Reduced mapping - Revision history

FloraC: Generator-chain mapping for higher ranks

2026-02-07T10:36:55Z

Generator-chain mapping for higher ranks

← Older revision		Revision as of 10:36, 7 February 2026
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	The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many [[period]]s should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.		The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many [[period]]s should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.

	A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the ''first'' row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}~~. This is the form of reduced mapping used in [[Template: Infobox regtemp]] on this wiki~~. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.		A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the ''first'' row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.

			It can be generalized to higher-rank temperaments by writing each row the same way. For instance, the reduced mapping of marvel is {{val\| 1; 1 0 2 -1; 0 1 2 -3 }}. This is the form of reduced mapping used in [[Template: Infobox regtemp]] on this wiki.

	== HKM's standard reduced mapping ==		== HKM's standard reduced mapping ==

Hkm: /* Hkm's standard reduced mapping */

2025-08-27T23:54:22Z

Hkm's standard reduced mapping

← Older revision		Revision as of 23:54, 27 August 2025
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	A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the ''first'' row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox regtemp]] on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.		A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the ''first'' row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox regtemp]] on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.

	== ~~Hkm~~'s standard reduced mapping ==		== HKM's standard reduced mapping ==
	To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the midnatssol temperament, which is		To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the midnatssol temperament, which is

FloraC: Restore some markup & update some links

2025-08-08T17:59:00Z

Restore some markup & update some links

← Older revision		Revision as of 17:59, 8 August 2025
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	== Generator-chain mapping ==		== Generator-chain mapping ==
	The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many ~~periods~~ should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.		The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many [[period]]s should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.

	A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the first row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox ~~Regtemp~~]] on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.		A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the ''first'' row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox regtemp]] on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.

	== Hkm's standard reduced mapping ==		== Hkm's standard reduced mapping ==

FloraC: Restore the old reduced mapping, which has some recognition in the community and is notably used in Template: Infobox Regtemp. The new "standard reduced mapping" isn't standard until accepted by the community; note the author as a means of distinction for now.

2025-08-08T17:50:09Z

Restore the old reduced mapping, which has some recognition in the community and is notably used in Template: Infobox Regtemp. The new "standard reduced mapping" isn't standard until accepted by the community; note the author as a means of distinction for now.

← Older revision		Revision as of 17:50, 8 August 2025
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	The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many periods should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.		The most basic reduced mapping is sometimes called ''generator-chain mapping'', which is simply the mapping without the first row. For instance, the mapping of magic is {{mapping\| 1 0 2 -1 \| 0 5 1 12 }}, and its generator-chain mapping is {{val\| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction\|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many periods should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a [[monzo]] to find its number of generator steps in the temperament.

	== ~~Standard~~ reduced mapping ==		A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the first row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is {{val\| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox Regtemp]] on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.
	To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament ~~"Midnatssol~~," which is
			== Hkm's standard reduced mapping ==
			To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the midnatssol temperament, which is

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17
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	Comma list: 289/288, 442/441, 561/560, 676/675		Comma list: 289/288, 442/441, 561/560, 676/675

	Mapping: ~~[⟨2~~ 0 1 3 7 -1 5~~], ⟨0~~ 2 1 1 -2 4 2~~], ⟨0~~ 0 2 1 3 2 0]]		Mapping: {{mapping\| 2 0 1 3 7 -1 5 \| 0 2 1 1 -2 4 2 \| 0 0 2 1 3 2 0 }}
	{\| class="wikitable"
	!Generator		{\| class="wikitable center-all"
	\|1st		! Generator
	\|2nd		\| 1st
	\|3rd		\| 2nd
			\| 3rd
	\|-		\|-
	!preimage		! preimage
	\|140/99		\| 140/99
	\|[[26/15]]		\| [[26/15]]
	\|[[10/7]]		\| [[10/7]]
	\|-		\|-
	!WE tuning		! WE tuning
	\|600.112		\| 600.112
	\|951.191		\| 951.191
	\|617.610		\| 617.610
	\|}		\|}
	We start with the mapping above, which must be in [[Hermite normal form]]. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]		We start with the mapping above, which must be in [[Hermite normal form]]. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]

