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

[[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but using an alternate format designed for commas. This format omits the degree (unison, 2nd, etc.). For example, [[Semaphore]] tempers out the zozo 2nd and is called the Zozo temperament. The name of the comma and the temperament is always capitalized, to distinguish it from the color. Thus zozo refers to all zozo ratios, whereas Zozo refers to one specific zozo ratio, 49/48.   

Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages [[Comma|here]], [[Medium commas|here]], [[Large commas|here]] and [[Unnoticeable comma|here]].

Exponent syllables like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''r<u>u</u>'''yo, Bi'''<u>zo</u>'''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'''.  

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.  

|204-294¢

|subtract 2

|add 2

|-

|subtract 1

|add 1

|-

|subtract 3

|subtract 1

|-

|996-1200¢

|add 2

|}

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.   

| --

| --

| --

| --

|-

|90-204¢

| --

| -bi

| --

|}

Multi-comma temperaments are named as a list of commas, e.g. Triyo & Ru. 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 prime limit of each comma must be higher than the one before.

#The comma list must be torsion-free.

#[[Odd limit|Double odd limit]] must be minimized.

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 & Zotrigu would become Gu & Laru, 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 Gu. Gu & Ru 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 Gu adds Zotrigu. This is called simply Gu, or Gu-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 Sawa + ya. The "+ ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. Lalawa & Ruyoyo. 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>:

Domination, 11/8 = A3

The color name can be derived from the comma list, and vice versa. The color name can be derived from the mapping matrix, and vice versa. However, inheritances have the same name.

4thward commas sharpen the 5th and 5thward ones flatten it. This indicates where in the scale tree compatible edos are likely to be. Thus temperaments that start with sa-, e.g. Sayo, tend to be compatible with sharp-5th edos 15, 17, 22, etc. And la- temperaments, e.g. Laru, tend to be compatible with edos 19, 21, 26, etc. Several caveats: central comma names like Triyo don't indicate 4thwd vs. 5thwd. Also, it's possible for a la- comma to be 4thwd if the color depth is >= 5 and the color is over, e.g. Laquinyo = Magic = (-10 -1 5). Sasa- or lala- commas are guaranteed to be 4thwd/5thwd up to color depth 11. Furthermore, Layo, while quite 5thwd, is compatible only with accurate-5th edos like 12, 24, etc.  

Color names are easier than [[Tour of Regular Temperaments|current temperament names]] for non-Anglophones. No need to learn to spell and pronounce obscure English words like porcupine, hedgehog and opossum. Color names are based on only those words that a first-year student of English would know, and spelling and pronunciation are greatly simplified.

Color names don't use mnemonics that rely on obscure facts, many with an implicit cultural bias, such as:

Furthermore, one doesn't have to guess what the significance of the numbers 57, 1964, 26, 007 or 1984 is.

Color names can be spoken without confusion, because there are no homonyms such as:

* Fifive/Fifives

*Ennealiminal/Ennealimmal/Ennealimmic/Ennealimnic

Temperaments have the same name as commas, reducing memorization, unlike current names, in which:

One last advantage: Color names are very flowing, and fun to say out loud. :)

There can be more than one way to name a comma. To avoid duplicate names, there are naming rules.  

* Avoid using the -a- delimiter if possible: z⁴gg = bizozogu, not quadzo-agugu.

Therefore if the color (the temperament name minus the magnitude) starts with an exponent word, and there's no -a- delimiter, that first exponent 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.  

There follows examples of remote colors, for illustration. These examples don't all correspond to musically useful temperaments.

gg = gugu (Bug)<br />

zgg = zogugu<br />

z⁹g = tritrizo-agu

if lu is not doubled or tripled, it just gets tacked onto the beginning:

If the 2nd prime could be merged with either the 1st prime or the 3rd prime, but not with both, it merges with whichever one has a larger exponent. Thus in 1uuz⁶g³, zo merges with the cubed prime, not the squared prime, to make lulu-trizozogu, not bilutrizo-atrigu.

if tho is not doubled or tripled, it just gets tacked onto the beginning: