# Color notation/Temperament names

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

## Definition

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.

The color defines a lattice row, and the magnitude (large, small, etc.) defines a **segment** of that row. A name without a magnitude, like Zozo, refers to the central segment. Each segment contains 7 ratios. The comma that is tempered out is usually the smallest in cents of those 7. If not, **-bi** or **-tri** is added to the end of the name to indicate that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi. Any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment, thus -bi and -tri are only used for very large commas.

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).

- Schismatic = Layo, Mavila = Layobi, Superpyth = Sayo, Meantone = Gu, Father = Gubi.
- Dicot = Yoyo, Immunity = Sasa-yoyo, Bug = Gugu, Diaschismic = Sagugu, Beatles = Sasa-gugu.
- Porcupine = Triyo, Augmented = Trigu, Laconic = Latrigu, Misty = Sasa-trigu.
- Negri = Laquadyo, Tetracot = Saquadyo, Vulture = Sasa-quadyo, Diminished = Quadgu.

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**yo, Bi

**zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented:**

__zo__**Sa**sa-

**gu**gu,

**Zo**zotri

**gu**.

Sometimes the smallest ratio in a segment is some other comma raised to some power. 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, only intervals are.

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 commas don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: the 2.3.5.7.11 subgroup with 81/80 tempered out is Gu + zala. Primes 2 and 3 are always assumed to be present in the subgroup, even if the commas don't contain them. They are never added, but are sometimes removed. Prime 3 is removed with the term "Nowa", and prime 2 with "Noca" (ca for clear). Thus 2.5.7 with 50/49 is named Biruyo Nowa. "Nowaca" removes both 2 and 3.

If the comma is wa, an edo is implied. For the most common cases of 5-edo, 7-edo and 12-edo, the temperament is named after the wa comma. Thus Blackwood is Sawa + ya, Whitewood is Lawa + ya, and Catler is Lalawa + za.

Any other wa comma is named using the Wa-N format. Thus Countercomp is Wa-41 + ya, not the difficult-to-decipher Tribisawa + ya. Note that multi-ring edos such as 10-edo can't be implied by a wa comma, and Wa-10 is not a valid comma name. However 10-edo can be created by a non-wa comma, or by a wa comma plus a non-wa comma, e.g. Sawa & Yoyo.

Temperaments are abbreviated with "T": Zozo = zzT, Triyo = y^{3}T, Gu & Rugu = g&rgT, Layobi = Ly#2T, Sawa + ya = sw+yT, Wa-41 + ya = w-41+yT, Gu + zala = g+z1aT, and Biruyo Nowa = rryy-wT.

More examples of temperaments:

- User:TallKite/Catalog of single-comma rank two temperaments with Color names
- 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 thirteen-limit rank two temperaments with Color names
- User:TallKite/Catalog of eleven-limit rank three temperaments with Color names
- Catalog of rank two temperaments (under construction)

## 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.

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:

If the midpoint
ratio is |
do this to the 3-exponent | ||
---|---|---|---|

if no suffix | if "-bi" suffix | if "-tri" suffix | |

0-204¢ | nothing | add 2 | subtract 3 |

204-294¢ | subtract 2 | nothing | add 2 |

294-498¢ | add 3 | subtract 2 | nothing |

498-702¢ | add 1 | add 3 | subtract 2 |

702-906¢ | subtract 1 | add 1 | add 3 |

906-996¢ | subtract 3 | subtract 1 | add 1 |

996-1200¢ | add 2 | subtract 3 | subtract 1 |

### 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.

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 __not__ 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.

if the
comma is |
and if S mod 7 is | ||||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | |

0-90¢ | -- | -- | -- | -- | -- | -- | -- |

90-204¢ | -- | -- | -- | -- | -bi | -bi | -- |

204-294¢ | -bi | -bi | -bi | -bi | -bi | -bi | -bi |

294-408¢ | -tri | -bi | -bi | -bi | -tri | -tri | -tri |

408-498¢ | -tri | -tri | -tri | -tri | -tri | -tri | -tri |

## Naming multi-comma temperaments

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.

