Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 623940729 - Original comment: ** |
Wikispaces>TallKite **Imported revision 623941279 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-17 07: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-17 07:27:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>623941279</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 15: | Line 15: | ||
For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth. | For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "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 | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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]]. | ||
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 > 3 (a **higher prime**), e.g. 81/80 or 64/63. 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 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 > 3 (a **higher prime**), e.g. 81/80 or 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**. | ||
| Line 52: | Line 52: | ||
A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. | ||
The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is | The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. | ||
=__Derivation__= | =__Derivation__= | ||
| Line 72: | Line 72: | ||
G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz | G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz | ||
<span style="display: block; text-align: center;">**The rank-2 pergen from the [(x, 0), (y, z)] matrix = {P8/x, (nz-y, x)/xz}**</span> | |||
Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix: | Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix: | ||
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 || | ||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 || | ||
| Line 113: | Line 112: | ||
Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. | Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. | ||
Most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. | Most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair. One possibility is **highs and lows**, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation. | ||
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result its temperament doesn't need ups and downs. | Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result its temperament doesn't need ups and downs. | ||
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction, up to quarter-splits. | ||
The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5 | The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma maps to the enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic. | ||
The **genchain** (chain of generators) | The **genchain** (chain of generators) in the table is only a short section of the full genchain. | ||
C - G implies ...Eb Bb F C G D A E B F# C#... | C - G implies ...Eb Bb F C G D A E B F# C#... | ||
C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... | C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... | ||
If the octave is split, the genchain is a **perchain** ("peer-chain") that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one. | If the octave is split, the genchain is a **perchain** ("peer-chain", chain of periods) that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one. | ||
An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead. | The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead. | ||
The table lists | The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination. | ||
(table is under construction) | (table is under construction) | ||
| Line 153: | Line 152: | ||
^1 = 33/32 ||= " || | ^1 = 33/32 ||= " || | ||
||= " ||= ^^d<span style="vertical-align: super;">3</span>2 ||= C^^ = B#<span style="vertical-align: super;">3</span> ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||= ||= " || | ||= " ||= ^^d<span style="vertical-align: super;">3</span>2 ||= C^^ = B#<span style="vertical-align: super;">3</span> ||= P8/2 = vAA4 = ^dd5 ||= C - F##v=Gbb^ - C ||= ||= " || | ||
||= (P8, P4/2) ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | ||= (P8, P4/2) ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | ||
^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*, | ^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*, | ||
| Line 202: | Line 200: | ||
/1 = 33/32 ||= " || | /1 = 33/32 ||= " || | ||
||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/3, P5 | ||= {P8/3, P5) ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||= 12, 15, 18b*, 21, | ||
24*, 27, 30* || | 24*, 27, 30* || | ||
||= {P8, P4/3 | ||= {P8, P4/3) ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = ^^m2 = vM2 ||= C - Dv - Eb^ - F ||= porcupine ||= 13b, 14*, 15, 21*, | ||
22, 28*, 29, 30* || | 22, 28*, 29, 30* || | ||
||= {P8, P5/3 | ||= {P8, P5/3) ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||= 15*, 16, 20*, 21, | ||
25*, 26, 30*, 31 || | 25*, 26, 30*, 31 || | ||
||= {P8, P11/3 | ||= {P8, P11/3) ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= ||= || | ||
||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^P4 = vvP5 ||= C F^ Cv F ||= ||= " || | ||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^P4 = vvP5 ||= C F^ Cv F ||= ||= " || | ||
||= {P8/3, P4/2 | ||= {P8/3, P4/2) ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span>M ``=`` D# ||= P8/3 = ^^m3 | ||
P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | ||
C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= ||= 15, 18b*, 24, 30 || | C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= ||= 15, 18b*, 24, 30 || | ||
||= {P8/3, P5/2 | ||= {P8/3, P5/2) ||= ||= ||= ||= ||= ||= 18b, 24, 30 || | ||
||= {P8/2, P4/3 | ||= {P8/2, P4/3) ||= ||= ||= ||= ||= ||= 14, 22, 28*, 30* || | ||
||= {P8/2, P5/3 | ||= {P8/2, P5/3) ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5 | ||
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C | P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C | ||
C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* || | C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* || | ||
| Line 223: | Line 221: | ||
P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | ||
C - /D - \F - G ||= ||= " || | C - /D - \F - G ||= ||= " || | ||
||= {P8/2, P11/3 | ||= {P8/2, P11/3) ||= ||= ||= ||= ||= ||= || | ||
||= {P8/3, P4/3 | ||= {P8/3, P4/3) ||= ||= ||= ||= ||= ||= 15, 21, 30* || | ||
