Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624811623 - Original comment: ** |
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<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>2018-01- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-13 00:57:51 UTC</tt>.<br> | ||
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One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2. | One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2. | ||
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a | Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly. | ||
Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further%20Discussion-Chord%20names%20and%20scale%20names|Chord names and scale names]] below. | Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further%20Discussion-Chord%20names%20and%20scale%20names|Chord names and scale names]] below. | ||
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third-5th ||= 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 || | third-5th ||= 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 || | ||
||= (P8, P11/3) | ||= (P8, P11/3) | ||
third-11th ||= ^<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 ||= | third-11th ||= ^<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 ||= small triple amber || | ||
||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= | ||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= " || | ||
||= (P8/3, P4/2) | ||= (P8/3, P4/2) | ||
third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 | third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4 | ||
P4/2 = v<span style="vertical-align: super;">3</span> | P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3 ||= C - Eb^^ - Avv - C | ||
C - | C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F ||= || | ||
||= " ||= ^<span style="vertical-align: super;">3</span>d2, | |||
\\m2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B# | |||
C``//`` = Db ||= P8/3 = vM3 = ^^d4 | |||
P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C | |||
C - D/=Eb\ - F ||= || | |||
||= (P8/3, P5/2) | ||= (P8/3, P5/2) | ||
third-8ve, half-5th ||= | third-8ve, half-5th ||= ^<span style="vertical-align: super;">3</span>d2 | ||
\\A1 ||= C^<span style="vertical-align: super;">3</span> ``=`` B# | |||
C``//`` = C# ||= P8/3 = vM3 = ^^d4 | |||
P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C | |||
C - Eb/=E\ - G ||= || | |||
||= (P8/2, P4/3) | ||= (P8/2, P4/3) | ||
half-8ve, third-4th ||= | half-8ve, third-4th ||= ^^d2 | ||
\<span style="vertical-align: super;">3</span>A1 ||= C^^ ``=`` Dbb | |||
C/<span style="vertical-align: super;">3</span> ``=`` C# ||= P8/2 = vA4 = ^d5 | |||
P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C | |||
C - D\ - Eb/ - F ||= || | |||
||= (P8/2, P5/3) | ||= (P8/2, P5/3) | ||
half-8ve, | half-8ve, | ||
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||= (P8/2, P11/3) | ||= (P8/2, P11/3) | ||
half-8ve, | half-8ve, | ||
third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= | third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5 | ||
P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5 ||= C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C | |||
C - F^^ - Cvv - F ||= || | |||
||= (P8/3, P4/3) | ||= (P8/3, P4/3) | ||
third- | third- | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | </span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | ||
Approaching pergens in a | Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd. | ||
Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | ||
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The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div> | The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:34 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:68:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:36 --><u><strong>Definition</strong></u></h1> | ||
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<br /> | <br /> | ||
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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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | 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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:38 --><u>Derivation</u></h1> | ||
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For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br /> | For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br /> | ||
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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, the multigen2. 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 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). 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 &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 /> | 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, the multigen2. 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 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). 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 &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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:40 --><u>Applications</u></h1> | ||
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One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.<br /> | One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.<br /> | ||
<br /> | <br /> | ||
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a | Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.<br /> | ||
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Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See <a class="wiki_link" href="/pergen#Further%20Discussion-Chord%20names%20and%20scale%20names">Chord names and scale names</a> below.<br /> | Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See <a class="wiki_link" href="/pergen#Further%20Discussion-Chord%20names%20and%20scale%20names">Chord names and scale names</a> below.<br /> | ||
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<td style="text-align: center;">C - F#v - Cb^ - F<br /> | <td style="text-align: center;">C - F#v - Cb^ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">small triple amber<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,471: | Line 1,485: | ||