### Choosing the commas

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.
- The choice of commas must allow elimination of commas via downward inheritances.
- 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 #1 makes a comma list that, if viewed as a matrix, has zeros in the upper right corner. Thus each comma's rightmost nonzero number is a pivot of the matrix. The mapping matrix always has zeros in the lower left, thus each row's leftmost nonzero number is a pivot. Every prime is either a comma pivot or a mapping pivot. The sign of the pivot is unimportant, so we'll define the pivot as the absolute value of the number in the matrix. As long as the comma matrix has no torsion (rule #2) and the mapping matrix isn't contorted, the product of the commas' pivots equals the product of the mappings' pivots. This number is called the temperament's **pivot product**. Torsion always makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. In particular, one can identify torsion in the comma list and remove it. (But if the products are the same, it's possible that there is both torsion __and__ contorsion, which is bad. So one can't rely on unequal pivot products to detect torsion.)

A comma's pivot is the absolute value of the last number in the comma's monzo. The color name of a comma indicates its pivot directly: it's the number of times the first color occurs. Sagugu has a pivot of 2, as does Biruyo. Both Rugu and Zotrigu have 1, and Trizo-agugu has 3. For wa commas, the pivot is the edo: Sawa has a pivot of 5. For multi-comma temperaments, the pivot product is the product of each comma's pivot. Sagugu & Latrizo = 2·3 = 6, Gu & Biruyo = 1·2 = 2, etc. Thus the color name directly indicates the pivot product.

For a rank-2 temperament with primes 2 and 3 both being mapping pivots, the pivot product indicates how many chains of 5ths are in the temperament's lattice, i.e. the amount of splitting in the pergen. A pivot product of 2 means something is split in half, e.g. Yoyo is half-fifth and Sagugu is half-octave. Triyo splits something into 3 parts. Neither Ru nor Layobi split anything. 4 means either one thing is split into quarters (e.g. Quadgu), or two things are split into halves (e.g. Zozo & Lulu).

Some double-split pergens have more splitting than the pivot product implies, thus a "quad-" comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). But M2 = P5 + P5 - P8, and P5 = P8/2 + M2/2 = 1 period + 2 generators. Thus P5 has a genspan of 2, and the mapping's pivot product is 2 x 2 = 4. And indeed Laquadlo's lattice has 4 chains of 5ths. For a pergen (P8/m, (a,b)/n), where (a,b) is the multigen, the pivot product is m·n/|b|. Pergens with an imperfect multigen (|b| > 1) are the only pergens where the pergen's splitting is more than the pivot product implies. Fortunately imperfect pergens are fairly rare, only about 3% of all rank-2 pergens. For a rank-3 pergen (P8/m, (a,b)/n, (a',b',c')/n'), the pivot product is m·n·n'/|b·c'|.

Eliminating torsion means minimizing the commas' pivots. For example, Quadgu & Quadru has a comma pivot product of 16, but the pergen is (P8/4, P5), which means the mapping's pivot product is only 4. Since the ya comma is fixed, the solution is to add/subtract some number of ya commas to the yaza comma to get a new yaza comma that can be simplified. Quadgu plus Quadru equals Quadrugu, which simplifies to Rugu. Quadgu & Rugu has no torsion, and is a better name than Quadgu & Quadru.

Because of rule #2, __the color name always indicates strong vs. weak upward extensions__. A strong extension always has the same pivot product, and a weak extension never does. Thus a strong upward extension always adds a comma with a pivot of 1, and a weak upward extension always adds a comma with a pivot > 1. (See "Issues" for downward 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. This is because Gugu & Zozo has torsion. We can't change the ya comma, because rule #1 completely determines the 1st comma. Instead we change the 2nd one, and 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 avoiding torsion 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 has a lower odd limit.

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.

### Inheriting temperament names

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.

Triyo implies Ru, and Triyo & Ru is called Triyo-d. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is Lasepyo, or Lasepyo-de.

Extensions can be downward (adding lower primes) as well as upward. Every two-comma temperament (i.e. codimension = 2) can be viewed as an extension in either direction. For example, Sayo & Ru is an upward extension of Sayo, and also a downward extension of Ru. These both happen to be not only strong extensions but also the best strong extensions, and this extension could be called either Sayo-d or Ru-c. But the smaller prime is preferred, so it's called Sayo-d. 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 (and is) called Zozo. And Sagugu & Zozo is weak both ways, so it can't be shortened.