||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv A^ C ||= diminished, | ||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv A^ C ||= diminished, | ||
| Line 404: | Line 402: | ||
For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.<br /> | For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.<br /> | ||
<br /> | <br /> | ||
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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 <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>.<br /> | ||
<br /> | <br /> | ||
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 <strong>higher prime</strong>), e.g. 81/80 or 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called <strong>unsplit</strong>.<br /> | 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 <strong>higher prime</strong>), e.g. 81/80 or 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called <strong>unsplit</strong>.<br /> | ||
| Line 657: | Line 655: | ||
A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so.<br /> | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so.<br /> | ||
<br /> | <br /> | ||
The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is | The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:23:&lt;h1&gt; --><h1 id="toc1"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:23 --><u>Derivation</u></h1> | <!-- ws:start:WikiTextHeadingRule:23:&lt;h1&gt; --><h1 id="toc1"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:23 --><u>Derivation</u></h1> | ||
| Line 677: | Line 675: | ||
G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br /> | G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br /> | ||
<br /> | <br /> | ||
<span style="display: block; text-align: center;"><strong>The rank-2 pergen from the [(x, 0), (y, z)] matrix = {P8/x, (nz-y, x)/xz}</strong></span><br /> | |||
< | |||
</ | |||
<br /> | |||
Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br /> | Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br /> | ||
| Line 955: | Line 944: | ||
Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.<br /> | Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.<br /> | ||
<br /> | <br /> | ||
Most rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. | Most rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. Certain rank-2 temperaments require another additional pair. One possibility is <strong>highs and lows</strong>, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.<br /> | ||
<br /> | <br /> | ||
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result its temperament doesn't need ups and downs.<br /> | Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result its temperament doesn't need ups and downs.<br /> | ||
<br /> | <br /> | ||
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction, up to quarter-splits.<br /> | ||
<br /> | <br /> | ||
The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5 | The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma maps to the enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.<br /> | ||
<br /> | <br /> | ||
The <strong>genchain</strong> (chain of generators) | The <strong>genchain</strong> (chain of generators) in the table is only a short section of the full genchain.<br /> | ||
C - G implies ...Eb Bb F C G D A E B F# C#...<br /> | C - G implies ...Eb Bb F C G D A E B F# C#...<br /> | ||
C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...<br /> | C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...<br /> | ||
If the octave is split, the genchain is a <strong>perchain</strong> (&quot;peer-chain&quot;) that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.<br /> | If the octave is split, the genchain is a <strong>perchain</strong> (&quot;peer-chain&quot;, chain of periods) that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.<br /> | ||
<br /> | <br /> | ||
An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.<br /> | The table shows compatible edos. An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.<br /> | ||
<br /> | <br /> | ||
The table lists | The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.<br /> | ||
<br /> | <br /> | ||
(table is under construction)<br /> | (table is under construction)<br /> | ||
| Line 1,101: | Line 1,090: | ||
</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,303: | Line 1,276: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P5 | <td style="text-align: center;">{P8/3, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br /> | ||
| Line 1,320: | Line 1,293: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8, P4/3 | <td style="text-align: center;">{P8, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>A1<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">3</span>A1<br /> | ||
| Line 1,337: | Line 1,310: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8, P5/3 | <td style="text-align: center;">{P8, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>m2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">3</span>m2<br /> | ||
| Line 1,354: | Line 1,327: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8, P11/3 | <td style="text-align: center;">{P8, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>dd2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">3</span>dd2<br /> | ||
| Line 1,386: | Line 1,359: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P4/2 | <td style="text-align: center;">{P8/3, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>A2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>A2<br /> | ||
| Line 1,404: | Line 1,377: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P5/2 | <td style="text-align: center;">{P8/3, P5/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,420: | Line 1,393: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/2, P4/3 | <td style="text-align: center;">{P8/2, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,436: | Line 1,409: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/2, P5/3 | <td style="text-align: center;">{P8/2, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2<br /> | ||
| Line 1,474: | Line 1,447: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/2, P11/3 | <td style="text-align: center;">{P8/2, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,490: | Line 1,463: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P4/3 | <td style="text-align: center;">{P8/3, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||