Projective tuning space - Revision history

ArrowHead294: Update links

2024-11-02T18:39:27Z

Update links

← Older revision		Revision as of 18:39, 2 November 2024
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	\| \| [[File:pts-2-3-5-e2-twtop-tlin.jpg\|alt=pts-2-3-5-e2-twtop-tlin.jpg\|x250px\|pts-2-3-5-e2-twtop-tlin.jpg]]		\| [[File:pts-2-3-5-e2-twtop-tlin.jpg\|alt=pts-2-3-5-e2-twtop-tlin.jpg\|x250px\|pts-2-3-5-e2-twtop-tlin.jpg]]
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	== See also ==		== See also ==
	* [[Gallery of projective tuning space images]]		* [[Gallery of projective tuning space images]]
	* [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: exploring~~ temperaments#~~Projective_tuning_space\|~~Projective tuning space~~, from~~ Dave Keenan & Douglas Blumeyer's ~~guide to RTT~~]]: a very detailed and beginner-friendly explanation (with many supplementary diagrams) of how to understand, read, and use these diagrams		* [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Projective tuning space\|Dave Keenan & Douglas Blumeyer's run-down on projective spaces]]: a very detailed and beginner-friendly explanation (with many supplementary diagrams) of how to understand, read, and use these diagrams

	== External links ==		== External links ==
	* [~~https~~://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline		* [http://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline
	* [~~https~~://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])		* [http://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Cmloegcmluin: /* See also */

2024-09-02T18:41:34Z

Fredg999: Added Wikipedia box, misc. edits

2023-01-15T03:42:01Z

Added Wikipedia box, misc. edits

← Older revision		Revision as of 03:42, 15 January 2023
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	Projective tuning space is the ~~[https://en.wikipedia.org/wiki/Projectivization~~ projectivization] of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.		{{Wikipedia\|Projectivization}}
			'''Projective tuning space''' is the projectivization of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

	== Quotes ==		== Quotes ==

Fredg999: Added link to app, misc. edits

2023-01-15T03:40:32Z

Added link to app, misc. edits

← Older revision		Revision as of 03:40, 15 January 2023
Line 1:		Line 1:
	Projective tuning space is the [https://en.wikipedia.org/wiki/Projectivization projectivization] of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.		Projective tuning space is the [https://en.wikipedia.org/wiki/Projectivization projectivization] of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

	=Quotes=		== Quotes ==
	"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich\|Paul Erlich]]		"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich\|Paul Erlich]]

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	= See also =		== See also ==
			* [[Gallery of projective tuning space images]]

	* [~~[Gallery_of_projective_tuning_space_images\|Gallery of projective tuning space images]~~]		== External links ==
			* [https://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline
	* [https://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])		* [https://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Cmloegcmluin: /* See also */

2022-03-29T20:25:22Z

Fredg999: Categories

2022-02-28T04:53:50Z

Sintel: Fix link

2021-12-19T14:39:09Z

Fix link

← Older revision		Revision as of 14:39, 19 December 2021
Line 1:		Line 1:
	Projective tuning space is the [~~http~~://en.wikipedia.org/wiki/~~Projectiavization~~ projectivization] of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.		Projective tuning space is the [https://en.wikipedia.org/wiki/Projectivization projectivization] of ordinary [[Vals_and_Tuning_Space\|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

	=Quotes=		=Quotes=

Cmloegcmluin: remove temporary link to defactoring

2021-11-12T23:07:09Z

remove temporary link to defactoring

← Older revision		Revision as of 23:07, 12 November 2021
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	Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically ''val space'', not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what ''tuning space'' is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as {{val\|1200 1900 2800}}. You could think of the difference this way: in val space, the val {{val\|6 10 14}} would be further away from the origin from {{val\|3 5 7}}, hidden exactly behind it; however in tuning space both of these could be represented by {{val\|1200 2000 2800}} (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. {{val\|1200.12 2000.20 2800.28}} or {{val\|1199.88 1999.80 2799.72}}).		Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically ''val space'', not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what ''tuning space'' is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as {{val\|1200 1900 2800}}. You could think of the difference this way: in val space, the val {{val\|6 10 14}} would be further away from the origin from {{val\|3 5 7}}, hidden exactly behind it; however in tuning space both of these could be represented by {{val\|1200 2000 2800}} (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. {{val\|1200.12 2000.20 2800.28}} or {{val\|1199.88 1999.80 2799.72}}).

	"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that [[contorted]] ~~(or [[enfactored]])~~ tunings — whose mappings are not in lowest terms — are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE		"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that [[contorted]] tunings — whose mappings are not in lowest terms — are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE

	{\| class="wikitable"		{\| class="wikitable"

Cmloegcmluin: add link for defactoring

2021-09-29T19:37:49Z

add link for defactoring

← Older revision		Revision as of 19:37, 29 September 2021
Line 6:		Line 6:
	Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically ''val space'', not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what ''tuning space'' is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as {{val\|1200 1900 2800}}. You could think of the difference this way: in val space, the val {{val\|6 10 14}} would be further away from the origin from {{val\|3 5 7}}, hidden exactly behind it; however in tuning space both of these could be represented by {{val\|1200 2000 2800}} (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. {{val\|1200.12 2000.20 2800.28}} or {{val\|1199.88 1999.80 2799.72}}).		Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically ''val space'', not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what ''tuning space'' is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as {{val\|1200 1900 2800}}. You could think of the difference this way: in val space, the val {{val\|6 10 14}} would be further away from the origin from {{val\|3 5 7}}, hidden exactly behind it; however in tuning space both of these could be represented by {{val\|1200 2000 2800}} (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. {{val\|1200.12 2000.20 2800.28}} or {{val\|1199.88 1999.80 2799.72}}).

	"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that "contorted" tunings (whose mappings are not in lowest terms) are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE		"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that [[contorted]] (or [[enfactored]]) tunings — whose mappings are not in lowest terms — are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE

	{\| class="wikitable"		{\| class="wikitable"

Cmloegcmluin: /* Quotes */ clarify quote from Paul

2021-05-10T18:44:08Z

Quotes: clarify quote from Paul

← Older revision		Revision as of 18:44, 10 May 2021
Line 3:		Line 3:
	=Quotes=		=Quotes=
	"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich\|Paul Erlich]]		"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich\|Paul Erlich]]

			Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically ''val space'', not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what ''tuning space'' is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as {{val\|1200 1900 2800}}. You could think of the difference this way: in val space, the val {{val\|6 10 14}} would be further away from the origin from {{val\|3 5 7}}, hidden exactly behind it; however in tuning space both of these could be represented by {{val\|1200 2000 2800}} (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. {{val\|1200.12 2000.20 2800.28}} or {{val\|1199.88 1999.80 2799.72}}).

	"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that "contorted" tunings (whose mappings are not in lowest terms) are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE		"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that "contorted" tunings (whose mappings are not in lowest terms) are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE

← Older revision		Revision as of 04:53, 28 February 2022
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	[[Gallery_of_projective_tuning_space_images\|Gallery of projective tuning space images]]		[[Gallery_of_projective_tuning_space_images\|Gallery of projective tuning space images]]

			[[Category:Regular temperament theory]]

← Older revision		Revision as of 18:41, 2 September 2024
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	== See also ==		== See also ==
	* [[Gallery of projective tuning space images]]		* [[Gallery of projective tuning space images]]
			* [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Projective_tuning_space\|Projective tuning space, from Dave Keenan & Douglas Blumeyer's guide to RTT]]: a very detailed and beginner-friendly explanation (with many supplementary diagrams) of how to understand, read, and use these diagrams

	== External links ==		== External links ==

← Older revision		Revision as of 20:25, 29 March 2022
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	= See also =		= See also =

	[[Gallery_of_projective_tuning_space_images\|Gallery of projective tuning space images]]		* [[Gallery_of_projective_tuning_space_images\|Gallery of projective tuning space images]]
			* [https://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]