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 without the degree (unison, 2nd, etc.). For example, Semaphore is the Zozo temperament. The name of the temperament and the comma is always capitalized, to distinguish it from the color. Thus zozo refers to all zozo ratios, whereas Zozo refers to one specific zozo ratio, the zozo 2nd = zz2 = 49/48. 

The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. A name without a magnitude, e.g. Zozo, refers to the central segment. Each segment contains 7 ratios. The comma that is tempered out is the smallest in cents of those 7.  

Words like large, small, double, etc. are abbreviated, to make the names a reasonable length.

* Double = '''bi-''' ("bee"), triple = '''tri-''' ("tree"), quadruple = '''quad-''', quintuple = '''quin-''', septuple = '''sep-'''

* '''-bi''' or '''-tri''' at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi 

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]].

# [[Schismatic]] = Layo, [[Mavila]] = Layobi, [[Superpyth]] = Sayo, [[Meantone]] = Gu, [[Father]] = Gubi.

# [[Dicot]] = Yoyo, [[Immunity family|Immunity]] = Sasa-yoyo, [[Bug]] = Gugu, [[Diaschismic]] = Sagugu = sggT, [[Beatles]] = Sasa-gugu.

# [[Porcupine]] = Triyo, [[Augmented]] = Trigu, [[Laconic family|Laconic]] = Latrigu, [[Misty comma|Misty]] = Sasa-trigu.

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-''' (as in "<u>le</u>git"), 13-fold = '''the-''' (as in "<u>the</u>saurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.

Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z³g³ and Trizo-agu = z³g. The "a" in la and sa acts as a delimiter: Trilayo = L³y and Trila-triyo = L³y³. More examples of temperaments: 

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 smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma bigger 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¢, 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.

Sometimes the smallest ratio in a segment is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)², tempering it out would simply result in the Zogu temperament. Thus there is no Bizogu temperament, although there is a Bizogubi one.

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. 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". For example, tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo.

If the commas 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 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, y³T = 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. If the comma is < 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.

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 set, in order of priority:

#The prime limit of each comma must be higher than the one before.

#The color name must indicate strong vs. weak extensions, if possible.

#The choice of commas must allow elimination of commas via upward and/or downward inheritances.

#Double odd limit must be minimized.

The first rule completely determines the first comma, except for the edo problem (see Issues below).

The color name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, Ru and Ruyoyo split nothing, etc. Some double-splits are false doubles, which means a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). The comma splits the 8ve "by accident".  

More importantly, the color name differentiates between strong extensions and weak extensions. Gugu = 27/25, and Zozo = 49/48, and each one is (P8, P4/2). Combining both commas, Gugu & Zozo is a bad name, because it looks like a weak extension of Gugu (and of Zozo) when it is actually strong. Instead we call it Gugu & Zogu. The Zogu comma is 21/20, so this name also has the advantage of using a lower odd-limit comma. However, often the effect of implying the right kind of extension is to raise the odd limit. For example, Pajara is Sagugu & Ru (2048/2025 & 64/63), not Sagugu & Biruyo, even though the Biruyo comma 50/49 is a lower 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 (lowest badness) strong (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 is Gu & Zotrigu. This is called simply Gu, or perhaps yaza Gu or 7-limit Gu. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma set. In the table below, it's written as Gu (& Zotrigu). Any combination of the Gu and Zotrigu commas, e.g. Ruyoyo, makes the same extension, so Gu could be said to imply Ruyoyo as well. But such a comma will have a higher odd limit, and isn't part of the name, so the canonical best za extension for Gu is Zotrigu.

Triyo implies Ru, and Triyo & Ru is called simply Triyo, or perhaps yaza Triyo. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is yazala Lasepyo, or simply Lasepyo.

Extensions can be downward (adding lower primes) as well as upward. Every two-comma (i.e. codimension = 2) temperament can be viewed as a strong or weak extension in either direction. For example, Sayo & Ru is a strong extension of Sayo, and also of Ru. These both happen to be the best strong extensions, and Sayo & Ru could be called either yaza Sayo or yaza Ru. But the upward extension always takes priority, so Sayo & Ru is called Sayo. Often strong extensions are not possible in one or both directions, because each comma individually creates a different pergen. For example, Gu & Zozo is upwardly weak but downwardly strong, so it can't be called Gu, but it can be called Zozo. And Sagugu & Zozo is weak both ways, so it can't be shortened, it's always called Sagugu & Zozo.  

Rule #3 says that if the upward extension is weak and the downward extension is the best, the name must reflect that by excluding the lower prime. For example, za Liese is called Latriru, after its comma |-9 11 0 -3>. The best downward extension of Liese has commas 81/80, 686/675 (z³gg) and 1029/1000 (z³g³), all lower odd limit than the Latrilu comma. But Yaza Liese is called neither Gu & Trizo-agugu nor Gu & Trizogu, but Latriru.

