Douglas Blumeyer's RTT How-To: Difference between revisions

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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}}}}.

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: 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. 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]]. 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". 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: 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. 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]]. 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". 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".

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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}}}}.
-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".