[*Possible refinement of this: given two commas that are each the strongest extension of the other, and having to choose just one to name the temperament, choose not the lower prime, but the prime with the simplest mapping. Simplest means fewest steps on the genchain from some 3-limit interval. For example, yazala Orwell has mapping [(1 0 3 1 3) (0 7 -3 8 2)]. We have a choice of Lasepyo, Sepru or Laseplo. The genchain mappings for 5, 7 and 11 are -3, 8 and 2. 5/4 is 3 steps away from P1, 7/6 is 1 step from P5, and 11/8 is 2 steps from P1. Thus 7/6 is closest, and Orwell is named Sepru-ce. Another example: yaza Superpyth has commas Sayo and Ru, and mapping [(1 1 -3 4) (0 1 9 -2)]. Here 5/4 and 7/4 both coincide with a 3-limit interval, so instead we use the numbers 9 and -2 and choose 7/4, and name the temperament Ru-c.*]

Rule #3 says that if the upward extension is weak and the downward extension is not only strong but also the best, the name must reflect that by excluding the lower prime. Thus 2.3.5.7 in effect becomes 2.3.7.5. For example, za Liese is called Latriru, after its comma (-9 11 0 -3). The best downward extension of Liese has commas 81/80 and 686/675 (z^{3}gg). Both are lower odd limit than the Latriru comma, thus without rule #3 7-limit Liese would be called Gu & Trizo-agugu. But then excluding the Gu comma would make Trizo-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza.

To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru. Thus 7-limit Liese is called Latriru-c.

Some rank-2 temperaments have wa commas, which imply edos. Every edo implies other commas, which are simply the best strong extension of the wa 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 by rule #4). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. Tweaks 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 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.

### Identifying vanishing commas

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.

For example, consider the Quadgu & Rugu temperament. Does the Zotrigu comma vanish? Remove zo by adding rugu to get quadgu. Remove gu by subtracting quadgu to get wa. Yes, it vanishes. Does the Biruyo comma vanish? Biruyo = ruruyoyo. Remove ru by subtracting rugu twice to get quadyo. Remove yo by adding quadgu to get wa. Yes, it vanishes. Does the Ruyoyo comma vanish? Remove ru by subtracting rugu to get triyo. Adding quadgu gives gu, so the comma can't be reduced to wa, and hence doesn't vanish.

Sometimes removing colors returns a false positive, because the prime limit is reduced to 3, but not necessarily to 1. In other words, the final wa interval may not be the wa unison. But this test never gives false negatives. If the comma's color can't be reduced to wa, the comma definitely does not vanish.

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.)

### Issues

__SELECTING THE COMMA SET__:

For some temperaments, the commas' odd limits are much smaller if one changes the order of higher primes: 2.3.5.7 becomes 2.3.7.5. This means the first comma is za and the second one is yaza. The 2nd comma's pivot is the ya-exponent.

For example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo. Miracle is Lala-tribiyo & Ruyoyo, but could be Latrizo & Ruyoyo.

A strong downward extension always removes the original name if the new comma's pivot is > 1. A strong upward extension never removes it.

Squares is Laquadru = (P8, P11/4). Sidi adds the Yoyo comma, (P8, P5/2) which is also (P8, P11/2). Sidi is a strong extension of Laquadru, but it's called Yoyo & Zozoyo, so it doesn't look like a strong extension, or even a weak one. Adding a lower prime with a similar pergen changes the higher prime's comma. za Orwell is Sepru, yaza Orwell is Lasepyo (& Ruyoyo).

Beep = Gugu + Zozo = Gugu (& Zogu). It's named after the badder of the two commas, so that the less bad comma can be the best extension. We use bad commas in order to get fewer commas.

To isolate each prime's effect on the temperament, put the comma list in IRREF form.

