Ploidacot/Monocot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=1|Pergen=[P8, P5]|Forms=5, 7, 12|Title=Monocot|Wedgie=1}}'''Monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth, and the period is a [[2/1]] octave. In other words, it is the standard chain of fifths, repeating every octave. Monocot temperaments usually generate the (mos)diatonic scale ([[5L 2s]]) and one of its chromatic children ([[5L 7s]] or [[7L 5s]]) (though there are exotempered exceptions such as [[mavila]] ([[2L 5s]]) and [[trienstonian]] ([[5L 3s]])) | {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=1|Pergen=[P8, P5]|Forms=5, 7, 12|Title=Monocot|Wedgie=1}}'''Monocot''' is a temperament archetype where the generator is a [[3/2]] perfect fifth, and the period is a [[2/1]] octave. In other words, it is the standard chain of fifths, repeating every octave. Monocot temperaments usually generate the (mos)diatonic scale ([[5L 2s]]) and one of its chromatic children ([[5L 7s]] or [[7L 5s]]) (though there are exotempered exceptions such as those of [[mavila]] ([[2L 5s]]) and [[trienstonian]] ([[5L 3s]])). They are very commonly used and are a well-explored category of temperaments, some of them dating to centuries ago (like meantone). | ||
== Intervals and notation == | == Intervals and notation == | ||
| Line 89: | Line 89: | ||
== Temperament interpretations == | == Temperament interpretations == | ||
Monocot temperaments | Monocot temperaments equate a number of stacked fifths to some other just interval-class of the 5-limit or greater. | ||
=== Low accuracy === | |||
=== Mavila === | === Mavila === | ||
[[Mavila]] is an exotemperament that flattens the fifth beyond the diatonic range (to around 670–680{{c}}) so that the "major third" | [[Mavila]] is an exotemperament that flattens the fifth beyond the diatonic range (to around 670–680{{c}}) so that the "major third" maps flatter than the "minor third", meaning that the "minor third" can be mapped to 5/4 and the "major third" can be mapped to 6/5. This has the effect of "swapping" major and minor intervals from what they are in meantone. This means that the normal chain-of-fifths interval names stop making much sense (but see in [[Mavila#Notation]]). | ||
=== Deeptone === | === Deeptone === | ||
[[Deeptone]] flattens the fifth to around 690{{c}}, so the major third | [[Deeptone]] flattens the fifth to around 690{{c}}, so the major third maps to [[16/13]] and the augmented third to [[5/4]], with apotomes mapped so small that they can work as sixth-tones. | ||
=== Trienstonian === | |||
[[Trienstonian]] is an exotemperament where the fifth is tuned so sharp that the major sixth maps to [[7/4]]. This is thus tuned best with tunings around [[5edo]] or sharper, which generate oneirotonic (5L 3s), not diatonic scales. | |||
=== Medium accuracy === | |||
==== Meantone ==== | |||
[[Meantone]] can be seen as the most "basic" 5-limit monocot temperament, which flattens the fifth slightly to around 696{{c}} so that the major third maps to [[5/4]]. Consequently, the minor third maps to [[6/5]], and the major sixth maps to [[5/3]]. | |||
In the 7-limit, the augmented sixth becomes 7/4 (often called ''subminor seventh''), resulting in [[septimal meantone]]. If the fifth can be flattened further (to around 693{{c}}), the diminished seventh maps sharper than the augmented sixth. Then, it makes sense to map the diminished seventh as 7/4, resulting in [[flattone]] temperament. If it is tuned sharper, close to 700c, 7/4 can be mapped to the minor seventh resulting in [[dominant]]. | |||
==== Superpyth ==== | |||
[[Superpyth]] maps the minor seventh to [[7/4]], which suggests a tuning of the perfect fifth at around 710{{c}}. [[5/4]] can then be mapped to the augmented second (for standard superpyth) or the double-augmented unison (for [[ultrapyth]], which also extends to prime 13 by mapping the major third to [[13/10]]). | |||
=== | === High accuracy === | ||
==== Schismic ==== | |||
[[Schismic]] (in the 5-limit called ''helmholtz'') has [[5/4]] be mapped to the submajor third (diminished fourth, an interval of about 384{{c}} when justly tuned), where the [[pythagorean comma]] [super/sub-] now can act as a [[syntonic comma]] accidental. This makes it an incredibly accurate [[microtemperament]], and it naturally extends to prime 19 by finding 19/16 at the minor third (called [[nestoria]]). | |||
In the 7-limit, [[7/4]] can be simply mapped to a subminor seventh (doubly diminished octave, an interval of about 973{{c}} when justly tuned) with lower accuracy compared to helmholtz; this extension is called [[garibaldi]]. In the >11-limit, 11/8 can be found at a hyperfourth (3x-augmented second), and 13/8 at a hyperminor sixth (3x-augmented fourth); this extension is called [[cassandra]]. | |||
[[ | |||
[[Pontiac]] instead maps it to a more complex triply sub major sixth (pentuply augmented third), but with practically no loss of accuracy compared to schismic. Interpretations to other primes can be found very deep in the monocot chain, as in [[ponta]]. | |||
=== | ==== Gary ==== | ||
[[ | [[Gary]] can be seen as the no-5 restriction of cassandra, keeping the maps of 7/4 and 11/8, so the pythagorean comma is equal to the septimal comma. Unlike schismic, it achieves far greater accuracy compared to cassandra, qualifying as a microtemperament. Additionally, Its major third also maps with incredible accuracy to [[19/15]]. | ||
=== | ==== Cotoneum ==== | ||
[[ | [[Cotoneum]] finds a way to extend gary into the whole >11-limit within monocot with only a slight loss accuracy, though with very high complexity. It finds very complex approximations of 5/4 and 13/8 as -49 and +61 fifths, where the [[41-comma]] now can act as an [[aberschisma]] accidental. | ||
Latest revision as of 21:57, 22 July 2026
| Pergen | [P8, P5] |
| Numeral form | 1-cot |
| Pure generator size | 498.04 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 5, 7, 12 |
| Characteristic multival entry | 1 |
Monocot is a temperament archetype where the generator is a 3/2 perfect fifth, and the period is a 2/1 octave. In other words, it is the standard chain of fifths, repeating every octave. Monocot temperaments usually generate the (mos)diatonic scale (5L 2s) and one of its chromatic children (5L 7s or 7L 5s) (though there are exotempered exceptions such as those of mavila (2L 5s) and trienstonian (5L 3s)). They are very commonly used and are a well-explored category of temperaments, some of them dating to centuries ago (like meantone).
