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]])) ~~and as such they~~ are very commonly used and are a well-explored category of temperaments.

{{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 ==

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]].

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]].

{| class="wikitable"

Line 50:

| '''0'''

| '''C'''

| Perfect unison ~~/ Perfect octave~~

| Perfect unison

|-

| 1

| 498.05

| 498.04

| F

| Perfect fourth

Line 63:

|-

| 3

| 294.14

| 294.13

| Eb

| Minor third

Line 83:

|-

| 7

| 1,086.32

| 1,086.31

| Cb

| Diminished octave

Line 89:

== Temperament interpretations ==

Monocot temperaments ~~involve equating~~ a number of stacked fifths to some other just interval of the 5-limit or greater.

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{{c}}) so that the "major third" is flatter than the "minor third", meaning that the "minor third" can be ~~assigned~~ to 5/4 and the "major third" can be ~~assigned~~ 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]]).

[[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]] flattens the fifth to around 690{{c}}, so the major third ~~is a~~ [[16/13]] and the augmented third is [[5/4]].

[[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.

=== ~~Meantone~~ ===

=== Medium accuracy ===

[[Meantone]] can be seen as the most "basic" monocot temperament, which flattens the fifth slightly to around 696{{c}} so that the major third is [[5/4]]. Consequently, the minor third becomes [[6/5]], and the major sixth becomes [[5/3]].

~~In the 7-limit, the augmented sixth becomes 7/4, resulting in~~ [[~~septimal meantone~~]]~~. Alternately~~, the fifth ~~can be flattened further (~~to around ~~693~~{{c}}) so that the ~~diminished seventh is sharper than the augmented sixth~~. ~~Then~~, ~~it makes sense~~ to ~~interpret the diminished seventh as 7~~/4, ~~resulting in~~ [[~~flattone~~]] ~~temperament~~.

==== 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]].

~~=== Schismic ===~~

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]].

[[~~Schismic~~]] ~~interprets~~ the ~~diminished fourth, an interval of about 384~~{{c}} ~~when justly tuned~~, as [[~~5/4~~]]. ~~This makes~~ it ~~a very accurate microtemperament~~, ~~and it naturally extends~~ to ~~prime 19 by finding 19~~/~~16 at~~ the minor ~~third (called~~ [[~~nestoria~~]]).

In the 7~~-limit~~, the ~~doubly-diminished octave, an interval of about 973~~{{c}} ~~when justly tuned, can be interpreted as~~ [[7/4]] ~~with a small loss of accuracy compared~~ to 5-~~limit schismic; this is called~~ [[~~garibaldi~~]].

==== 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]]).

=== ~~Superpyth~~ ===

=== High accuracy ===

[[Superpyth]] interprets the minor seventh as [[7/4]], which suggests a tuning of the perfect fifth at around 710{{c}}. [[5/4]] can then be seen as the augmented second (for standard superpyth) or the double-augmented unison (for [[ultrapyth]], which also extends to prime 13 by setting the major third equal to [[13/10]]).

=== ~~Trienstonian~~ ===

==== Schismic ====

[[~~Trienstonian~~]] is an ~~exotemperament~~ where the ~~fifth is sharpened so that~~ the ~~major sixth is~~ [[7/4]]. ~~This~~ is ~~thus tuned best~~ with ~~tunings around or sharp~~ of [[~~5edo~~]], ~~which do not generate~~ a ~~(usable) diatonic scale~~.

[[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]].

[[~~Category:Ploidacot~~]]

==== 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.

@@ Line 1: / Line 1: @@
-{{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&nbsp;2s]]) and one of its chromatic children ([[5L&nbsp;7s]] or [[7L&nbsp;5s]]) (though there are exotempered exceptions such as [[mavila]] ([[2L&nbsp;5s]]) and [[trienstonian]] ([[5L&nbsp;3s]])) and as such they are very commonly used and are a well-explored category of temperaments.
+{{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&nbsp;2s]]) and one of its chromatic children ([[5L&nbsp;7s]] or [[7L&nbsp;5s]]) (though there are exotempered exceptions such as those of [[mavila]] ([[2L&nbsp;5s]]) and [[trienstonian]] ([[5L&nbsp;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]].
+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]].
 {| class="wikitable"
@@ Line 50: / Line 50: @@
 | '''0'''
 | '''C'''
-| Perfect unison / Perfect octave
+| Perfect unison
 |-
 | 1
-| 498.05
+| 498.04
 | F
 | Perfect fourth
@@ Line 63: / Line 63: @@
 |-
 | 3
-| 294.14
+| 294.13
 | Eb
 | Minor third
@@ Line 83: / Line 83: @@
 |-
 | 7
-| 1,086.32
+| 1,086.31
 | Cb
 | Diminished octave
@@ Line 89: / Line 89: @@
 == Temperament interpretations ==
-Monocot temperaments involve equating a number of stacked fifths to some other just interval of the 5-limit or greater.
+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{{c}}) so that the "major third" is flatter than the "minor third", meaning that the "minor third" can be assigned to 5/4 and the "major third" can be assigned 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]]).
+[[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]] flattens the fifth to around 690{{c}}, so the major third is a [[16/13]] and the augmented third is [[5/4]].
+[[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.
-=== Meantone ===
+=== Medium accuracy ===
-[[Meantone]] can be seen as the most "basic" monocot temperament, which flattens the fifth slightly to around 696{{c}} so that the major third is [[5/4]]. Consequently, the minor third becomes [[6/5]], and the major sixth becomes [[5/3]].
-In the 7-limit, the augmented sixth becomes 7/4, resulting in [[septimal meantone]]. Alternately, the fifth can be flattened further (to around 693{{c}}) so that the diminished seventh is sharper than the augmented sixth. Then, it makes sense to interpret the diminished seventh as 7/4, resulting in [[flattone]] temperament.
+==== 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]].
-=== Schismic ===
+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]].
-[[Schismic]] interprets the diminished fourth, an interval of about 384{{c}} when justly tuned, as [[5/4]]. This makes it a very accurate microtemperament, and it naturally extends to prime 19 by finding 19/16 at the minor third (called [[nestoria]]).
-In the 7-limit, the doubly-diminished octave, an interval of about 973{{c}} when justly tuned, can be interpreted as [[7/4]] with a small loss of accuracy compared to 5-limit schismic; this is called [[garibaldi]].
+==== 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]]).
-=== Superpyth ===
+=== High accuracy ===
-[[Superpyth]] interprets the minor seventh as [[7/4]], which suggests a tuning of the perfect fifth at around 710{{c}}. [[5/4]] can then be seen as the augmented second (for standard superpyth) or the double-augmented unison (for [[ultrapyth]], which also extends to prime 13 by setting the major third equal to [[13/10]]).
-=== Trienstonian ===
+==== Schismic ====
-[[Trienstonian]] is an exotemperament where the fifth is sharpened so that the major sixth is [[7/4]]. This is thus tuned best with tunings around or sharp of [[5edo]], which do not generate a (usable) diatonic scale.
+[[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]].
-[[Category:Ploidacot]]
+==== 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.