*There could be a rule that if two primes make the same pergen, choose the one who's IRREF comma has the lowest double odd limit to head up the subgroup. Thus Yoyo + Lulu = 2.3.5.11 = Yoyo (& Luyo) as before, but Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Before, it was Trisa-yoyo (& Saluyo), so this new rule makes a shorter name. But Beep remains 2.3.5.7 = Gugu (& Zogu), which we don't want, because the Gugu comma is so high-error.*

*Or we could choose the IRREF comma that has the lowest badness. This makes Yoyo + Lulu = 2.3.11.5 = Lulu (& Luyo), Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Gugu + Zogu = Zozo (& Zogu). But sometimes the names in parentheses are NOT the best extensions, and they can't be dropped.*

Old names: Hemififths = P5/2 = Sasa-zozo ==> Trisa-yoyo ==> Lulu ==> Thuthu. All commas have the same pergen. Lulu = 243/242, Thuthu = 512/507. Ordering the primes by odd limit of the commas makes a 2.3.11.13.7.5 temperament, called Lulu (& Thulu & Saluzo/Tholuluzo & Saluyo/Tritho-aquadlu-ayo/Luzozogu/Thuzozogu).

Combining 2 commas: an upward ext must equal a downward extension: A + B must equal B + A

Yoyo + Lulu = Yoyo (& Loyo), because Lulu is the best ila extension of Yoyo, so the name Yoyo is inherited.

Lulu + Yoyo must also be Yoyo (& Loyo). Yoyo is a strong but not best downward extension of Lulu. it steals the name, removing Lulu from the list to avoid torsion. Lulu is "dis-inherited". The name reflects the worst commas, not the best ones.

If we treat it as 2.3.11.5, Lulu + Yoyo = Lulu & Loyo, strong but not best. Yoyo + Lulu is "downward", Lulu steals the name.

Triyo + Trirubi = Triyo & Rugu, strong but not best upwd ext

Trirubi + Triyo = Triyo & Rugu, best ext of Trirubi but NOT best ext of Triyo. Trirubi + Triyo could be Trirubi (& Rugu) if viewed as 2.3.7.5.

*Best extension = IRREF comma makes same pergen, has least double odd limit? No, makes Gu (& Ru). Can't ignore error. Has least badness? No, Triyo + Ru = Triyo (& Ru), not the same pergen but still the best ext. Two low badness commas can make a high-badness temperament.*

best up & best down: Vulture Sasa-quadyo + Saquadru = Sasa-quadyo (& Saquadru)

best up & strong down: Yoyo + Lulu = Yoyo (& Loyo)

best up & weak down: Triyo + Ru = Triyo (& Ru) ............. Sagugu + Ru = Sagugu (& Ru)

strong up & best down:

strong up & strong down: Gu + Ru = Gu & Rugu

strong up & weak down: Triyo + Rugu = Triyo & Rugu

weak up & best down: Liese, Gu + Latruru = (Gu &) Latriru .............. Gu + Laquadru = (Gu &) Laquadru

weak up & strong down: Gu + Zozo = (Gu &) Zozo

weak up & weak down: Triyo + Zozo = Triyo & Zozo

__DEFINITION OF BADNESS__:

The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The Cangwu badness is sqrt (k*complexity^2 + (complexity*error)^2) for a weighting parameter k. 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 directly from the name.

When two different extensions could both arguably be considered the best, depending on the exact metric, one way to resolve the matter is to not allow either one to inherit the name.

The best metric for naming purposes is one that tends to give the same inheritances that have already been agreed on. This hasn't been determined yet.

__AN ATTEMPT TO NAME MEANTONE STRONG EXTENSIONS (AKA MEANTONE'S IMMEDIATE FAMILY) WITH TWEAKS AKA WARTS__:

2.3.5.7

The badness is from the xenwiki page on the meantone family. I just took the only 5 strong extensions listed and ranked them by badness.

Meantone-d is septimal, 7/4 = A6, Badness: 0.0170

Meantone-dd is dominant, 7/4 = m7, Badness: 0.0207

Meantone-ddd is sharptone, 7/4 = M6, Badness: 0.0248

Meantone-dddd is flattone, 7/4 = d7, Badness: 0.0386

Meantone-ddddd is Plutus, 7/4 = M7, Badness: 0.0453

2.3.5.11

Unfortunately the page doesn't list any 2.3.5.11 strong extensions at all, so I don't know the badnesses. So I just guessed at the rankings.

Meantone-e is unidecimal, 11/8 = AA3

Meantone-ee is meanpop, 11/8 = dd5

Meantone-eee is Meanenneadecal, 11/8 = A4

Meantone-eeee is Meanundeci or Meanertone, 11/8 = P4

11-limit dominant, 11/8 = d5

Domination, 11/8 = A3

## Advantages of color names

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.

A temperament's color name is fairly concise. Assuming a reasonable prime-limit, if the comma's numerator has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables. Thus the spoken color name is generally much shorter than the spoken ratio.

The color name indicates the prime subgroup. For example, Ruyoyo (225/224, Marvel) is yaza (2.3.5.7) because it contains 2 explicit colors ru and yo (7 and 5) and 2 implicit colors wa and clear (3 and 2). For explicit colors, each color pair (yo/gu, zo/ru, ilo/lu etc.) indicates a single prime. For example, Sagugu & Biruyo has only 2 explicit color pairs, and is yaza.

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. __Don't subtract plusses__. sw+yT (3 colors minus 1 comma) is 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 indicates the pivot product, and thus 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 color name also indicates splitting of colors other than wa. For example, Ruyoyo equates every zo ratio with a yoyo ratio. Every other yoyo ratio is some yo ratio doubled, so every other zo ratio is halved. The zo ratio may need to be widened by an 8ve, so actually every other voicing of every other zo ratio is halved. Likewise every other ru ratio equals two gu ratios. For example, two yo 3rds equals a zo 6th, and two gu 2nds equals a ru 2nd.

The color name of a multi-comma temperament creates an easy test to see if some other comma vanishes, see above.

The length of the color name is a rough indication of the comma's 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 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.

The color name indicates the cents of the comma only very loosely. Without an ending -bi, the comma is 0-204¢. If ending with -bi, the comma is 90-408¢, if with -tri, it's 294-612¢, and if with -quad it's 498-702¢.

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 > 90¢ but somewhat far away, and is medium damage. Sasa-quadyo is < 204¢ and quite far away, and low damage.

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.

### Advantages over current temperament names

Color names are easier than 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:

- 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

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:

- Squares/Skwares
- Srutar/Shrutar
- Sensei/Sensi
- Sensis/Sensus
- Wurschmidt/Worschmidt/Whirrschmidt
- Fifive/Fifives
- Ennealiminal/Ennealimmal/Ennealimmic/Ennealimnic

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

- The schisma creates Helmholtz
- The diaschisma creates Srutal
- The semicomma creates Orson
- The gamelisma creates Slendric

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

## Rules for naming remote colors

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

- Adjacent exponents are always listed largest first: tribi- not bitri-.
- Bibi- is never used, use quad- instead.
- Avoid using the -a- delimiter if possible: z
^{4}gg = 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'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.

Bi- is not used with primary colors (zogugu not zobigu, and zozotrigu not bizo-atrigu), unless preceded by another exponent (tribigu not trigugu). However bi- is always used with primary colors of two or more syllables (bitwetho not twethotwetho). Bi- is always used with compound colors, to indicate the GCD: bizogugu not zozoquadgu.

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

### Bicolored examples

gg = gugu (Bug)

zgg = zogugu

zzgg = bizogu

zzg = zozogu

g^{3} = trigu (Augmented)

zg^{3} = zotrigu (Starling)

zzg^{3} = zozotrigu

z^{3}g^{3} = trizogu

z^{3}gg = trizo-agugu

z^{3}g = trizo-agu

g^{4} = quadgu (Diminished)

zg^{4} = zoquadgu

zzg^{4} = bizogugu

z^{3}g^{4} = trizo-aquadgu

z^{4}g^{4} = quadzogu

z^{4}g^{3} = quadzo-atrigu

z^{4}gg = bizozogu (Breedsmic)

z^{4}g = quadzo-agu

g^{5} = quingu

zg^{5} = zoquingu

zzg^{5} = zozoquingu

z^{3}g^{5} = trizo-aquingu

z^{4}g^{5} = quadzo-aquingu

z^{5}g^{5} = quinzogu

z^{5}g^{4} = quinzo-aquadgu

z^{5}g^{3} = quinzo-atrigu

z^{5}gg = quinzo-agugu

z^{5}g = quinzo-agu

g^{6} = tribigu

zg^{6} = zotribigu

zzg^{6} = bizotrigu

z^{3}g^{6} = trizogugu

z^{4}g^{6} = bizozotrigu

z^{5}g^{6} = quinzo-atribigu

z^{6}g^{6} = tribizogu

z^{6}g^{5} = tribizo-aquingu

z^{6}g^{4} = tribizo-aquadgu*

z^{6}g^{3} = trizozogu

z^{6}gg = tribizo-agugu*

z^{6}g = tribizo-agu

g^{7} = sepgu

zg^{7} = zosepgu

zzg^{7} = zozosepgu

z^{3}g^{7} = trizo-asepgu

z^{4}g^{7} = quadzo-asepgu

z^{5}g^{7} = quinzo-asepgu

z^{6}g^{7} = tribizo-asepgu

z^{7}g^{7} = sepzogu

z^{7}g^{6} = sepzo-atribigu

z^{7}g^{5} = sepzo-aquingu

z^{7}g^{4} = sepzo-aquadgu

z^{7}g^{3} = sepzo-atrigu

z^{7}gg = sepzo-agugu

z^{7}g = sepzo-agu

g^{8} = quadbigu

zg^{8} = zoquadbigu

zzg^{8} = bizoquadgu

z^{3}g^{8} = trizo-aquadbigu

z^{4}g^{8} = quadzogugu

z^{5}g^{8} = quinzo-aquadbigu

z^{6}g^{8} = tribizo-aquadbigu*

z^{7}g^{8} = sepzo-aquadbigu

z^{8}g^{8} = quadbizogu

z^{8}g^{7} = quadbizo-asepgu

z^{8}g^{6} = quadbizo-atribigu*

z^{8}g^{5} = quadbizo-aquingu

z^{8}g^{4} = quadzozogu

z^{8}g^{3} = quadbizo-atrigu

z^{8}gg = quadbizo-agugu*

z^{8}g = quadbizo-agu

g^{9} = tritrigu

zg^{9} = zotritrigu

zzg^{9} = zozotritrigu

z^{3}g^{9} = trizotrigu

z^{4}g^{9} = quadzo-atritrigu

z^{5}g^{9} = quinzo-atritrigu

z^{6}g^{9} = trizozotrigu

z^{7}g^{9} = sepzo-atritrigu

z^{8}g^{9} = quadbizo-atritrigu

z^{9}g^{9} = tritrizogu

z^{9}g^{8} = tritrizo-aquadbigu

z^{9}g^{7} = tritrizo-asepgu

z^{9}g^{6} = tritrizo-atribigu*

z^{9}g^{5} = tritrizo-aquingu

z^{9}g^{4} = tritrizo-aquadgu

z^{9}g^{3} = tritrizo-atrigu*

z^{9}gg = tritrizo-agugu

z^{9}g = tritrizo-agu

### Tricolored examples

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

1uzgg = luzogugu

1uzzgg = lubizogu

1uzzg = luzozogu

1uzg^{3} = luzotrigu

etc.

1uuzg = luluzogu

1uugg = bilugu

1uuzgg = luluzogugu

1uuzzzgg = biluzogu

1uuzzg = biluzo-agu

1uuzg^{3} = luluzotrigu

1uuzzg^{3} = biluzo-atrigu

1uuz^{3}g^{3} = lulutrizogu

1uuz^{3}gg = lulutrizo-agugu

1uuz^{3}g = lulutrizo-agu

1u^{3}zg = trilu-azogu

1u^{3}zgg = trilu-azogugu

1u^{3}zzgg = trilu-abizogu

1u^{3}zzg = trilu-azozogu

1u^{3}zg^{3} = trilu-azotrigu

1u^{3}zzg^{3} = trilu-azozotrigu

1u^{3}z^{3}g^{3} = triluzogu

1u^{3}z^{3}gg = triluzo-agugu

1u^{3}z^{3}g = triluzo-agu

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 1uuz^{6}g^{3}, zo merges with the cubed prime, not the squared prime, to make lulu-trizozogu, not bilutrizo-atrigu.

### Quadricolored examples

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

3o1uzg = tholuzogu

3o1uzgg = tholuzogugu

3o1uzzgg = tholubizogu

3o1uuzzg = thobiluzo-agu

etc.

3oo1uzg = thotholuzogu

3oo1uzgg = thotholuzogugu

3oo1uzzg = thotholuzozogu

3oo1uzzgg = thotholubizogu

3oo1uuzg = bitholu-azogu

3oo1uuzgg = bitholu-azogugu

3oo1uuzzg = bitholuzo-agu

3oo1uuzzgg = bitholuzogu

etc.