseudoinverse]] of ''S'', ''S''<sup>+</sup>, and the corresponding smonzo is given by
	~~The following section goes over ways to convert between~~ the ~~two types~~ of ~~monzos without looking at~~ the ~~monzo's value as a rational.~~

	~~In order to convert a subgroup monzo to a standard monzo (without converting it to a rational number), look at the monzo for each basis element of the subgroup monzo. For example, 2.3.13/5 [1 -1 1⟩:~~		$$\vec m_G = S^+ \vec m$$

	2 = [1 0 0 ~~0 0 0⟩~~		For example, consider the subgroup generated by the [[The Archipelago\|barbados triad]], 1–13/10–3/2–2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is {{monzo\| 2 -3 2 0 0 2 }}, and left-multiplying this by ''S''<sup>+</sup> gives the smonzo {{monzo\| 2 -3 2 }}. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired.

	~~3 = [0 1 0 0 0 0⟩~~		Here is a tutorial that goes through the steps if you need to calculate them by hand.

	13/5 ~~= [0 0~~ -1 ~~0 0 1⟩~~		{{Databox\|Tutorial\|
			Say we want to convert the subgroup monzo 2.3.13/5 {{monzo\| 1 -1 1 }} to a standard monzo, look at the monzo for each basis element of the subgroup monzo:

	~~Now, we take each of these monzos and multiply each of the entries by the corresponding entry in the subgroup~~ monzo.		2 {{=}} {{monzo\| 1 0 0 0 0 0 }}

	~~[1 0~~ 0 ~~0 0 0⟩ * 1 = [~~1 0 0 0 0 0⟩		3 {{=}} {{monzo\| 0 1 0 0 0 0 }}

	[0 1 0 0 0 0⟩ * -1 ~~= [~~0 -1 ~~0 0 0 0⟩~~		13/5 {{=}} {{monzo\| 0 0 -1 0 0 1 }}

	~~[0 0 -1 0 0 1⟩ * 1 = [0 0 -1 0 0 1⟩~~		Now, we take each of these monzos and multiply each of the entries by the corresponding entry in the subgroup monzo.

	~~Now, we add all the entries together vertically, to get:~~		{{monzo\| 1 0 0 0 0 0 }} × 1 {{=}} {{monzo\| 1 0 0 0 0 0 }}

	~~26/15 = [~~1 -1 -1 0 0 ~~1⟩.~~		{{monzo\| 0 1 0 0 0 0 }} × (-1) {{=}} {{monzo\| 0 -1 0 0 0 0 }}

	~~To go back from [1 -1~~ -1 0 0 ~~1⟩ to our subgroup~~ monzo~~, the most reasonable option is to repeatedly subtract and add the monzos for our subgroup basis elements until we reach the unison [~~0 0 0 0 ~~0 0⟩.~~		{{monzo\| 0 0 -1 0 0 1 }} × 1 {{=}} {{monzo\| 0 0 -1 0 0 1 }}

	~~[1 -1 -1 0 0 1⟩ - [0 0 -1 0 0 1⟩ + [0 1 0 0 0 0⟩ - [1 0 0 0 0 0⟩ = [0 0 0 0 0 0⟩~~		Now, we add all the entries together vertically, to get:

	~~If we keep track of how many times we subtract each basis~~ monzo ~~(negative for adding it), it's~~ 1 ~~for 2,~~ -1 ~~for 3, and 1 for 13/5. Thus, we re~~-~~derive our subgroup monzo [~~1 -1 1⟩.		26/15 {{=}} {{monzo\| 1 -1 -1 0 0 1 }}.

	~~=== Linear algebra ===~~		To go back from {{monzo\| 1 -1 -1 0 0 1 }} to our subgroup monzo, the most reasonable option is to repeatedly subtract and add the monzos for our subgroup basis elements until we reach the unison {{monzo\| 0 0 0 0 0 0 }}.
	~~Mathematically speaking~~, ~~if '''m'''<sub>''G''</sub>~~ is ~~an smonzo of~~ the subgroup ~~''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a~~ basis ~~for the subgroup ''G'', then~~ the ~~corresponding~~ monzo ~~'''m''' is given by~~

	~~$$\vec m~~ = ~~S\vec m_G$$~~		{{monzo\| 1 -1 -1 0 0 1 }} - {{monzo\| 0 0 -1 0 0 1 }} + {{monzo\| 0 1 0 0 0 0 }} - {{monzo\| 1 0 0 0 0 0 }} {{=}} {{monzo\| 0 0 0 0 0 0 }}

	~~Conversely~~, ~~if '''m'''~~ is ~~a monzo~~ for ~~an interval in the subgroup~~, ~~then we can take the [[pseudoinverse]] of ''S''~~, ~~''S''<sup>+<~~/~~sup>~~, ~~and the corresponding smonzo is given by~~		If we keep track of how many times we subtract each basis monzo (negative for adding it), it is 1 for 2, -1 for 3, and 1 for 13/5. Thus, we re-derive our subgroup monzo {{monzo\| 1 -1 1 }}.
			}}

	~~$$\vec m_G~~ = ~~S^+ \vec m$$~~		=== In elementary math ===
			An elementary way to convert between standard monzos and subgroup monzos is to evaluate the monzo as a rational number, then start factoring out either primes or subgroup basis elements. To convert 2.3.13/5 {{monzo\| 1 -1 1 }} to a standard monzo, we first look at the interval itself, which is 26/15. You can see that there is {{nowrap\| 2 × 13 }} on the top, and then {{nowrap\| 3 × 5 }} on the bottom, and so the correct monzo is {{monzo\| 1 -1 -1 0 0 1 }}.

