Kite's color notation/Temperament names: Difference between revisions
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== Explanation == | == Explanation == | ||
[[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but without the degree (unison, 2nd, etc.). For example, Semaphore is the Zozo temperament. The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. Each segment contains 7 ratios. The comma that is tempered out is assumed to be the smallest in cents of those 7. Any possible color combined with any possible magnitude defines a comma, and thus a temperament, but not all of these will be musically useful. | [[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but without the degree (unison, 2nd, etc.). For example, Semaphore is the Zozo temperament. Examples: | ||
* [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]] | |||
* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]] | |||
* [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] | |||
The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. Each segment contains 7 ratios. The comma that is tempered out is assumed to be the smallest in cents of those 7. Any possible color combined with any possible magnitude defines a comma, and thus a temperament, but not all of these will be musically useful. | |||
Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | ||
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Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' ("eh") for exponent: | Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' ("eh") for exponent: | ||
* 11-fold = '''le-''' ("leh"), 13-fold = '''the-''' (as in "<u>the</u>saurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc. | * 11-fold = '''le-''' ("leh"), 13-fold = '''the-''' (as in "<u>the</u>saurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc. | ||
To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah eh ee oh oo. Quin is an exception. In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Number words like bi or tri are always unaccented. Quad may optionally be spoken as "kwah", and quin as "kwih" or "kwee". To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''. | To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah eh ee oh oo. Quin is an exception. In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Number words like bi or tri are always unaccented. Quad may optionally be spoken as "kwah", and quin as "kwih" or "kwee" or "keen". To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''. | ||
Any comma < 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma > 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi must be appended to the name. If a comma is 90-204¢, let S = the sum of all the numbers in the monzo except the first one. If and only if S mod 7 is 4 or 5, 256/243 can be subtracted without changing the magnitude, and the comma is the 2nd smallest ratio. Any 204-294¢ comma is -bi, and any 408-498¢ comma is -tri. | Any comma < 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma > 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi must be appended to the name. If a comma is 90-204¢, let S = the sum of all the numbers in the monzo except the first one. If and only if S mod 7 is 4 or 5, 256/243 can be subtracted without changing the magnitude, and the comma is the 2nd smallest ratio. Any 204-294¢ comma is -bi, and any 408-498¢ comma is -tri. | ||
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La means both large and 11-all, and sa means both small and 17-all. To avoid confusion, large and small should never be abbreviated unless part of a longer word. La is also the La note in solfege, and Sa is the tonic in saregam. The meaning will always be clear from context. Notes are never large or small. | La means both large and 11-all, and sa means both small and 17-all. To avoid confusion, large and small should never be abbreviated unless part of a longer word. La is also the La note in solfege, and Sa is the tonic in saregam. The meaning will always be clear from context. Notes are never large or small. | ||
Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments. The comma list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. See | Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments. The comma list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. See below for further discussion. | ||
If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma, because "12-edo" is more informative than "Lalawa". Tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo. | If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma, because "12-edo" is more informative than "Lalawa". Tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo. | ||
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If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: [[Blackwood]] is 5-edo + ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is Gu + zala. | If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: [[Blackwood]] is 5-edo + ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is Gu + zala. | ||
A non-wa comma can also imply an edo, but the temperament name doesn't use edos. Tempering out 128/125 from 2.3.5 makes Trigu, not 3-edo + wa. This avoids | A non-wa comma can also imply an edo, but the temperament name doesn't use edos. Tempering out 128/125 from 2.3.5 makes Trigu, not 3-edo + wa. This avoids two commas having the same name, e.g. 256/243 is 5-edo and |-14 0 0 5> is Laquinzo. | ||
Temperaments can be abbreviated much as colors are: zzT = Zozo, y<sup>3</sup>T = Triyo, g&rgT = Gu & Rugu, Ly#2T = Layobi, and g+z1aT = Gu + zala. | Temperaments can be abbreviated much as colors are: zzT = Zozo, y<sup>3</sup>T = Triyo, g&rgT = Gu & Rugu, Ly#2T = Layobi, and g+z1aT = Gu + zala. | ||
