Mike's lecture on vector spaces and dual spaces - Revision history

FloraC: Undo revision 203568 by VectorGraphics (talk). This is Mike's essay

2025-06-25T14:41:41Z

Undo revision 203568 by VectorGraphics (talk). This is Mike's essay

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	{{~~main~~\|Mike's lectures on regular temperament theory}}		{{Main\|Mike's lectures on regular temperament theory}}
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	= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =		= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =

VectorGraphics at 04:30, 25 June 2025

2025-06-25T04:30:35Z

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	{{main\|Mike's lectures on regular temperament theory}}		{{main\|Mike's lectures on regular temperament theory}}
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	= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =		= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =

ArrowHead294 at 15:44, 31 March 2025

2025-03-31T15:44:12Z

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ArrowHead294: /* 1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO) */

2024-11-11T16:30:48Z

1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)

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	# First you go down 4 octaves, at a rate of 12 steps per octave, putting you underwater at −48 steps.		# First you go down 4 octaves, at a rate of 12 steps per octave, putting you underwater at −48 steps.
	# Then, you go up 4 tritaves, times 19 steps per tritave, giving you 76 steps. This lands you at a net of {{nowrap\|−48 + 76 {{=}} 28}} steps.		# Then, you go up 4 tritaves, times 19 steps per tritave, giving you 76 steps. This lands you at a net of {{nowrap\|−48 + 76 {{=}} 28}} steps.
	~~<nowiki />~~# Finally, you go down one 5/1, times 28 steps per 5/1, putting you down 28 more steps. This lands you at a net of {{nowrap\|28 − 28 steps {{=}} 0}}.		# Finally, you go down one 5/1, times 28 steps per 5/1, putting you down 28 more steps. This lands you at a net of {{nowrap\|28 − 28 steps {{=}} 0}}.

	So, if you mechanically work out the way that you'd compute how many steps 81/80 is in 12-EDO, you get 0 steps, meaning you're back at 1/1 and hence tempered out. No surprise there.		So, if you mechanically work out the way that you'd compute how many steps 81/80 is in 12-EDO, you get 0 steps, meaning you're back at 1/1 and hence tempered out. No surprise there.

ArrowHead294 at 16:19, 11 November 2024

2024-11-11T16:19:30Z

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	= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =		= <u>EPISODE 1: Vector Spaces and Dual Spaces</u> =
	If you haven't seen monzos or vals before and are totally confused, please read the pages on [[monzos\|Monzos]] and [[Vals\|Vals]] first!		If you haven't seen monzos or vals before and are totally confused, please read the pages on [[monzos\|Monzos]] and [[Vals\|Vals]] first!

ArrowHead294: Markup formatting

2024-11-11T16:19:18Z

Markup formatting

Show changes

Fredg999: Categories

2023-03-06T21:07:00Z

Fredg999: Fredg999 moved page Mike's Lecture on Vector Spaces and Dual Spaces to Mike's lecture on vector spaces and dual spaces: WP:NCCAPS

2023-03-06T21:06:33Z

Fredg999 moved page Mike's Lecture on Vector Spaces and Dual Spaces to Mike's lecture on vector spaces and dual spaces: WP:NCCAPS

← Older revision	Revision as of 21:06, 6 March 2023
(No difference)

Inthar: /* 1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACE[1]. */idk if slight is the right word, since Z modules not necessarily being free is a big difference!

2021-01-05T11:25:49Z

1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACE[1].: idk if slight is the right word, since Z modules not necessarily being free is a big difference!

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	-----		-----
	==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. ~~The differences are slight and for~~ practical purposes, this technicality is completely accounted for by tools like the [http://x31eq.com/temper/ x31eq temperament finder] --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperament theory we usually want non-contorted temperaments.</ref>.==		==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. For practical purposes, this technicality is completely accounted for by tools like the [http://x31eq.com/temper/ x31eq temperament finder] --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperament theory we usually want non-contorted temperaments.</ref>.==

	For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors:		For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors:

Inthar: /* 1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACETechnically, monzos don't form a vector space but a "Z-module", because monzos only take integer coefficients and Z, the set of integers, is a ring but not a field. Similarly vals are a dual Z-module to the monzos. For practical purposes, this technicality is completely accounted for by tools like the x31eq temperament finder --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperamen...

2020-12-07T04:31:55Z

/* 1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACETechnically, monzos don't form a vector space but a "Z-module", because monzos only take integer coefficients and Z, the set of integers, is a ring but not a field. Similarly vals are a dual Z-module to the monzos. For practical purposes, this technicality is completely accounted for by tools like the x31eq temperament finder --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperamen...

← Older revision		Revision as of 04:31, 7 December 2020
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	-----		-----
	==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. ~~For~~ practical purposes, this technicality is completely accounted for by tools like the [http://x31eq.com/temper/ x31eq temperament finder] --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperament theory we usually want non-contorted temperaments.</ref>.==		==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. The differences are slight and for practical purposes, this technicality is completely accounted for by tools like the [http://x31eq.com/temper/ x31eq temperament finder] --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperament theory we usually want non-contorted temperaments.</ref>.==

	For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors:		For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors:

← Older revision		Revision as of 21:07, 6 March 2023
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	<references/>		<references/>

			[[Category:Mike's lectures on regular temperament theory]]