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

(Note that Kite has developed a completely different way to name temperaments that looks somewhat similar to color notation, but uses [[Pergen|pergens]] and his [[List of uniform solfeges for pergens|uniform solfege]] instead. See [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]] for examples.)

Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages ([[Small comma]], [[Medium comma]], [[Large comma]] and [[Unnoticeable comma]]).

Exponent syllables like bi or tri are always unaccented. The final "-ti" is too. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''r<u>u</u>'''yoti, Bi'''<u>zo</u>'''zoguti. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''Sa'''sa-'''gu'''guti, '''Zo'''zotri'''gu'''ti.  

More examples of temperaments:

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.  

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

{| class="wikitable"

|-

|-

|-

| nothing

|-

|-

|-

| add 3

|-

| 906-996¢

| add 1

|-

| subtract 3

| subtract 1

|}

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.   

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.   

{| class="wikitable"

|-

|-

| --

| --

| --

|-

|  --

| --

| --

|-

| -bi

| -bi

| -bi

|-

| -tri

| -bi

| -tri

|-

| -tri

| -tri

|}

Multi-comma temperaments are named as a list of commas, e.g. Triyo & Ruti. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments.

Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma list, in order of priority:

# The choice of commas must allow elimination of commas via downward inheritances.

Rule #1 ensures linear independence. It completely determines the first comma. Given two yaza commas, one can always derive the ya comma by combining the two commas such that the za component becomes zero. For example, take Ruyoyoo and Biruyo. Subtract Ruyoyo twice from Biruyo to get Sagugu. Next take Latrizo and Biruyo. The za-exponents are 3 and -2 respectively, so two Latrizos plus three Biruyos make a ya comma, Latribiyo.

Rule #3 is justified in the next section. Rule #4 is needed to ensure a unique comma list. An alternative rule would require the comma list to be in Hermite normal form, but with negative pivots allowed to ensure that the comma's cents are positive. But this would result in more obscure commas. For example, Gu & Zotriguti would become Gu & Laruti, and 126/125 would become 59049/57344. This is far less useful musically, thus rule #4 uses the double odd limit.  

Multi-comma temperament names can get quite long. To shorten them, certain extensions inherit the name of what they are extended from. The best (i.e. lowest badness) strong (i.e. same pergen) extension of a temperament inherits the name of the temperament. Thus every temperament implies certain other commas. Consider extensions of Guti. Gu & Ruti is a strong extension, but not the best strong extension, so nothing is inherited and the name can't be shortened. The best extension of Guti adds Zotrigu. This is called za Guti, or Guti-d. The "d" is analogous to '''tweaks''' aka edo warts and indicates prime 7. But unlike tweaks, "-d" is the best extension, and "-dd" is the 2nd best. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma list.  

Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is Sawati + ya. The "+ ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. Lalawa & Ruyoyoti. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied.

Rule #2 ensures that every vanishing comma is some combination of those in the list. This allows an easy way to check if a given comma is tempered out. Repeatedly reduce the prime limit of the comma in question by adding/subtracting the appropriate comma from the list. If the prime limit can be reduced to 1, the comma vanishes. The color name indicates what needs to be subtracted.

Thus once the color is reduced to wa, a 2nd test is needed. If you know the cents of each of the commas on the list as well as the one being tested, you can simply keep rough track of the cents as you add and subtract commas. If it's roughly zero, the comma vanishes. If you know each comma's 3-exponent, you can simply add and subtract those instead, and check that the end result is zero. (Presumably the commas won't add up to an entire octave.)

<u>SELECTING THE COMMA SET</u>: