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Ploidacot

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The ploidacot system is a classification of rank-2 temperaments based on how a temperament divides the intervals of Pythagorean tuning. A particularly simple case is if a temperament divides its 3/2 interval into n steps, it can be called an n-cot tuning. More generally, ploidacots are written as m-ploid s-sheared n-cot, with m- and n- often replaced by greek numeral prefixes, such as mono-, di-, tri-, etc. (and m-ploid omitted entirely if the octave is not split), and "s-sheared" replaced by a greek letter, such as alpha-, beta-, etc. (or omitted entirely if s = 0).

The "ploid" number of a temperament refers to how many equal parts, or periods the octave is divided into, and the "cot" number refers to how many generator steps of the temperament are needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. Temperaments where 3/2 is a whole number of ploids are written as acot. However, stacking n cots sometimes doesn't reach 3/2, but instead an interval s ploids above 3/2. There are infinitely many possible values of s, but for the sake of ploidacot, s takes its residue modulo n (which is the same for all possible cots), and is an integer between 0 and n - 1 inclusive.

For example, meantone is monocot because it is does not split the octave, and is generated by the perfect fifth. Kleismic is alpha-hexacot, since it does not split the octave, but splits 3/1, which is one octave above 3/2, into six equal parts (~317 ¢ each). Pajara is diploid monocot, since it is generated by the fifth and splits the octave in two 600 ¢ halves. Shrutar is diploid alpha-dicot, since it splits the octave in half, and splits the interval 600 ¢ above 3/2 (~1300 ¢) into two ~650 ¢ halves. Note that in shrutar the interval one ploid above 3/2 is ~1300 ¢ and not 3/1, since the octave is split into two 600 ¢ ploids.

It is similar to the pergen, and is a canonical naming scheme for pergens of rank-2 temperaments of 2.3.(…) subgroups in that every such pergen corresponds to a unique name in the ploidacot system.

The ploidacot system was developed by Praveen Venkataramana.

Specification

Ploids

Any rank-2 temperament of the 2.3.… subgroup has an octave, and it may split the octave into a number of parts, or periods, called ploids. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (not *monoploid), which can be omitted.

Cots

If 3/2 is represented by a linearly independent element to the ploid, there is a number of ploids which when added to 3/2 gives the interval which is split into the largest number of parts, namely generators, by the temperament. Each of these parts is called a cot or cotyledon. The ploidacot system uses Greek letters (alpha-, beta-, etc.) to describe the smallest nonnegative number of ploids that should be added to 3/2 to form a whole number of cots. If the number is zero, it is left empty. The number of cots is then indicated by a Greek numeral prefix. Temperaments that do not divide the fifth are called monocot (not *haplocot). The full specification of cots is thus a (possibly empty) Greek letter prefix, followed by a Greek numeral prefix, and -cot.

Temperaments where the image of 3/2 is a whole number of ploids are called acot.

Greek letter prefixes

The Greek letter prefixes follow the ancient gematria/isopsephic system, detailed below:

Greek letter prefixes in ploidacot
Number n 1 2 3 4 5 6 7 8 9
Prefix n alpha beta gamma delta epsilon digamma/wau zeta eta theta
10n iota kappa lambda mu nu xi omicron pi qoppa
n + 10 iota-alpha iota-beta iota-gamma iota-delta iota-epsilon iota-digamma/iota-wau iota-zeta iota-eta iota-theta

Prefixes for numbers between 21 and 99 are constructed the same way as number words in English, for instance 21 is kappa-alpha and 99 is qoppa-theta.

Alternatively, Arabic numerals may be used in place of the Greek alphabetical and numeric prefixes, with the word "sheared" or its equivalent in other languages used in place of the alphabetic prefixes, so a diploid epsilon-heptacot system may be referred to as a 2-ploid 5-sheared 7-cot system.

Properties

  • For n-cot systems there are exactly n settings of shear, or number of ploids to add to the step that represents the interval class of 3. The possible values of shear are 0, 1, 2, …, (n − 1). For example, the tricot systems are tricot (0-sheared), alpha-tricot (1-sheared), and beta-tricot (2-sheared). There is not a *gamma-tricot since that would be equivalent to tricot.