It's fairly easy to find the color name for a temperament, except for multi-comma temperaments. If there's only one comma, and it's < 90¢, the name can be found directly from the monzo. The color is obvious. The magnitude is the sum of all the exponents except the 2-exponent, divided by 7 and rounded off. | |||
It's a little harder to find the comma(s) from the color name. The 3-exponent can be found by summing commas. For example, to find the sagugu comma, start by adding two gu commas. This makes |-8 8 -2>, which is unfortunately large, not small. Correct the magnitude by adding or subtracting a centswise-small wa interval. Since we want to traverse two segments, the pythagorean comma is ideal, because it's double large. Subtracting it makes 2*g1 - LLw-2 = |11 -4 -2>, which is indeed small. These commas are all under 25¢, so two of one minus another must be < 90¢, and this must be the smallest ratio in the sagugu segment, and the one we're looking for. | |||
The Triyo comma can be found by subtracting three gu commas from some wa interval. The pythagorean comma is too small at 24¢, so try the large wa unison Lw1 = |-11 7>, aka the apotome. This makes |1 -5 3>, which is indeed central. The cents of Lw1 - 3*g1 is a semitone minus 3 small commas, roughly a quartertone. Again, this is < 90¢, so it must be the smallest ratio in the segment. | |||
== Naming multi-comma temperaments == | == Naming multi-comma temperaments == | ||
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Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is 5-edo + Ya. The "+ Ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. 12-edo & Ruyoyo. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied. | Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is 5-edo + Ya. The "+ Ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. 12-edo & Ruyoyo. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied. | ||
=== Issues === | |||
QUARTER-SPLIT PROBLEM: | QUARTER-SPLIT PROBLEM: | ||
Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it Rurugu & Quadru? | Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it Rurugu & Quadru? Quadru & Rurugu? (Breaks rule #1) | ||
Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it Lologu & Quadlo? | Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it Lologu & Quadlo? Quadlo & Lologu? | ||
Clyde: Tribiyo & Sasa-quadtrizo = (P8, W<sup>4</sup>P4/12) | Clyde: Tribiyo & Sasa-quadtrizo = (P8, W<sup>4</sup>P4/12) | ||
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The taxicab distance and the cents together roughly indicate the damage of the temperament. Gubi is > 90¢ and not far away, and thus high damage. Layobi is medium damage, and Sasa-quadyo is low damage. | The taxicab distance and the cents together roughly indicate the damage of the temperament. Gubi is > 90¢ and not far away, and thus high damage. Layobi is medium damage, and Sasa-quadyo is low damage. | ||
One last advantage: Color names are very flowing, and fun to say out loud. :) | One last advantage: Color names are very flowing, and fun to say out loud. :) | ||
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* The "a" in la and sa acts as a delimiter: trilayo = L<sup>3</sup>y and trila-triyo = L<sup>3</sup>y<sup>3</sup>. | * The "a" in la and sa acts as a delimiter: trilayo = L<sup>3</sup>y and trila-triyo = L<sup>3</sup>y<sup>3</sup>. | ||
* Avoid using the -a- delimiter if possible: z<sup>4</sup>gg = bizozogu, not quadzo-agugu. | * Avoid using the -a- delimiter if possible: z<sup>4</sup>gg = bizozogu, not quadzo-agugu. | ||
Therefore if the name (minus the magnitude) starts with a multiplier word, and there's no -a- delimiter, that first multiplier word usually indicates the color GCD and thus the [[Pergen|pergen's]] split(s). e.g. bizozogu = (P8, P5/2, /1). In the list of colors below, an asterisk marks cases where this isn't possible, and the GCD is not obvious. | Therefore if the name (minus the magnitude) starts with a multiplier word, and there's no -a- delimiter, that first multiplier word usually indicates the color GCD and thus the [[Pergen|pergen's]] split(s). e.g. bizozogu = (P8, P5/2, /1). In the list of colors below, an asterisk marks cases where this isn't possible, and the GCD is not obvious. See below for a possible solution. | ||
Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity. Hyphens are inserted before every -a- delimiter and after the magnitude (after the final la- or sa-). However, the hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are unhyphenated. | Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity. Hyphens are inserted before every -a- delimiter and after the magnitude (after the final la- or sa-). However, the hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are unhyphenated. | ||
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''z<sup>6</sup>g<sup>4</sup> = tribizo-aquadgu* = biatrizo-agugu?<br />'' | ''z<sup>6</sup>g<sup>4</sup> = tribizo-aquadgu* = biatrizo-agugu?<br />'' | ||
''z<sup>6</sup>gg = tribizo-agugu* = biatrizo-agu?<br />'' | ''z<sup>6</sup>gg = tribizo-agugu* = biatrizo-agu?<br />'' | ||
z<sup>6</sup>g<sup>8</sup> = tribizo-aquadbigu* = biatrizo-aquadgu?<br /> | ''z<sup>6</sup>g<sup>8</sup> = tribizo-aquadbigu* = biatrizo-aquadgu?<br /> | ||
z<sup>8</sup>g<sup>6</sup> = quadbizo-atribigu* = biaquadzo-atrigu?<br /> | z<sup>8</sup>g<sup>6</sup> = quadbizo-atribigu* = biaquadzo-atrigu?<br /> | ||
z<sup>8</sup>gg = quadbizo-agugu* = biaquadzo-agu?<br /> | z<sup>8</sup>gg = quadbizo-agugu* = biaquadzo-agu?<br /> | ||
z<sup>9</sup>g<sup>6</sup> = tritrizo-atribigu* = triatrizo-agugu?<br /> | z<sup>9</sup>g<sup>6</sup> = tritrizo-atribigu* = triatrizo-agugu?<br /> | ||
z<sup>9</sup>g<sup>3</sup> = tritrizo-atrigu* = triatrizo-agu? | z<sup>9</sup>g<sup>3</sup> = tritrizo-atrigu* = triatrizo-agu?'' | ||
=== Tricolored examples === | === Tricolored examples === | ||