Kite's thoughts on pergens: Difference between revisions

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==~~'''~~Definition~~'''~~==

=='''Definition'''==

A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

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For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using ~~[[Ups_and_Downs_Notation|~~ups and downs]]. See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]].

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using ups and downs (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and Downs Notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

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'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored.

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating ~~the~~ piece, either the pergen notation or the EDO notation can be used. ~~The former is more general~~, ~~because~~ the ~~piece can be easily played in other supporting EDOs~~. ~~But~~ it ~~may require the musician~~ to mentally double every up and down ~~symbol~~.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has E = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has E = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has E = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has E = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

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If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the ~~simpler~~ the pergen they make. Examples::

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

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A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of ~~those three~~ edos defines (P8, P5/2).

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

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 __FORCETOC__
-==<u>'''Definition'''</u>==
+=='''<u>Definition</u>'''==
 A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.
@@ Line 9: / Line 9: @@
 For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on "semi-fourth", is of course half-fourth.
-Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using [[Ups_and_Downs_Notation|ups and downs]]. See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]].
+Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using ups and downs (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and Downs Notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.
 The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.
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 '''<u>Notating a pergen tuned to an EDO</u>'''
-If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored.
+If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.
-When notating the piece, either the pergen notation or the EDO notation can be used. The former is more general, because the piece can be easily played in other supporting EDOs. But it may require the musician to mentally double every up and down symbol.
+When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.
-Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4.
+Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.
+Half-5th has E = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.
+Half-4th has E = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.
+Third-4th has E = v<sup>3</sup>A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has E = v<sup>3</sup>m2. Doubled EDOs are the same as half-4th's tripled EDOs.
+The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.
 <u>'''Pergens Within An EDO'''</u>
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 If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.
-The closer two edos are in the scale tree, the simpler the pergen they make. Examples::
+The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:
 {| class="wikitable"
@@ Line 3,876: / Line 3,884: @@
 |}
-A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).
+A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).
 ==Array Keyboards (unfinished)==