Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625870419 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-04 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-04 02:29:47 UTC</tt>.<br> | ||
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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc. | If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc. | ||
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 | For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its 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, (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 fewer than perhaps 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]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]]. | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps 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]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]]. | ||
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In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of 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 comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...). | In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of 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 comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...). | ||
=__Derivation__= | =__Derivation__= | ||
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||~ 3/1 ||= 0 ||= 1 ||= -1 || || | ||~ 3/1 ||= 0 ||= 1 ||= -1 || || | ||
||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 || | ||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 || | ||
Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen | Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). | ||
The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. | |||
=__Applications__= | =__Applications__= | ||
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For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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. | For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 | 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 this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See [[pergen#Further%20Discussion-Notating%20unsplit%20pergens|Notating unsplit pergens]] below. | ||
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. | ||
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||= d<span style="vertical-align: super;">4</span>3 ||= C - Eb<span style="vertical-align: super;">5</span> ||= -31 ||= 31-edo ||= ~697¢ ||= 697-720¢ ||= 600-697¢ ||= upped || | ||= d<span style="vertical-align: super;">4</span>3 ||= C - Eb<span style="vertical-align: super;">5</span> ||= -31 ||= 31-edo ||= ~697¢ ||= 697-720¢ ||= 600-697¢ ||= upped || | ||
||= etc. ||= ||= ||= ||= ||= ||= ||= || | ||= etc. ||= ||= ||= ||= ||= ||= ||= || | ||
=__Further Discussion__= | =__Further Discussion__= | ||
==Extremely large multigens== | ==Extremely large multigens== | ||
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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different. | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different. | ||
==Finding a notation for a pergen== | ==Finding a notation for a pergen== | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | ||
==Chord names and scale names== | ==Chord names and scale names== | ||
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Equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, pajara's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C7(^d5,v3). It might be voiced C Gb^ Bb Ev. The Gb^ - Ev interval is an awkward vvA6, even though it's the same interval as C - Bb. Spelling Gb^ as F#v makes a more natural m7 interval. But that would change the Gb^ - Bb = vM3 interval to F#v - Bb, an awkward ^d4. To indicate all this, the chord could be named C7([^d5=vA4]v3). Likewise, injera's dom7b5 chord can be named C,v7(vd5=^A4). | Equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, pajara's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C7(^d5,v3). It might be voiced C Gb^ Bb Ev. The Gb^ - Ev interval is an awkward vvA6, even though it's the same interval as C - Bb. Spelling Gb^ as F#v makes a more natural m7 interval. But that would change the Gb^ - Bb = vM3 interval to F#v - Bb, an awkward ^d4. To indicate all this, the chord could be named C7([^d5=vA4]v3). Likewise, injera's dom7b5 chord can be named C,v7(vd5=^A4). | ||
==Tipping points and sweet spots== | ==Tipping points and sweet spots== | ||
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An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80. | An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80. | ||
==Notating unsplit pergens== | ==Notating unsplit pergens== | ||
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A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 would require ups and downs, if spelling 7/4 as a m7. An unsplit temperament with two commas may require double-pair notation to avoid spelling the 4:5:6:7 chord something like C Fb G A#. | A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 would require ups and downs, if spelling 7/4 as a m7. An unsplit temperament with two commas may require double-pair notation to avoid spelling the 4:5:6:7 chord something like C Fb G A#. | ||
==Notating rank-3 pergens== | ==Notating rank-3 pergens== | ||
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(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is | (2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is | ||
==Notating Blackwood-like pergens== | ==Notating Blackwood-like pergens== | ||
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12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E) | 12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E) | ||
17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \ | 17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \ | ||
==Pergens and MOS scales== | ==Pergens and MOS scales== | ||
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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | ||
==Pergens and EDOs== | ==Pergens and EDOs== | ||
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||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||
==Supplemental materials== | ==Supplemental materials== | ||
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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens. | The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens. | ||
==Various proofs (unfinished)== | ==Various proofs (unfinished)== | ||
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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister. | Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister. | ||
==Miscellaneous Notes== | ==Miscellaneous Notes== | ||
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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multi-gen</strong>. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multi-gen</strong>. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | ||
<br /> | <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 | For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its 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, (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 fewer than perhaps 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>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a>.<br /> | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps 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>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a>.<br /> | ||
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<br /> | <br /> | ||