Extensions

Omega extension

The Greek letter omega, proposed by Godtone, is used for −1. ("Contra" has also been used in place of omega.) This simplifies the classification of certain temperaments, e.g. porcupine, which instead of beta-tricot can be omega-tricot, as splitting the interval 4/3 into three is arguably more intuitive than splitting the interval 6. This effectively shifts the possible values of shear to -1, 0, 1, …, (n − 2) if n ≥ 3.

Note that omega should only be used with n ≥ 3. When n = 1, there is only monocot. When n = 2, alpha-dicot is preferred over omega-dicot. Omega-based names are also not preferred when dealing with temperaments that split the octave, as they may be confusing - for instance, diploid alpha-tricot splits 4/3 in three while diploid beta-tricot splits 3/1 in three.

No-twos or no-threes temperaments

The ploidacot system, similarly to pergens, relies on the presence of a 3-limit, i.e. 2.3 subgroup, spine, but its defining principles can be easily applied to a 2.5, 3.5, 3.7, etc. spine instead, and in the case of ploidacot, the "cot" suffix is simply replaced with a different suffix indicating the family of intervals being cloven. The existing extensions are "seph" for 5/4 with octave equivalence, and "gem" for 7/3 with tritave equivalence (note that 3.7 is preferred over 3.5 since 9/7 and 7/3 generate a much more commonly used structure in tritave systems, i.e. Lambda, than 5/3 and 9/5).

For instance, in the 2.5.7 subgroup, didacus can be labeled as "diseph", because its generator divides 5/4 in two, and llywelyn can be labeled as "alpha-heptaseph" because seven generators make up 5/2. In the tritave world, BPS (3.5.7) is "monogem" as its generator is 9/7, while mintaka (3.7.11) is alpha-trigem as its generator (of ~21/11) splits 7/1 in three.

Even if 3 is included in a given temperament, the ploidaseph framework may occasionally be more useful than the ploidacot framework, in cases where the mapping of 3 is very complex and the structure of the temperament therefore deprioritizes prime 3. Hemiwürschmidt, a strong extension of the aforementioned didacus, has a ploidacot of beta-hexadecacot as it divides 6/1 into sixteen generators; while trismegistus has a ploidacot of epsilon-pentadecacot as it maps 96/1 to fifteen generators. Each of these has a more intuitizable expression in terms of 2.5 intervals, which are much simpler in the respective temperaments: hemiwürschmidt is diseph and trismegistus is alpha-triseph (one-third 5/2).

Combining ploidacots and ploidasephs determines its 5-limit properties; for instance, meantone can be labeled as "monocot beta-tetraseph" because four generators make up 5/1 while the generator represents 3/2, and valentine can be labeled as "enneacot pentaseph" because five generators make up 5/4 and nine of them make up 3/2.

Examples

The ploidacots of most common temperaments can be intuitively derived from a basic understanding of its mapping. Meantone and helmholtz are monocot since they have a period of a whole octave and are generated by the perfect fifth. Dicot is dicot since it has a period of a whole octave and splits the perfect fifth in two. Semaphore has a period of a whole octave and splits the perfect twelfth in two. It requires one period to add to the fifth to make it a twelfth, and one is alpha. So it is alpha-dicot.

For a more complex example, let us consider sensi and its weak extension bison. Sensi splits 6/1 in seven. It requires two periods to the fifth to reach 6/1, and two is beta. So it is beta-heptacot. Bison splits the period of sensi in two. As a result, it now requires four periods to the fifth to reach 6/1, and four is delta. So it is diploid delta-heptacot.

Below is a list of ploidacots for common temperaments

Notation

TODO: Come up with canonical ups and downs notation systems for pergen squares

List of ploidacots

Acot

Monocot

Dicot

Tricot

Tetracot

Pentacot

Hexacot

Heptacot

Octacot

Enneacot

Decacot

>10 cots

See also

  • Wedgie – a mathematical generalization of the concept of ploidacots that uniquely characterizes a temperament
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