Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 624272165 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624350869 - 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> | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-02 06:30:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624350869</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | < 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | ||
^1 = 64/63 ||= " || | ^1 = 64/63 ||= " || | ||
||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^P4 = vP5 ||= C - F^=Gv - C ||= 128/121 | ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^P4 = vP5 ||= C - F^=Gv - C ||= 128/121 | ||
^1 = 33/32 ||= " || | ^1 = 33/32 ||= " || | ||
||= (P8, P4/2) | ||= (P8, P4/2) | ||
half-fourth ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | half-fourth ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | ||
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23, 24, 25*, 28*, 29, | 23, 24, 25*, 28*, 29, | ||
30* || | 30* || | ||
||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= | ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||= " || | ||
||= (P8, P5/2) | ||= (P8, P5/2) | ||
half-fifth ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira | half-fifth ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira | ||
^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*, | ^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*, | ||
24, 27, 28*, 30*, 31 || | 24, 27, 28*, 30*, 31 || | ||
||= (P8/2, P4/2) | ||= (P8/2, P4/2) | ||
half- | half- | ||
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C - Eb^=Ev - G, | C - Eb^=Ev - G, | ||
C - F#v/=Gb^\ - C, | C - F#v/=Gb^\ - C, | ||
C - F^/=Gv\ - C ||= | C - F^/=Gv\ - C ||= 49/48 & 128/121 | ||
^1 = 33/32 | ^1 = 33/32 | ||
/1 = 64/63 ||= 14, 18b, 20*, | /1 = 64/63 ||= 14, 18b, 20*, | ||
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P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C, | P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C, | ||
C - D/=Eb\ - F, | C - D/=Eb\ - F, | ||
C - Eb^/=Ev\ - G ||= | C - Eb^/=Ev\ - G ||= 2048/2025 | ||
& 49/48 | |||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 64/63 ||= " || | /1 = 64/63 ||= " || | ||
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P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | ||
C - Eb/=E\ - G, | C - Eb/=E\ - G, | ||
C - Dv/=Eb^\ - F ||= | C - Dv/=Eb^\ - F ||= 2048/2025 | ||
& 128/121 | |||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 33/32 ||= " || | /1 = 33/32 ||= " || | ||
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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 ||= ||= || | ||
||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^P4 = vvP5 ||= C - F^ - Cv - F ||= ||= " || | ||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^P4 = vvP5 ||= C - F^ - Cv - F ||= ||= " || | ||
||= (P8/3, P4/2) ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 | ||= (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 | |||
P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | ||
C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= ||= 15, 18b*, 24, 30 || | C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= ||= 15, 18b*, 24, 30 || | ||
||= (P8/3, P5/2) ||= v<span style="vertical-align: super;">6</span>m3 ||= | ||= (P8/3, P5/2) | ||
||= (P8/2, P4/3) ||= v<span style="vertical-align: super;">6</span>d4 ||= | third-8ve, half-5th ||= v<span style="vertical-align: super;">6</span>m3 ||= C^<span style="vertical-align: super;">6</span> ``=`` Eb ||= ||= ||= ||= 18b, 24, 30 || | ||
||= " ||= v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4 ||= | ||= (P8/2, P4/3) | ||
||= (P8/2, P5/3) ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5 | half-8ve, third-4th ||= v<span style="vertical-align: super;">6</span>d4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb ||= ||= ||= ||= 14, 22, 28*, 30* || | ||
||= " ||= v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb3 ||= ||= ||= ||= || | |||
||= (P8/2, P5/3) | |||
half-8ve, | |||
third-5th ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5 | |||
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C | P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C | ||
C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* || | C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* || | ||
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P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | ||
C - /D - \F - G ||= ||= " || | C - /D - \F - G ||= ||= " || | ||
||= (P8/2, P11/3) ||= v<span style="vertical-align: super;">6</span>M2 ||= | ||= (P8/2, P11/3) | ||
||= (P8/3, P4/3) ||= ||= ||= ||= ||= ||= 15, 21, 30* || | half-8ve, | ||
third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= ||= ||= ||= || | |||
||= (P8/3, P4/3) | |||
third- | |||
everything ||= ||= ||= ||= ||= ||= 15, 21, 30* || | |||
||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv A^ C ||= diminished, | ||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv A^ C ||= diminished, | ||
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||= {P8/4, P4/4} ||= ||= ||= ||= ||= ||= || | ||= {P8/4, P4/4} ||= ||= ||= ||= ||= ||= || | ||
Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just. | Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just. | ||
The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the "rungspan") is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed. | The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the "rungspan") is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed. | ||
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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth. | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth. | ||
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | ||
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Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. For example, consider (P8/5, P5). One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. Because the enharmonic E is multiplied by 2, it will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs equals the count. Equipped with the period and the enharmonic, the perchain is easily found: | Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. For example, consider (P8/5, P5). One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. Because the enharmonic E is multiplied by 2, it will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs equals the count. Equipped with the period and the enharmonic, the perchain is easily found: | ||
<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span> | <span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span> | ||
Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is: | |||
<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^=C#v<span style="vertical-align: super;">3</span> -- Dv -- Eb^ -- Fb^<span style="vertical-align: super;">3</span>=Evv -- F</span> | <span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^=C#v<span style="vertical-align: super;">3</span> -- Dv -- Eb^ -- Fb^<span style="vertical-align: super;">3</span>=Evv -- F</span> | ||
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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 a 3rd or a 4th, as with (P8/2, P4/3). | 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 a 3rd or a 4th, as with (P8/2, P4/3). | ||
Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/4, P5/3) unreduces to (P8/4, m6/12). The bare alternate generator is m6/12 = [8,5]/12 = [1,0] = A1. The bare enharmonic is [8,5] - 12*[1,0] = [-4,5] = a 6th diminished 11 times! This would make for a very confusing notation. However, [8,5]/12 can be rounded off to [-1,0] = d1. The bare enharmonic becomes [8,5] - 12*[-1,0] = [20,5] = 4*[4,1] = AA2, which is awkward but still an improvement. The period is v3M3 and the generator is ^4m2. | |||
