Kite's color notation/Temperament names: Difference between revisions

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* [[Catalog of rank two temperaments]] (coming eventually)

== Finding the comma from the name and vice versa ==

=== Finding the comma ===

Every ratio can be named either as a standard interval or as a comma/temperament, e.g. 128/125 is both the trigu 2nd and the Trigu comma. The latter is awkward for low-odd-limit ratios: 5/4 would be the Yobi "comma" and 6/5 would be the Gutri "comma". But the former is awkward for high odd-limit ratios, because there will be many 2nds and 3rds and even 4ths, and many of them will be negative. So the latter name is used for commas, for brevity. Unfortunately, this makes identifying the comma from the name a little more work.

~~First~~ find the ratio of the midpoint of the segment. For this ratio, the sum of all the monzo exponents except the 2-exponent is a multiple of 7. For example, the gu midpoint is 6/5, and the sayoyo midpoint is (10 -9 2). Then find the cents of ~~this ratio~~, and ~~use~~ this chart, which is based on the 3-limit Dorian scale:

If the monzo is (a b c d...) then all but a and b are obvious from the color name. Next find the ratio of the midpoint of the segment. For this ratio, the sum of all the monzo exponents except the 2-exponent is a multiple of 7. For example, the gu midpoint is 6/5, and the sayoyo midpoint is (10 -9 2).

Let M be the color name's magnitude (where L = 1, LL = 2, s = -1, etc.) and let S be the sum of c, d, etc. Then the midpoint's monzo is (a 7M-S c d...), where a is chosen to octave-reduce the ratio to < 2/1. The 7 ratios of the segment are found by letting b range from 7M-S-3 to 7M-S+3. Then find the cents of all 7 ratios and sort them by the cents. The comma is the smallest cents, unless it uses the -bi suffix (2nd smallest) or -tri (3rd smallest).

An alternative method uses only the cents of the midpoint, and uses this chart, which is based on the 3-limit Dorian scale:

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~~It's fairly easy to find~~ the ~~color~~ name ~~from the comma.~~ The color is obvious from the monzo. Let S be the sum of all the monzo exponents except the 2-exponent. The magnitude is S divided by 7 and rounded off. The color and the magnitude define the segment.

=== Finding the name ===

The color is obvious from the monzo. Let S be the sum of all the monzo exponents except the 2-exponent. The magnitude is S divided by 7 and rounded off. The color and the magnitude define the segment.

Brute force method to find the suffix: find the cents of all 7 ratios in the segment, sort them by cents, and find the input ratio's place in the list.

~~Any~~ comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma larger than 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi or -tri must be appended to the name. If a comma is 90-204¢, and 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.

Alternate method: any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma larger than 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi or -tri must be appended to the name. If a comma is 90-204¢, and 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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 * [[Catalog of rank two temperaments]] (coming eventually)
 == Finding the comma from the name and vice versa ==
+=== Finding the comma ===
 Every ratio can be named either as a standard interval or as a comma/temperament, e.g. 128/125 is both the trigu 2nd and the Trigu comma. The latter is awkward for low-odd-limit ratios: 5/4 would be the Yobi "comma" and 6/5 would be the Gutri "comma". But the former is awkward for high odd-limit ratios, because there will be many 2nds and 3rds and even 4ths, and many of them will be negative. So the latter name is used for commas, for brevity. Unfortunately, this makes identifying the comma from the name a little more work.
-First find the ratio of the midpoint of the segment. For this ratio, the sum of all the monzo exponents except the 2-exponent is a multiple of 7. For example, the gu midpoint is 6/5, and the sayoyo midpoint is (10 -9 2). Then find the cents of this ratio, and use this chart, which is based on the 3-limit Dorian scale:
+If the monzo is (a b c d...) then all but a and b are obvious from the color name. Next find the ratio of the midpoint of the segment. For this ratio, the sum of all the monzo exponents except the 2-exponent is a multiple of 7. For example, the gu midpoint is 6/5, and the sayoyo midpoint is (10 -9 2).
+Let M be the color name's magnitude (where L = 1, LL = 2, s = -1, etc.) and let S be the sum of c, d, etc. Then the midpoint's monzo is (a 7M-S c d...), where a is chosen to octave-reduce the ratio to < 2/1. The 7 ratios of the segment are found by letting b range from 7M-S-3 to 7M-S+3. Then find the cents of all 7 ratios and sort them by the cents. The comma is the smallest cents, unless it uses the -bi suffix (2nd smallest) or -tri (3rd smallest).
+An alternative method uses only the cents of the midpoint, and uses this chart, which is based on the 3-limit Dorian scale:
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-It's fairly easy to find the color name from the comma. The color is obvious from the monzo. Let S be the sum of all the monzo exponents except the 2-exponent. The magnitude is S divided by 7 and rounded off. The color and the magnitude define the segment.
+=== Finding the name ===
+The color is obvious from the monzo. Let S be the sum of all the monzo exponents except the 2-exponent. The magnitude is S divided by 7 and rounded off. The color and the magnitude define the segment.
+Brute force method to find the suffix: find the cents of all 7 ratios in the segment, sort them by cents, and find the input ratio's place in the list.
-Any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma larger than 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi or -tri must be appended to the name. If a comma is 90-204¢, and 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.
+Alternate method: any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma larger than 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi or -tri must be appended to the name. If a comma is 90-204¢, and 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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