Intervals and notation
Monocot is the only ploidacot to have an agreed-upon, fully unambiguous scheme for interval and note names, which can be found at chain-of-fifths notation.
| # | Steps | Notation | Name |
|---|---|---|---|
| −7 | 113.69 | C# | Augmented unison |
| −6 | 611.73 | F# | Augmented fourth |
| −5 | 1,109.78 | B | Major seventh |
| −4 | 407.82 | E | Major third |
| −3 | 905.87 | A | Major sixth |
| −2 | 203.91 | D | Major second |
| −1 | 701.96 | G | Perfect fifth |
| 0 | 0 | C | Perfect unison |
| 1 | 498.04 | F | Perfect fourth |
| 2 | 996.09 | Bb | Minor seventh |
| 3 | 294.13 | Eb | Minor third |
| 4 | 792.18 | Ab | Minor sixth |
| 5 | 90.22 | Db | Minor second |
| 6 | 588.27 | Gb | Diminished fifth |
| 7 | 1,086.31 | Cb | Diminished octave |
Temperament interpretations
Monocot temperaments equate a number of stacked fifths to some other just interval-class of the 5-limit or greater.
Low accuracy
Mavila
Mavila is an exotemperament that flattens the fifth beyond the diatonic range (to around 670–680 ¢) so that the "major third" maps flatter than the "minor third", meaning that the "minor third" can be mapped to 5/4 and the "major third" can be mapped to 6/5. This has the effect of "swapping" major and minor intervals from what they are in meantone. This means that the normal chain-of-fifths interval names stop making much sense (but see in Mavila#Notation).
Deeptone
Deeptone flattens the fifth to around 690 ¢, so the major third maps to 16/13 and the augmented third to 5/4, with apotomes mapped so small that they can work as sixth-tones.
Trienstonian
Trienstonian is an exotemperament where the fifth is tuned so sharp that the major sixth maps to 7/4. This is thus tuned best with tunings around 5edo or sharper, which generate oneirotonic (5L 3s), not diatonic scales.
Medium accuracy
Meantone
Meantone can be seen as the most "basic" 5-limit monocot temperament, which flattens the fifth slightly to around 696 ¢ so that the major third maps to 5/4. Consequently, the minor third maps to 6/5, and the major sixth maps to 5/3.
In the 7-limit, the augmented sixth becomes 7/4 (often called subminor seventh), resulting in septimal meantone. If the fifth can be flattened further (to around 693 ¢), the diminished seventh maps sharper than the augmented sixth. Then, it makes sense to map the diminished seventh as 7/4, resulting in flattone temperament. If it is tuned sharper, close to 700c, 7/4 can be mapped to the minor seventh resulting in dominant.
Superpyth
Superpyth maps the minor seventh to 7/4, which suggests a tuning of the perfect fifth at around 710 ¢. 5/4 can then be mapped to the augmented second (for standard superpyth) or the double-augmented unison (for ultrapyth, which also extends to prime 13 by mapping the major third to 13/10).
High accuracy
Schismic
Schismic (in the 5-limit called helmholtz) has 5/4 be mapped to the submajor third (diminished fourth, an interval of about 384 ¢ when justly tuned), where the pythagorean comma [super/sub-] now can act as a syntonic comma accidental. This makes it an incredibly accurate microtemperament, and it naturally extends to prime 19 by finding 19/16 at the minor third (called nestoria).
In the 7-limit, 7/4 can be simply mapped to a subminor seventh (doubly diminished octave, an interval of about 973 ¢ when justly tuned) with lower accuracy compared to helmholtz; this extension is called garibaldi. In the >11-limit, 11/8 can be found at a hyperfourth (3x-augmented second), and 13/8 at a hyperminor sixth (3x-augmented fourth); this extension is called cassandra.
Pontiac instead maps it to a more complex triply sub major sixth (pentuply augmented third), but with practically no loss of accuracy compared to schismic. Interpretations to other primes can be found very deep in the monocot chain, as in ponta.
Gary
Gary can be seen as the no-5 restriction of cassandra, keeping the maps of 7/4 and 11/8, so the pythagorean comma is equal to the septimal comma. Unlike schismic, it achieves far greater accuracy compared to cassandra, qualifying as a microtemperament. Additionally, Its major third also maps with incredible accuracy to 19/15.
Cotoneum
Cotoneum finds a way to extend gary into the whole >11-limit within monocot with only a slight loss accuracy, though with very high complexity. It finds very complex approximations of 5/4 and 13/8 as -49 and +61 fifths, where the 41-comma now can act as an aberschisma accidental.