<span style="display: block; text-align: center;">P1 -- v3M3 -- v6A5=^6d4 -- ^3m6 -- P8 | |||
</span><span style="display: block; text-align: center;">C -- Ev3 -- G#v6=Fb^6 -- Ab^3 -- C | |||
</span><span style="display: block; text-align: center;">P1 -- ^4m2 -- ^8d3=v4A4 -- P5 | |||
</span><span style="display: block; text-align: center;">C -- Db^4 -- F#v4 -- G | |||
</span> | |||
==Alternate enharmonics== | ==Alternate enharmonics== | ||
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[//Question: what if there are highs and lows?//]</pre></div> | [//Question: what if there are highs and lows?//]</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 names</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 names</title></head><body><!-- ws:start:WikiTextHeadingRule:25:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:25 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:51:&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:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multi-gens">Extremely large multi-gens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><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:59 --><!-- ws:start:WikiTextTocRule:60: --><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:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><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:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 2em;"><a href="#toc11"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextHeadingRule:27:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:27 --><u><strong>Definition</strong></u></h1> | ||
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The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:29:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:29 --><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 other 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.<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 other 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.<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 multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multi-gen 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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth 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 multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multi-gen 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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:31 --><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. Meantone (mean = average, tone = major 2nd) implies y3/2 = (5/4)/2. Semisixth (aka sensei) implies y6/2 = (5/3)/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is.<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. Meantone (mean = average, tone = major 2nd) implies y3/2 = (5/4)/2. Semisixth (aka sensei) implies y6/2 = (5/3)/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is.<br /> | ||
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<td style="text-align: center;">C - F^=Gv - C<br /> | <td style="text-align: center;">C - F^=Gv - C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">128/121 | <td style="text-align: center;">128/121<br /> | ||
^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
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<td style="text-align: center;">C - D#v=Ebb^ - F<br /> | <td style="text-align: center;">C - D#v=Ebb^ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(-22,-11,2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
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<td style="text-align: center;">14*, 17, 18b, 20*, 21*,<br /> | <td style="text-align: center;">14*, 17, 18b, 20*, 21*,<br /> | ||
24, 27, 28*, 30*, 31<br /> | 24, 27, 28*, 30*, 31<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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C - F^/=Gv\ - C<br /> | C - F^/=Gv\ - C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">49/48 &amp; 128/121<br /> | ||
^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
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C - Eb^/=Ev\ - G<br /> | C - Eb^/=Ev\ - G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">2048/2025<br /> | ||
&amp; 49/48<br /> | |||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
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C - Dv/=Eb^\ - F<br /> | C - Dv/=Eb^\ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">2048/2025<br /> | ||
&amp; 128/121<br /> | |||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
/1 = 33/32<br /> | /1 = 33/32<br /> | ||
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<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P4/2)<br /> | <td style="text-align: center;">(P8/3, P4/2)<br /> | ||
third-8ve, half-4th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>A2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>A2<br /> | ||
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<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P5/2)<br /> | <td style="text-align: center;">(P8/3, P5/2)<br /> | ||
third-8ve, half-5th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>m3<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>m3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:014:``=`` -->=<!-- ws:end:WikiTextRawRule:014 --> Eb<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P4/3)<br /> | <td style="text-align: center;">(P8/2, P4/3)<br /> | ||
half-8ve, third-4th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>d4<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>d4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:015:``=`` -->=<!-- ws:end:WikiTextRawRule:015 --> Fb<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> Fb3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P5/3)<br /> | <td style="text-align: center;">(P8/2, P5/3)<br /> | ||
half-8ve,<br /> | |||
third-5th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2<br /> | ||
</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:017:``=`` -->=<!-- ws:end:WikiTextRawRule:017 --> 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 /> | ||
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</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:018:``///`` -->///<!-- ws:end:WikiTextRawRule:018 --> = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
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<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P11/3)<br /> | <td style="text-align: center;">(P8/2, P11/3)<br /> | ||
half-8ve,<br /> | |||
third-11th<br /> | |||
</td> | </td> | ||
<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;"><br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,560: | Line 1,535: | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P4/3)<br /> | <td style="text-align: center;">(P8/3, P4/3)<br /> | ||
third-<br /> | |||
everything<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,595: | Line 1,572: | ||
<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:020:``=`` -->=<!-- ws:end:WikiTextRawRule:020 --> 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 /> | ||
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<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:021:``=`` -->=<!-- ws:end:WikiTextRawRule:021 --> 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 1,628: | Line 1,605: | ||
<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:022:``=`` -->=<!-- ws:end:WikiTextRawRule:022 --> 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 1,644: | Line 1,621: | ||
<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:023:``=`` -->=<!-- ws:end:WikiTextRawRule:023 --> 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 1,660: | Line 1,637: | ||
<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:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P12/4 = vP4 = ^<span style="vertical-align: super;">3</span>M3<br /> | <td style="text-align: center;">P12/4 = vP4 = ^<span style="vertical-align: super;">3</span>M3<br /> | ||