Hkm: /* Standard reduced mapping */

2025-04-28T19:28:06Z

Standard reduced mapping

← Older revision		Revision as of 19:28, 28 April 2025
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	== Standard reduced mapping ==		== Standard reduced mapping ==

	~~{{todo\|inline=1\|correct\|comment=this is only unique in the rank 2 case atm}}~~
	To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is		To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is

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	\|617.610		\|617.610
	\|}		\|}
	We start with the mapping above, which must be ~~an upper triangular matrix~~. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]		We start with the mapping above, which must be in [[Hermite normal form]]. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]

	Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 \|" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> \|".) Thus we have ⟨2 \| 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]		Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 \|" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> \|".) Thus we have ⟨2 \| 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]

Hkm: /* Standard reduced mapping */

2025-04-28T19:16:44Z

Standard reduced mapping

← Older revision		Revision as of 19:16, 28 April 2025
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	== Standard reduced mapping ==		== Standard reduced mapping ==

	[[todo\|inline=1\|correct\|comment=this is only unique in the rank 2 case atm]]		{{todo\|inline=1\|correct\|comment=this is only unique in the rank 2 case atm}}
	To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is		To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is

Hkm: /* Standard reduced mapping */

2025-04-28T19:16:13Z

Standard reduced mapping

← Older revision		Revision as of 19:16, 28 April 2025
Line 5:		Line 5:

	== Standard reduced mapping ==		== Standard reduced mapping ==

			[[todo\|inline=1\|correct\|comment=this is only unique in the rank 2 case atm]]
	To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is		To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is

Hkm: /* Standard reduced mapping */

2025-04-28T01:21:16Z

Standard reduced mapping

← Older revision		Revision as of 01:21, 28 April 2025
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	Then we reduce the generator sizes to become less than the period, then take the period-complement of every generator except the first if it is larger than half the period. At this point we now note that the sizes of the second and third generators in cents have became 249c and 18c. This gives us ⟨2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 '-2,0]		Then we reduce the generator sizes to become less than the period, then take the period-complement of every generator except the first if it is larger than half the period. At this point we now note that the sizes of the second and third generators in cents have became 249c and 18c. This gives us ⟨2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 '-2,0]

	Then we mark the just intervals in the subgroup after prime 13 or after the first prime we skipped. We also replace our ~~bra-ket~~ notation with standard [ ] brackets. Our final reduced mapping is [2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 17:'-2,0]		Then we mark the just intervals in the subgroup after prime 13 or after the first prime we skipped. We also replace our ⟨ ] notation with standard [ ] brackets. Our final reduced mapping is [2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 17:'-2,0]

	[[Category:Mapping]]		[[Category:Mapping]]
	[[Category:Terms]]		[[Category:Terms]]

Hkm: /* Standard reduced mapping */

2025-04-28T01:20:38Z

Standard reduced mapping

← Older revision		Revision as of 01:20, 28 April 2025
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	== Standard reduced mapping ==		== Standard reduced mapping ==
	To make a mapping even more concise and readable, we can do the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is		To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

Hkm: there shouldn't be options if we really want uniqueness

2025-04-28T01:20:11Z

there shouldn't be options if we really want uniqueness

← Older revision		Revision as of 01:20, 28 April 2025
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	We start with the mapping above, which must be an upper triangular matrix. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]		We start with the mapping above, which must be an upper triangular matrix. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]

	Then we ~~optionally~~ mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 \|" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> \|".) Thus we have ⟨2 \| 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]		Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 \|" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> \|".) Thus we have ⟨2 \| 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]

	Then we reduce the generator sizes to become less than the period, then take the period-complement of every generator except the first if it is larger than half the period. At this point we now note that the sizes of the second and third generators in cents have became 249c and 18c. This gives us ⟨2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 '-2,0]		Then we reduce the generator sizes to become less than the period, then take the period-complement of every generator except the first if it is larger than half the period. At this point we now note that the sizes of the second and third generators in cents have became 249c and 18c. This gives us ⟨2 \| -2,0 '-1,2 -1,3 2,3 '-4,2 '-2,0]