Some rank-2 temperaments have 3-limit commas, which are written as edos. Every edo implies other commas, which are simply the best strong extension of the 3-limit temperament to higher primes. 12-edo implies Gu and Ru. 5-edo implies Gubi and Zo (and also Ru, but Zo is the canonical comma). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. '''Tweaks''' aka warts change the implied comma: 22c-edo implies Gu and Ru. [''needs checking: The best extension sometimes creates tweaks, e.g. 12-edo's best 11-limit extension is 33/32, not 729/704, thus 12-edo becomes 12e-edo.'']

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.

SELECTING THE COMMA SET:

It would be simpler to select commas solely on prime limit and double odd limit. It's not always possible to indicate the correct amount of splitting in the pergen, e.g. Clyde is Tribiyo & Zozoyo. The name implies a 6-fold split, but the pergen is (P8, W⁴P4/12). One can deduce the 12-fold because the 1st comma equates 6-fold yo to wa, and the 2nd comma equates ruru to yo, thus 12-fold ru equals wa. To get an explicit 12-fold split, one could have a za and a yaza comma: Sasa-tribizo & Zozoyo.

Even simpler than prime limit and double odd limit would be the Hermite Normal Form, or IRREF (Integral Reduced Row Echelon Form), or even LLL reduction (Lenstra, Lenstra, and Lovász).

The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The latter uses various weighting parameters. So the definition will inevitably be somewhat arbitrary. The best extension will also not be obvious from merely examining the commas, but will require lengthy computations. This removes one of the main advantages of color names, that the comma set, and hence the mapping, the pergen, etc., can be derived from the name.

QUARTER-SPLIT PROBLEM:

Clyde: Tribiyo & Sasa-quadtrizo = (P8, W⁴P4/12)

Also a ninth-split problem, sixteenth-split problem, etc.

EDO PROBLEM:

4-edo & Yoyo = (P8/4) implies a different pergen. 9/8 halves the octave, Yoyo halves the 5th, so make it 2-edo & Yoyo?

The color name also indicates the rank of the temperament. Ruyoyo is rank-3 because 4 colors minus 1 comma = rank-3. Sagugu & Biruyo is 4 color pairs minus 2 commas = rank-2. <u>Subtract edos, but not plusses</u>. 12edo&ryyT (4 colors minus 1 edo and 1 comma) is rank-2. 5edo+yT (3 colors minus 1 edo) is also rank-2. Primes 2 and 3 are assumed present in the temperament even if they are not present in the comma. Biruyo is yaza and rank-3, and Biruyo Nowa is yaza nowa and rank-2.

The color name also hints at the [[pergen]]. The name only indicates the amount of splitting, not which wa interval is split. Because Sagugu has gu twice, it halves something, in this case the 8ve. Zozo halves the 4th, Bizozogu halves the 5th, and Latrizo splits the 5th into three parts. A name with a tribi color either splits something into six parts, or splits something into two and something else into three. (This is one rationale for using tribi and not hexa, to show the possibilities.) A strong extension of a temperament has the same pergen, and a weak extension has a different one. Thus adding either 2 or 3 to the subgroup is a weak extension. For example, Gu & Biruyo must be a weak extension of Gu, and a strong extension of Biruyo. The commas in a multi-comma temperament name are chosen to indicate strong & weak extensions.

The length of the color name is a rough indication of the comma's [[Commas by taxicab distance|taxicab distance]] in the lattice. Each la- or sa- adds on average 7 steps on the three-axis. Each yo or gu adds a step on the five-axis, each zo/ru adds a seven-axis step, etc. If [[Commas by taxicab distance|triangularized]] taxicab distance is desired, let over-colors (yo, zo, ilo, etc.) cancel under-colors of smaller primes (gu, ru, etc.), and let under-colors cancel smaller over-colors.

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 safe 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 conventional 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.

* Heinz ketchup uses 57 varieties of pickles

* The Beatles toured the US in 1964

* Injera is an Ethiopean bread, and the Ethiopean alphabet has 26 letters

* James Bond is agent 007

* Mavila is a Chopi village

* Orwell wrote "1984", in which Winston, Big Brother and Doublethink appear

* Squares/Skwares

* Srutar/Shrutar

* Sensei/Sensi

* Sensis/Sensus

* Wurschmidt/Worschmidt/Whirrschmidt

== Rules for naming remote colors ==

* Adjacent multipliers are always listed largest first: tribi- not bitri-.

* Bibi- is never used, use quad- instead.

Therefore if the color (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.  

Bi- is not used with primary colors (zogugu not zobigu, and zozotrigu not bizo-atrigu), unless preceded by another multiplier (tribigu not trigugu). Bi- is always used with compound colors, to indicate the GCD: bizogugu not zozoquadgu.  

=== Bicolored examples ===

 

 

=== Tricolored examples ===

=== Quadricolored examples ===