Kite's thoughts on pergens: Difference between revisions

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Overwhelmed? See '''[https://Tallkite.com/misc files/notation guide for rank-2 pergens.pdf TallKite.com/misc_files/notation guide for rank-2 pergens.pdf]''' for practical notation examples.

''See also: [[Rank-2 temperaments by mapping of 3]]''

__FORCETOC__

==Definition==

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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 aka Sagugu and Injera aka Gu & Biruyo 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'''.

The largest category contains all single-comma rank-2 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'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

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==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for ~~P =~~ 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them.

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyo, where ^1 equals 64/63 minus 81/80.

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This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate enharmonics for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''tallkite.com/misc_files/notation guide for rank-2 pergens.pdf''']

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>tallkite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"

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square mapping

lifts and drops (the / and \ symbols)

enharmonic, EI

enharmonic interval, EI

uninflected

genchain

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Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[PraveenVenkataramana|Praveen Venkataramana]].

== ~~Addenda~~ ==

== Addendum (late 2023) ==

WORK IN PROGRESS

=== The EI(s) define the pergen ===

The pergen can be derived directly from the EI(s). Thus the EI(s) define both the pergen and the notation. An EI can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EI is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

A double-pair example uses the 2.3.^./ subgroup...

One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike conventional 2.3.^ monzos, the first two numbers are usually fairly small, and the second number is never negative. This facilitates one's search.

=== Arrow commas ===

The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EI is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EI of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

=== The three commas ===

Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EI. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EI turns out to be.

The EI always equals the VC plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. (Sometimes the EI has a negative stepspan, or is a diminished unison, and needs to be inverted.)

In our Triyo example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EI always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EI. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so 135/128 or the schisma are possibilities. But either of these would lead one further fourthward in the lattice to a very remote EI (^3AA1 and v3d44 respectively), making a very awkward notation.

(TO DO: answer the question, why is it ^3AA1 and not v3AA1?)

Next let's specify the AC, experiment with the VC and see what the EI turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EI (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EI for (P8/2, P5).

More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0], and adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Gugu/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2.

Again, this is the recommended EI.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.

(TO DO: apply to multi-comma temperaments)

[[Category:Regular temperament theory]]

[[Category:Notation]]

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+Overwhelmed? See '''[https://Tallkite.com/misc&#x20;files/notation&#x20;guide&#x20;for&#x20;rank-2&#x20;pergens.pdf TallKite.com/misc_files/notation guide for rank-2 pergens.pdf]''' for practical notation examples.
 ''See also: [[Rank-2 temperaments by mapping of 3]]''
 __FORCETOC__
 ==Definition==
@@ Line 11: / Line 14: @@
 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 aka Sagugu and Injera aka Gu & Biruyo 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'''.
+The largest category contains all single-comma rank-2 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'''.
 Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.
@@ Line 1,138: / Line 1,141: @@
 ==Ratio and cents of the accidentals==
-The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for P = 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them.
+The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.
 If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyo, where ^1 equals 64/63 minus 81/80.
@@ Line 3,994: / Line 3,997: @@
 This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate enharmonics for many pergens.
-[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''tallkite.com/misc_files/notation guide for rank-2 pergens.pdf''']
+[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>tallkite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']
 {| class="wikitable" style="text-align:center;"
@@ Line 4,353: / Line 4,356: @@
 square mapping<br />
 lifts and drops (the / and \ symbols)<br />
-enharmonic, EI<br />
+enharmonic interval, EI<br />
 uninflected<br />
 genchain<br />
@@ Line 4,415: / Line 4,418: @@
 Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[PraveenVenkataramana|Praveen Venkataramana]].
-== Addenda ==
+== Addendum (late 2023) ==
+WORK IN PROGRESS
+=== The EI(s) define the pergen ===
+The pergen can be derived directly from the EI(s). Thus the EI(s) define both the pergen and the notation. An EI can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.
+For example, if the EI is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).
+A double-pair example uses the 2.3.^./ subgroup...
+One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v<sup>3</sup>m2 = [1 1 -3], etc. Unlike conventional 2.3.^ monzos, the first two numbers are usually fairly small, and the second number is never negative. This facilitates one's search.
+=== Arrow commas ===
+The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.
+Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EI is v<sup>3</sup>A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EI of v<sup>3</sup>A1, the vAC is v(81/80) or [-4 4 -1 -1].
+=== The three commas ===
+Thus when we consider a single-comma temperament along with its notation, there are <u>three</u> commas of interest, the VC, the vAC and the EI. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.
+Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EI turns out to be.
+The EI always equals the VC plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. (Sometimes the EI has a negative stepspan, or is a diminished unison, and needs to be inverted.)
+In our Triyo example, 250/243 plus 3 downed syntonic commas = v<sup>3</sup>A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EI always has a zero prime-5-count.
+Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EI. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so 135/128 or the schisma are possibilities. But either of these would lead one further fourthward in the lattice to a very remote EI (^<sup>3</sup>AA1 and v<sup>3</sup>d<sup>4</sup>4 respectively), making a very awkward notation.
+(TO DO: answer the question, why is it ^<sup>3</sup>AA1 and not v<sup>3</sup>AA1?)
+Next let's specify the AC, experiment with the VC and see what the EI turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EI (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EI for (P8/2, P5).
+More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0], and adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^<sup>5</sup>dd2. Gugu/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2.
+Again, this is the recommended EI.
+Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.
+(TO DO: apply to multi-comma temperaments)
 [[Category:Regular temperament theory]]
 [[Category:Notation]]