Ploidacot
The ploidacot system is a classification of rank-2 temperaments based on how a temperament can be thought of as a union of copies of Pythagorean tuning. It is similar to the pergen, and is a canonical naming scheme for pergens of rank-2 temperaments of the 2.3.… subgroup 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:
| Number n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Prefix | n | alpha | beta | gamma | delta | epsilon | digamma | 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-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. 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.
Non-octave temperaments
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Todo: expand |
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
- Meantone and helmholtz are haploid monocot
- Mohajira and dicot are dicot
- Bug and semaphore are alpha-dicot
- Shrutar is diploid alpha-dicot
- Ennealimmal is enneaploid dicot
- Hemiennealimmal is octodecaploid (18-ploid) dicot
- Slendric, mothra, and rodan are tricot
- Tricot is alpha-tricot
- Porcupine is beta-tricot
- Hedgehog is diploid alpha-tricot
- Tetracot is tetracot
- Squares is beta-tetracot
- Bleu is pentacot
- Magic is alpha-pentacot
- Amity is gamma-pentacot
- Miracle is hexacot
- Hanson is alpha-hexacot
- Harry is diploid delta-hexacot
- Orwell is alpha-heptacot
- Sensi is beta-heptacot
- Vishnu is diploid epsilon-heptacot
- Octacot is octacot
- Würschmidt is beta-octacot
- Valentine is enneacot
- Sycamore is hendecacot
- Chromo is tridecacot
- Pajara and injera are diploid
- Antitonic is diploid acot
- Augene is triploid
- Diminished is tetraploid
- Blackwood is pentaploid acot
- Whitewood is heptaploid acot
- Ennealimmal is enneaploid
- Compton is dodecaploid acot
Notation
- TODO: Come up with canonical ups and downs notation systems for pergen squares