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<br /> | <br /> | ||
Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; enharmonic, a conventional interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &quot;tipping point&quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>ups and downs may need to be swapped, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br /> | Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &quot;tipping point&quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>ups and downs may need to be swapped, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br /> | ||
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The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the &quot;rungspan&quot;) is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.<br /> | The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the &quot;rungspan&quot;) is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:33:&lt;h1&gt; --><h1 id="toc4"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:33 --><u>Further Discussion</u></h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:35:&lt;h2&gt; --><h2 id="toc5"><a name="Further Discussion-Extremely large multi-gens"></a><!-- ws:end:WikiTextHeadingRule:35 -->Extremely large multi-gens</h2> | ||
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So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen 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 multi-gens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen 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 multi-gens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:37:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:37 -->Singles and doubles</h2> | ||
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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 pergens 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.<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 pergens 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.<br /> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:39:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:39 -->Finding an example temperament</h2> | ||
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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*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*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*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.<br /> | ||
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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.<br /> | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.<br /> | ||
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This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48.<br /> | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.<br /> | ||
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A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br /> | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br /> | ||
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There are also alternate enharmonics, see below.<br /> | There are also alternate enharmonics, see below.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:41:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:41 -->Finding a notation for a pergen</h2> | ||
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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 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 basic concepts:<br /> | ||
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Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. For example, consider (P8/5, P5). One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. Because the enharmonic E is multiplied by 2, it will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's <strong>count</strong> is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs equals the count. Equipped with the period and the enharmonic, the perchain is easily found:<br /> | Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. For example, consider (P8/5, P5). One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. Because the enharmonic E is multiplied by 2, it will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's <strong>count</strong> is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs equals the count. Equipped with the period and the enharmonic, the perchain is easily found:<br /> | ||
<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span><br /> | <span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span><br /> | ||
Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:<br /> | |||
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<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^=C#v<span style="vertical-align: super;">3</span> -- Dv -- Eb^ -- Fb^<span style="vertical-align: super;">3</span>=Evv -- F</span><br /> | <span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^=C#v<span style="vertical-align: super;">3</span> -- Dv -- Eb^ -- Fb^<span style="vertical-align: super;">3</span>=Evv -- F</span><br /> | ||
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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 a 3rd or a 4th, 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 a 3rd or a 4th, as with (P8/2, P4/3).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/4, P5/3) unreduces to (P8/4, m6/12). The bare alternate generator is m6/12 = [8,5]/12 = [1,0] = A1. The bare enharmonic is [8,5] - 12*[1,0] = [-4,5] = a 6th diminished 11 times! This would make for a very confusing notation. However, [8,5]/12 can be rounded off to [-1,0] = d1. The bare enharmonic becomes [8,5] - 12*[-1,0] = [20,5] = 4*[4,1] = AA2, which is awkward but still an improvement. The period is v3M3 and the generator is ^4m2.<br /> | ||
<span style="display: block; text-align: center;">P1 -- v3M3 -- v6A5=^6d4 -- ^3m6 -- P8<br /> | |||
</span><span style="display: block; text-align: center;">C -- Ev3 -- G#v6=Fb^6 -- Ab^3 -- C<br /> | |||
</span><span style="display: block; text-align: center;">P1 -- ^4m2 -- ^8d3=v4A4 -- P5<br /> | |||
</span><span style="display: block; text-align: center;">C -- Db^4 -- F#v4 -- G<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:43:&lt;h2&gt; --><h2 id="toc9"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:43 -->Alternate enharmonics</h2> | |||
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For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2<br /> | For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2<br /> | ||
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(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | (P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:45:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:45 -->Alternate keyspans and stepspans</h2> | ||
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One might wonder, 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 7-edo 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. These edos would also work, 12-edo is merely the most convenient choice.<br /> | One might wonder, 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 7-edo 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. These edos would also work, 12-edo is merely the most convenient choice.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:49:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:49 -->Combining pergens</h2> | ||
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250/243 makes third-fourth, and 49/48 makes half-fourth, and tempering out both commas makes sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). General rules for combining pergens:<br /> | 250/243 makes third-fourth, and 49/48 makes half-fourth, and tempering out both commas makes sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). General rules for combining pergens:<br /> | ||