	~~For example~~, ~~consider~~ the subgroup ~~generated by the [[The Archipelago\|barbados triad]], 1–13/10–3/2–2~~. ~~The normal interval list for [13/10,~~ 3~~/2, 2] is [2, 3,~~ 13/5], ~~and~~ we ~~may use these just like the primes [2, 3, 5] of 5-limit just intonation~~. ~~We may convert intervals in~~ the ~~subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting~~ of ~~the monzos for 2, 3, and~~ 13/~~10. Now~~ the ~~monzo for 676~~/~~675 is~~ {{~~monzo~~\| ~~2 -3 2 0 0~~ 2 }}, and ~~left-multiplying this by ''S''<sup>+</sup> gives~~ the ~~smonzo~~ {{monzo\| 2 -~~3 2~~ }}~~. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired~~.		To convert back, knowing the subgroup 2.3.13/5, we evaluate this monzo as a fraction (26/15). Recognizing the factor of 13/5 is a little tricky, but you can rearrange the expression as (26/5)/3 to make it more visible. Once you factor it out you have {{nowrap\| 13/5 × 2 }} on top, and 3 on the bottom, which can be used to write the subgroup monzo {{monzo\| 1 -1 1 }}.

	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Val]]		[[Category:Val]]
	[[Category:Monzo]]		[[Category:Monzo]]

VectorGraphics at 05:37, 16 May 2025

2025-05-16T05:37:01Z

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	{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}

	A '''subgroup monzo''' is a monzo whose elements refer to powers of ~~arbitrary~~ "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.		A '''subgroup monzo''' is a monzo whose elements refer to powers of the "basis elements" of a JI subgroup (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

	A '''subgroup val''' is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.		A '''subgroup val''' is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.

VectorGraphics at 05:12, 16 May 2025

2025-05-16T05:12:48Z

← Older revision		Revision as of 05:12, 16 May 2025
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	A '''subgroup val''' is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.		A '''subgroup val''' is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.

			For example, the subgroup val 2.3.7 ⟨5 8 14] tells you that prime 2 is mapped to 5 steps, prime 3 is mapped to 8 steps, and prime 7 is mapped to 14 steps, without specifying a mapping for 5.

	For short, a subgroup monzo may be referred to as an '''smonzo''', and a subgroup val may be referred to as an '''sval'''.		For short, a subgroup monzo may be referred to as an '''smonzo''', and a subgroup val may be referred to as an '''sval'''.

VectorGraphics at 05:02, 16 May 2025

2025-05-16T05:02:13Z

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	{{Breadcrumb\|Monzo}}		{{Breadcrumb\|Monzo\|}}{{Breadcrumb\|Val\|}}

	A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.		A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

VectorGraphics at 05:00, 16 May 2025

2025-05-16T05:00:55Z

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			{{Breadcrumb\|Monzo}}

	A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.		A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

VectorGraphics: VectorGraphics moved page Smonzos and svals to Subgroup monzos and vals

2025-05-16T05:00:34Z

VectorGraphics moved page Smonzos and svals to Subgroup monzos and vals

← Older revision	Revision as of 05:00, 16 May 2025
(No difference)

VectorGraphics at 05:00, 16 May 2025

2025-05-16T05:00:21Z

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-Given a [[just intonation subgroup]], we can find a canonical form for its [[generator]]s by means of the [[Normal lists #Normal interval list|normal interval list]] which may be computed from any finite set of generators. In the case of the full ''p''-limit group for any prime ''p'', this consists of the primes from 2 to ''p'' in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and [[Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to ''p''. This generalization we may call the '''subgroup monzos''' and '''subgroup vals''', or '''smonzos''' and '''svals''' for short.
+A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.
 == Conversion ==
 In order to convert a subgroup monzo to a standard monzo (without converting it to a rational number), look at the monzo for each basis element of the subgroup monzo. For example, 2.3.13/5 [1 -1 1⟩:
@@ Line 22: / Line 33: @@
 /15 = [1 -1 -1 0 0 1⟩.
 Mathematically speaking, if '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by

VectorGraphics at 01:16, 15 May 2025

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 == Conversion ==
-If '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by
+In order to convert a subgroup monzo to a standard monzo (without converting it to a rational number), look at the monzo for each basis element of the subgroup monzo. For example, 2.3.13/5 [1 -1 1⟩:
 $$\vec m = S\vec m_G$$

VectorGraphics: "a smonzo" is incorrect

2025-05-15T01:08:37Z

"a smonzo" is incorrect

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	== Conversion ==		== Conversion ==
	If '''m'''<sub>''G''</sub> is a smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by		If '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by

	$$\vec m = S\vec m_G$$		$$\vec m = S\vec m_G$$