Douglas Blumeyer's RTT How-To - Revision history

Cmloegcmluin: this article has now been superseded by this series, which contains everything it used to contain and more, but improved

2022-12-08T19:28:03Z

this article has now been superseded by this series, which contains everything it used to contain and more, but improved

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Dave Keenan: /* Mapping-row-bases and comma bases */ Changed "Or at least that when" to "Or at least when"

2022-09-02T23:39:16Z

Mapping-row-bases and comma bases: Changed "Or at least that when" to "Or at least when"

← Older revision		Revision as of 23:39, 2 September 2022
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	Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.		Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.

	If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave Keenan:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in their writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since they first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least ~~that~~ when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".		If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave Keenan:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in their writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since they first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".

	And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".		And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

Dave Keenan: /* Mapping-row-bases and comma bases */ Changed "our" to "their" and "we" to "they".

2022-09-02T23:34:50Z

Mapping-row-bases and comma bases: Changed "our" to "their" and "we" to "they".

← Older revision		Revision as of 23:34, 2 September 2022
Line 487:		Line 487:
	Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.		Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.

	If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave Keenan:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in ~~our~~ writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since we first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least that when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".		If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave Keenan:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in their writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since they first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least that when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".

	And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".		And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

Dave Keenan: /* Mapping-row-bases and comma bases */ Corrected "Dave Keeanan" to "Dave Keenan"

2022-09-02T23:27:52Z

Mapping-row-bases and comma bases: Corrected "Dave Keeanan" to "Dave Keenan"

← Older revision		Revision as of 23:27, 2 September 2022
Line 487:		Line 487:
	Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.		Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.

	If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave ~~Keeanan~~:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in our writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since we first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least that when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".		If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps<ref>The following notes are adapted from research by Dave Keenan:<br><br>While it is true that, in mathematics generally, ''mapping'' and ''map'', when used as nouns, are synonyms, and both are synonymous with ''function''. But there is very little difference between an individual row of a mapping, and a mapping with only one row. So if we were to agree that, in RTT, only an individual row should be called a ''map'', and someone new to the field assumes that a map is the same as a mapping, then there are almost no consequences of that temporary confusion, if it can even be called confusion. For 12edo, its 5-limit ''map'' is ⟨12 19 28], and its 5-limit ''mapping'' is [⟨12 19 28]⟩. The mnemonic is simple: The shorter term applies to the smaller object. The difference rarely matters to anyone.<br><br>Dave Keenan is one of the founders of regular temperament theory along with Paul Erlich, Graham Breed, Gene Smith and others, since 1998. In online discussions of regular temperaments, and in our writings, all four of them have referred to any array of numbers whose units are "generators per prime", as a mapping, ever since we first referred to them as anything at all, which seems to have been in early 2001. Only rarely has this been shortened to "map" — typically only as a heading in tables of temperament data generated by Gene Ward Smith. But even Gene is on record as defining a "prime mapping" as a "list of vals", here: http://www.tonalsoft.com/enc/p/prime-mapping.aspx Evidence of this history of usage of map and mapping can be found in the [[YahooTuningListArchival\|Yahoo tuning groups archive]].<br><br>Most of the temperament data in the Xen Wiki was generated by Gene, so it is not surprising if it contained "map" as an abbreviation of "mapping".<br><br>In the Xen Wiki and Graham Breed's temperament finder and the tuning archives, the term "map" (and not "mapping") already consistently refers to an individual row of the form {{map\|...}}. This is in the case of a "[[tuning map]]", which maps from generators to cents. This is a map in "tuning space". By analogy, a row of a mapping is therefore a map in "temperament space", and so it would be perfectly consistent with existing terminology, to refer to a mapping-row or one-row mapping as a "temperament map" as opposed to a temperament mapping. So an unqualified "map" should be assumed to be a temperament map, not a tuning map. Or at least that when it is clear from the context that it is a temperament map, the qualifier "temperament" can be dropped.</ref>. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".

	And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".		And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

Cmloegcmluin: avoid using the term "normalize" where "put into normal form" can be used instead, to avoid conflict with other notions of normalization

2022-05-27T02:03:07Z

avoid using the term "normalize" where "put into normal form" can be used instead, to avoid conflict with other notions of normalization

← Older revision		Revision as of 02:03, 27 May 2022
Line 286:		Line 286:
	Why did I do this to you? Well, I decided map space was conceptually easier to introduce than tuning space. Paul himself prefers to think of this diagram as a projection of tuning space, however, so I don’t want to leave this material before clarifying the difference. Also, there are different helpful insights you can get from thinking of PTS as tuning space. Let’s consider those now.		Why did I do this to you? Well, I decided map space was conceptually easier to introduce than tuning space. Paul himself prefers to think of this diagram as a projection of tuning space, however, so I don’t want to leave this material before clarifying the difference. Also, there are different helpful insights you can get from thinking of PTS as tuning space. Let’s consider those now.