<td style="text-align: center;">C - F^ - Cv - F<br /> | <td style="text-align: center;">C - F^ - Cv - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,482: | Line 1,496: | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:013:``=`` -->=<!-- ws:end:WikiTextRawRule:013 --> D#<br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:013:``=`` -->=<!-- ws:end:WikiTextRawRule:013 --> D#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/3 = ^^m3<br /> | <td style="text-align: center;">P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4<br /> | ||
P4/2 = v<span style="vertical-align: super;">3</span> | P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C - Eb^^ - Avv - C<br /> | <td style="text-align: center;">C - Eb^^ - Avv - C<br /> | ||
C - | C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,492: | Line 1,506: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2,<br /> | ||
\\m2<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;"> | <td style="text-align: center;">C^<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:014:``=`` -->=<!-- ws:end:WikiTextRawRule:014 --> B#<br /> | ||
C<!-- ws:start:WikiTextRawRule:015:``//`` -->//<!-- ws:end:WikiTextRawRule:015 --> = Db<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">P8/3 = vM3 = ^^d4<br /> | ||
P4/2 = /M2 = \m3<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C - Ev - Ab^ - C<br /> | ||
C - D/=Eb\ - F<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,507: | Line 1,524: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/ | <td style="text-align: center;">(P8/3, P5/2)<br /> | ||
third-8ve, half-5th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br /> | ||
\\A1<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;"> | <td style="text-align: center;">C^<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> B#<br /> | ||
C<!-- ws:start:WikiTextRawRule:017:``//`` -->//<!-- ws:end:WikiTextRawRule:017 --> = C#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">P8/3 = vM3 = ^^d4<br /> | ||
P5/2 = /m3 = \M3<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C - Ev - Ab/ - C<br /> | ||
C - Eb/=E\ - G<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,522: | Line 1,543: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">& | <td style="text-align: center;">(P8/2, P4/3)<br /> | ||
half-8ve, third-4th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">^^d2<br /> | ||
\<span style="vertical-align: super;">3</span>A1<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;"> | <td style="text-align: center;">C^^ <!-- ws:start:WikiTextRawRule:018:``=`` -->=<!-- ws:end:WikiTextRawRule:018 --> Dbb<br /> | ||
C/<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> C#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
P4/3 = \M2 = <!-- ws:start:WikiTextRawRule:020:``//`` -->//<!-- ws:end:WikiTextRawRule:020 -->m2<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C - F#v=Gb^ - C<br /> | ||
C - D\ - Eb/ - F<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,542: | Line 1,568: | ||
<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 /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:021:``=`` -->=<!-- ws:end:WikiTextRawRule:021 --> B#<span style="vertical-align: super;">3</span><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5<br /> | <td style="text-align: center;">P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5<br /> | ||
| Line 1,560: | Line 1,586: | ||
</td> | </td> | ||
<td style="text-align: center;">C^^ = B#<br /> | <td style="text-align: center;">C^^ = B#<br /> | ||
C<!-- ws:start:WikiTextRawRule: | C<!-- ws:start:WikiTextRawRule:022:``///`` -->///<!-- ws:end:WikiTextRawRule:022 --> = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
| Line 1,578: | Line 1,604: | ||
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>M2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>M2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:023:``=`` -->=<!-- ws:end:WikiTextRawRule:023 --> D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5<br /> | ||
P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C<br /> | ||
C - F^^ - Cvv - F<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,622: | Line 1,650: | ||
<td style="text-align: center;">^<span style="vertical-align: super;">4</span>d2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">4</span>d2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> B#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br /> | <td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br /> | ||
| Line 1,637: | Line 1,665: | ||
<td style="text-align: center;">^<span style="vertical-align: super;">4</span>dd2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">4</span>dd2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:025:``=`` -->=<!-- ws:end:WikiTextRawRule:025 --> B##<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1<br /> | <td style="text-align: center;">P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1<br /> | ||
| Line 1,651: | Line 1,679: | ||
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>A1<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>A1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:026:``=`` -->=<!-- ws:end:WikiTextRawRule:026 --> C#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2<br /> | <td style="text-align: center;">P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2<br /> | ||
| Line 1,665: | Line 1,693: | ||
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>dd3<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>dd3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:027:``=`` -->=<!-- ws:end:WikiTextRawRule:027 --> Eb<span style="vertical-align: super;">3</span><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5<br /> | <td style="text-align: center;">P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5<br /> | ||
| Line 1,679: | Line 1,707: | ||
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>m2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>m2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule: | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:028:``=`` -->=<!-- ws:end:WikiTextRawRule:028 --> Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3<br /> | <td style="text-align: center;">P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3<br /> | ||