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of 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 comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of 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 comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | ||
<br /> | |||
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</table> | </table> | ||
Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen | Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). <br /> | ||
<br /> | |||
The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:61:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:61 --><u>Applications</u></h1> | <!-- ws:start:WikiTextHeadingRule:61:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:61 --><u>Applications</u></h1> | ||
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For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other 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, <strong>highs and lows</strong>, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 /> | For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other 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, <strong>highs and lows</strong>, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 | 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 this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20unsplit%20pergens">Notating unsplit pergens</a> below.<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.<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.<br /> | ||
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</table> | </table> | ||
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<!-- ws:start:WikiTextHeadingRule:65:&lt;h1&gt; --><h1 id="toc5"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:65 --><u>Further Discussion</u></h1> | <!-- ws:start:WikiTextHeadingRule:65:&lt;h1&gt; --><h1 id="toc5"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:65 --><u>Further Discussion</u></h1> | ||
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<!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Extremely large multigens</h2> | <!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Extremely large multigens</h2> | ||
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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br /> | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:75 -->Finding a notation for a pergen</h2> | <!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:75 -->Finding a notation for a pergen</h2> | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:79:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:79 -->Chord names and scale names</h2> | <!-- ws:start:WikiTextHeadingRule:79:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:79 -->Chord names and scale names</h2> | ||
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Equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, pajara's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C7(^d5,v3). It might be voiced C Gb^ Bb Ev. The Gb^ - Ev interval is an awkward vvA6, even though it's the same interval as C - Bb. Spelling Gb^ as F#v makes a more natural m7 interval. But that would change the Gb^ - Bb = vM3 interval to F#v - Bb, an awkward ^d4. To indicate all this, the chord could be named C7([^d5=vA4]v3). Likewise, injera's dom7b5 chord can be named C,v7(vd5=^A4).<br /> | Equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, pajara's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C7(^d5,v3). It might be voiced C Gb^ Bb Ev. The Gb^ - Ev interval is an awkward vvA6, even though it's the same interval as C - Bb. Spelling Gb^ as F#v makes a more natural m7 interval. But that would change the Gb^ - Bb = vM3 interval to F#v - Bb, an awkward ^d4. To indicate all this, the chord could be named C7([^d5=vA4]v3). Likewise, injera's dom7b5 chord can be named C,v7(vd5=^A4).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:81:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:81 -->Tipping points and sweet spots</h2> | <!-- ws:start:WikiTextHeadingRule:81:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:81 -->Tipping points and sweet spots</h2> | ||
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An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.<br /> | An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.7¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is a m2 raised by 81/80.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:83:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Notating unsplit pergens"></a><!-- ws:end:WikiTextHeadingRule:83 -->Notating unsplit pergens</h2> | <!-- ws:start:WikiTextHeadingRule:83:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Notating unsplit pergens"></a><!-- ws:end:WikiTextHeadingRule:83 -->Notating unsplit pergens</h2> | ||
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A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 would require ups and downs, if spelling 7/4 as a m7. An unsplit temperament with two commas may require double-pair notation to avoid spelling the 4:5:6:7 chord something like C Fb G A#.<br /> | A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 would require ups and downs, if spelling 7/4 as a m7. An unsplit temperament with two commas may require double-pair notation to avoid spelling the 4:5:6:7 chord something like C Fb G A#.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:85:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Notating rank-3 pergens"></a><!-- ws:end:WikiTextHeadingRule:85 -->Notating rank-3 pergens</h2> | <!-- ws:start:WikiTextHeadingRule:85:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Notating rank-3 pergens"></a><!-- ws:end:WikiTextHeadingRule:85 -->Notating rank-3 pergens</h2> | ||
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(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is<br /> | (2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Notating Blackwood-like pergens"></a><!-- ws:end:WikiTextHeadingRule:87 -->Notating Blackwood-like pergens</h2> | <!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Notating Blackwood-like pergens"></a><!-- ws:end:WikiTextHeadingRule:87 -->Notating Blackwood-like pergens</h2> | ||
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12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)<br /> | 12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)<br /> | ||
17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \<br /> | 17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \<br /> | ||
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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br /> | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:91 -->Pergens and EDOs</h2> | <!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:91 -->Pergens and EDOs</h2> | ||
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<!-- ws:start:WikiTextHeadingRule:93:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:93 -->Supplemental materials</h2> | <!-- ws:start:WikiTextHeadingRule:93:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:93 -->Supplemental materials</h2> | ||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br /> | The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:95:&lt;h2&gt; --><h2 id="toc20"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:95 -->Various proofs (unfinished)</h2> | <!-- ws:start:WikiTextHeadingRule:95:&lt;h2&gt; --><h2 id="toc20"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:95 -->Various proofs (unfinished)</h2> | ||
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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br /> | Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:97 -->Miscellaneous Notes</h2> | <!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:97 -->Miscellaneous Notes</h2> | ||