	The first key difference to notice is that we can ~~normalize~~ coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map\|3 5 7}} is located physically in front of {{map\|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map\|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map\|6 9 14}} is halfway between {{map\|3 5 7}} and {{map\|3 4 7}}, but in tuning space it is immediately obvious that {{map\|1200 1800 2800}} is halfway between {{map\|1200 2000 2800}} and {{map\|1200 1600 2800}}.		The first key difference to notice is that we can standardize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map\|3 5 7}} is located physically in front of {{map\|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map\|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map\|6 9 14}} is halfway between {{map\|3 5 7}} and {{map\|3 4 7}}, but in tuning space it is immediately obvious that {{map\|1200 1800 2800}} is halfway between {{map\|1200 2000 2800}} and {{map\|1200 1600 2800}}.

	So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 3k)'', we can see how every point is about the same distance from us.		So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 3k)'', we can see how every point is about the same distance from us.

	[[File:Tuning space version.png\|400px\|thumb\|right\|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space ~~normalizes itself nearby to~~ JI.]]		[[File:Tuning space version.png\|400px\|thumb\|right\|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space tends toward JI.]]

	The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{map\|6 10 14}} and {{map\|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{map\|1200 2000 2800}} and {{map\|1200 1800 2800}} you’ve got an infinitude of choices smoothly transitioning between each other; you’ve basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{map\|1200 1999.999 2800}} to {{map\|1200 1901.955 2800}} to {{map\|1200 1817.643 2800}} is along the way.		The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{map\|6 10 14}} and {{map\|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{map\|1200 2000 2800}} and {{map\|1200 1800 2800}} you’ve got an infinitude of choices smoothly transitioning between each other; you’ve basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{map\|1200 1999.999 2800}} to {{map\|1200 1901.955 2800}} to {{map\|1200 1817.643 2800}} is along the way.

Ninly: /* Tuning & pure octaves */Temperament is the ''first'' T in RTT.

2022-04-27T11:18:10Z

Tuning & pure octaves: Temperament is the ''first'' T in RTT.

← Older revision		Revision as of 11:18, 27 April 2022
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	If you think about it, you would never want to tune all the primes sharp at the same time, or all of them flat; if you care about this particular proportion of their tunings, why wouldn’t you shift them all in the same direction, toward accuracy, while maintaining that proportion? ''(see Figure 2e)''		If you think about it, you would never want to tune all the primes sharp at the same time, or all of them flat; if you care about this particular proportion of their tunings, why wouldn’t you shift them all in the same direction, toward accuracy, while maintaining that proportion? ''(see Figure 2e)''

	This matter of choosing the exact generator for a map is called '''tuning''', and if you’ll believe it, we won’t actually talk about that in detail again until much later. Temperament — the ~~second~~ ‘T’ in “RTT” — is the discipline concerned with choosing an interesting map, and tuning can remain largely independent from it. The temperament is only concerned with the fact that — no matter what exact size you ultimately make the generator — it is the case e.g. that 12 of them make a 2, 19 of them make a 3, and 28 of them make a 5. So, for now, whenever we show a value for g, assume we’ve given a computer a formula for optimizing the tuning to approximate all three primes equally well. As for us humans, let’s stay focused on tempering.		This matter of choosing the exact generator for a map is called '''tuning''', and if you’ll believe it, we won’t actually talk about that in detail again until much later. Temperament — the first ‘T’ in “RTT” — is the discipline concerned with choosing an interesting map, and tuning can remain largely independent from it. The temperament is only concerned with the fact that — no matter what exact size you ultimately make the generator — it is the case e.g. that 12 of them make a 2, 19 of them make a 3, and 28 of them make a 5. So, for now, whenever we show a value for g, assume we’ve given a computer a formula for optimizing the tuning to approximate all three primes equally well. As for us humans, let’s stay focused on tempering.