| Line 2,106: | Line 2,134: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc4"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:42 --><u>Further Discussion</u></h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc5"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:44 -->Extremely large multigens</h2> | ||
<br /> | <br /> | ||
So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:46 -->Singles and doubles</h2> | ||
<br /> | <br /> | ||
If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.<br /> | If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.<br /> | ||
| Line 2,120: | Line 2,148: | ||
A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:48 -->Finding an example temperament</h2> | ||
<br /> | <br /> | ||
To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br /> | To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br /> | ||
| Line 2,142: | Line 2,170: | ||
There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br /> | There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Ratio and cents of the accidentals"></a><!-- ws:end:WikiTextHeadingRule:50 -->Ratio and cents of the accidentals</h2> | ||
<br /> | <br /> | ||
In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.<br /> | In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.<br /> | ||
| Line 2,148: | Line 2,176: | ||
We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br /> | We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc9"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:52 -->Finding a notation for a pergen</h2> | ||
<br /> | <br /> | ||
There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | ||
| Line 2,175: | Line 2,203: | ||
A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | ||
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Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span><!-- ws:start:WikiTextRawRule: | Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span><!-- ws:start:WikiTextRawRule:029:``//`` -->//<!-- ws:end:WikiTextRawRule:029 -->ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br /> | ||
<span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span><span style="display: block; text-align: center;">C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F</span><br /> | <span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span><span style="display: block; text-align: center;">C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F</span><br /> | ||
One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.<br /> | One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.<br /> | ||
Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = <!-- ws:start:WikiTextRawRule: | Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = <!-- ws:start:WikiTextRawRule:030:``//`` -->//<!-- ws:end:WikiTextRawRule:030 -->d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^<!-- ws:start:WikiTextRawRule:031:``//`` -->//<!-- ws:end:WikiTextRawRule:031 -->d4.<br /> | ||
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<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | ||
This is a lot of math, but it only needs to be done once for each pergen.<br /> | This is a lot of math, but it only needs to be done once for each pergen.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:54 -->Alternate enharmonics</h2> | ||
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Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br /> | Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br /> | ||
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<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8<br /> | <span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8<br /> | ||
</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br /> | </span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br /> | ||
</span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule: | </span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:032:``//`` -->//<!-- ws:end:WikiTextRawRule:032 -->d3=\\A2 -- \M3 -- P4<br /> | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule: | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:033:``//`` -->//<!-- ws:end:WikiTextRawRule:033 -->=D#\\ -- E\ -- F</span><br /> | ||
Approaching pergens in a | Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br /> | ||
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Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | ||
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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, tend to 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. So alternate enharmonics are needed sometimes.<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, tend to 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. So alternate enharmonics are needed sometimes.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc11"><a name="Further Discussion-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:56 -->Alternate keyspans and stepspans</h2> | ||
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One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br /> | One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:58 -->Chord names and scale names</h2> | ||
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:60 -->Combining pergens</h2> | ||
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Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br /> | Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br /> | ||
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However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:62 -->Pergens and EDOs</h2> | ||
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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:64 -->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-something block.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.<br /> | ||
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(links coming soon)<br /> | (links coming soon)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:66 -->Misc notes</h2> | ||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule: | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule:3788:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3788 --><br /> | ||
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Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br /> | Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br /> | ||