	=== A multitude of maps ===		=== A multitude of maps ===

Cmloegcmluin: /* Mapping-row-bases and comma bases */ link to new article

2022-03-07T00:14:55Z

Mapping-row-bases and comma bases: link to new article

← Older revision		Revision as of 00:14, 7 March 2022
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	[[File:Different nestings.png\|400px\|thumb\|left\|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]		[[File:Different nestings.png\|400px\|thumb\|left\|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

	We can ~~extend our angle bracket notation (technically called~~ [[~~Wikipedia:Bra%E2%80%93ket_notation\|~~bra-ket notation~~, or Dirac~~ notation]]~~<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>)~~ to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra\|{{vector\|-4 4 -1}} {{vector\|-10 -1 5}}}}. Starting from the outside, the {{map\|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket\|{{map\|-4 -10}} {{map\|4 -1}} {{map\|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).		We can [[extended bra-ket notation\|extend our angle bracket notation]] to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra\|{{vector\|-4 4 -1}} {{vector\|-10 -1 5}}}}. Starting from the outside, the {{map\|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket\|{{map\|-4 -10}} {{map\|4 -1}} {{map\|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

	Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.		Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.

Cmloegcmluin: introduce and link to myself

2022-02-28T18:56:30Z

introduce and link to myself

← Older revision		Revision as of 18:56, 28 February 2022
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	~~This~~ is the reference I wish I had when I was learning RTT, or [[Regular temperament theory\|Regular Temperament Theory]]. There are other great resources out there, but this is how I would have liked to have learned it myself. I might say these materials lean more visual and geometric than others I've seen, and focus on elementary computation and representation rather than theory. It's not really a big picture introduction, it doesn't explore musical applications, and its algorithms are for humans, not computers. In any case, I hope others are able to benefit from these tools and explanations.		Hello there. My name is [[Douglas Blumeyer]], and this is the reference I wish I had when I was learning RTT, or [[Regular temperament theory\|Regular Temperament Theory]]. There are other great resources out there, but this is how I would have liked to have learned it myself. I might say these materials lean more visual and geometric than others I've seen, and focus on elementary computation and representation rather than theory. It's not really a big picture introduction, it doesn't explore musical applications, and its algorithms are for humans, not computers. In any case, I hope others are able to benefit from these tools and explanations.

	There's still a ton more to say here, though, and I hope to get to completing this material soon.		There's still a ton more to say here, though, and I hope to get to completing this material soon.

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:51Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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	[[Category:Tuning]]		[[Category:Tuning]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale ~~theory~~]]		[[Category:Scale]]

Cmloegcmluin: link to new page for Supports

2022-01-25T18:56:51Z

link to new page for Supports

← Older revision		Revision as of 18:56, 25 January 2022
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	But a more interesting way to think about this result involves treating {{vector\|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{vector\|-1 1 0}} = {{vector\|-4 4 0}}, and then moving down one pentave {{vector\|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one pentave is the same thing as moving by four fifths ''(see Figure 2b)''. One can make compelling music that [[Keenan's comma pump page\|exploits such harmonic mechanisms]].		But a more interesting way to think about this result involves treating {{vector\|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{vector\|-1 1 0}} = {{vector\|-4 4 0}}, and then moving down one pentave {{vector\|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one pentave is the same thing as moving by four fifths ''(see Figure 2b)''. One can make compelling music that [[Keenan's comma pump page\|exploits such harmonic mechanisms]].

	From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''~~supports~~''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT.		From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''[[support]]s''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT.

	But we’re still only talking about JI and EDOs. If you’re familiar with meantone as a historical temperament, you may be aware already that it is neither JI nor an EDO. Well, we’ve got a ways to go yet before we get there.		But we’re still only talking about JI and EDOs. If you’re familiar with meantone as a historical temperament, you may be aware already that it is neither JI nor an EDO. Well, we’ve got a ways to go yet before we get there.

← Older revision		Revision as of 00:14, 7 March 2022
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	[[File:Different nestings.png\|400px\|thumb\|left\|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]		[[File:Different nestings.png\|400px\|thumb\|left\|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

	We can ~~extend our angle bracket notation (technically called~~ [[~~Wikipedia:Bra%E2%80%93ket_notation\|~~bra-ket notation~~, or Dirac~~ notation]]~~<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>)~~ to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra\|{{vector\|-4 4 -1}} {{vector\|-10 -1 5}}}}. Starting from the outside, the {{map\|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket\|{{map\|-4 -10}} {{map\|4 -1}} {{map\|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).		We can [[extended bra-ket notation\|extend our angle bracket notation]] to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra\|{{vector\|-4 4 -1}} {{vector\|-10 -1 5}}}}. Starting from the outside, the {{map\|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket\|{{map\|-4 -10}} {{map\|4 -1}} {{map\|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

	Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.		Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra\|{{vector\|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket\|{{map\|19 30 44}}}}.