Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 622079045 - Original comment: ** |
Wikispaces>TallKite **Imported revision 622083245 - 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>2017-11-21 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-21 04:24:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>622083245</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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lence(s) ||~ genchain(s) ||~ examples ||~ compatible edos | lence(s) ||~ genchain(s) ||~ examples ||~ compatible edos | ||
(12-31 only) || | (12-31 only) || | ||
||= {P8, P5} ||= none ||= | ||= {P8, P5} ||= none ||= none ||= none ||= C - G ||= meantone ||= 12, 13b, 14*, 15*, 16, | ||
17, 18b*, 19, 20*, 21*, | 17, 18b*, 19, 20*, 21*, | ||
22, 23, 24*, 25*, 26, | 22, 23, 24*, 25*, 26, | ||
27, 28*, 29, 30*, 31 || | 27, 28*, 29, 30*, 31 || | ||
||~ halves ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ halves ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/2, P5} ||= P8/2 = vA4 = ^d5 ||= ^^d2 ||= C^^ = B# ||= C - F#v=Gb^ - C ||= srutal | ||= {P8/2, P5} ||= P8/2 = vA4 = ^d5 | ||
^1 = 81/80 ||= 12, 20*, 22, 24*, 30* || | (if 5th > 700¢) ||= ^^d2 ||= C^^ = B# ||= C - F#v=Gb^ - C ||= srutal | ||
^1 = 81/80 ||= 12, 14, 16, 18b, 20*, | |||
22, 24*, 26, 28*, 30* || | |||
||= " ||= P8/2 = ^A4 = vd5 | ||= " ||= P8/2 = ^A4 = vd5 | ||
(if 5th < 700¢) ||= vvd2 ||= C^^ = Db ||= C - F#^=Gbv - C ||= large deep red | (if 5th < 700¢) ||= vvd2 ||= C^^ = Db ||= C - F#^=Gbv - C ||= large deep red | ||
^1 = 64/63 ||= | ^1 = 64/63 ||= " || | ||
||= " ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C^^ = D ||= C - F^=Gv - C ||= ||= || | ||= " ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C^^ = D ||= C - F^=Gv - C ||= ||= || | ||
||= {P8, P4/2} ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= C^^ = Db ||= C - D^=Ebv - F ||= semaphore ||= | ||= {P8, P4/2} ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= C^^ = Db ||= C - D^=Ebv - F ||= semaphore ||= 14, 15*, 18b*, 19, 20*, | ||
||= " ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= C^^ = B## ||= C - D#v=Ebb^ - F ||= ||= | 23, 24, 25*, 28*, 29, | ||
30* || | |||
||= " ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= C^^ = B## ||= C - D#v=Ebb^ - F ||= ||= " || | |||
||= {P8, P5/2} ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C^^ = C# ||= C - Eb^=Ev - G ||= mohajira | ||= {P8, P5/2} ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C^^ = C# ||= 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} ||= P4/2 = /M2 = \m3 | ||= {P8/2, P4/2} ||= P4/2 = /M2 = \m3 | ||
P5/2 = ^m3 = vM3 | P5/2 = ^m3 = vM3 | ||
P8/2 = v/A4 = ^\d5 | P8/2 = v/A4 = ^\d5 | ||
= ^/P4 = v\P5 ||= \\m2, | = ^/P4 = v\P5 ||= \\m2, | ||
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^^\\d2, | ^^\\d2, | ||
vv\\M2 ||= C// = Db, | vv\\M2 ||= C// = Db, | ||
C^^ = C# | C^^ = C# | ||
C^^// = D ||= C - D/=Eb\ - F, | C^^// = D ||= C - D/=Eb\ - F, | ||
C - Eb^=Ev - G, | C - Eb^=Ev - G, | ||
C - F#v/=Gb^\ - C, | C - F#v/=Gb^\ - C, | ||
C - F^/=Gv\ - C ||= bb&aaT | C - F^/=Gv\ - C ||= bb&aaT | ||
^1 = 33/32 | ^1 = 33/32 | ||
/1 = 64/63 ||= 14, 18b, 20*, | /1 = 64/63 ||= 14, 18b, 20*, | ||
24, 28*, 30* || | 24, 28*, 30* || | ||
||= " ||= P8/2 = vA4 = ^d5 | ||= " ||= P8/2 = vA4 = ^d5 | ||
P4/2 = /M2 = \m3 | P4/2 = /M2 = \m3 | ||
P5/2 = ^/m3 = v\M3 ||= ^^d2, | P5/2 = ^/m3 = v\M3 ||= ^^d2, | ||
\\m2, | \\m2, | ||
vv\\A1 ||= C^^ = B# | vv\\A1 ||= C^^ = B# | ||
C// = Db, | C// = Db, | ||
C^^// = C# ||= C - F#v=Gb^ - C, | C^^// = C# ||= C - F#v=Gb^ - C, | ||
C - D/=Eb\ - F, | C - D/=Eb\ - F, | ||
C - Eb^/=Ev\ - G ||= sgg&bbT | C - Eb^/=Ev\ - G ||= sgg&bbT | ||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 64/63 ||= " || | /1 = 64/63 ||= " || | ||
||= " ||= P8/2 = vA4 = ^d5 | ||= " ||= P8/2 = vA4 = ^d5 | ||
P5/2 = /m3 = \M3 | P5/2 = /m3 = \M3 | ||
P4/2 =v/M2 = ^\m3 ||= ^^d2, | P4/2 =v/M2 = ^\m3 ||= ^^d2, | ||
\\A1, | \\A1, | ||
^^\\m2 ||= C^^ = B# | ^^\\m2 ||= C^^ = B# | ||
C// = C# | C// = C# | ||
C^^// = B## ||= C - F#v=Gb^ - C, | C^^// = B## ||= C - F#v=Gb^ - C, | ||
C - Eb/=E\ - G, | C - Eb/=E\ - G, | ||
C - Dv/=Eb^\ - F ||= sgg&aaT | C - Dv/=Eb^\ - F ||= sgg&aaT | ||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 33/32 ||= " || | /1 = 33/32 ||= " || | ||
||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 | ||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3</span> = B# ||= <span style="font-family: Arial,Helvetica,sans-serif;">C - Ev - Ab^ - C</span> ||= ||= 12, 15, 18b*, 21, | ||
24*, 27, 30* || | |||
27, 30* || | |||
||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3</span> = C# ||= C - Dv - Eb^ - F ||= ||= 13b, 14*, 15, 21*, | ||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3</span> = C# ||= C - Dv - Eb^ - F ||= ||= 13b, 14*, 15, 21*, | ||
22, 28*, 29, 30* || | 22, 28*, 29, 30* || | ||
||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3</span> = Db ||= C - D^ - Fv - G ||= ||= 15*, 16, 20*, 21, | ||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3</span> = Db ||= C - D^ - Fv - G ||= ||= 15*, 16, 20*, 21, | ||
25*, 26, 30*, 31 || | 25*, 26, 30*, 31 || | ||
||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> = B## ||= C - F#v - Cb^ - F ||= ||= | ||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> = B## ||= C - F#v - Cb^ - F ||= ||= || | ||
||= " ||= P11/3 = ^P4 = vvP5 ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3</span> = D ||= C F^ Cv F ||= ||= " || | ||= " ||= P11/3 = ^P4 = vvP5 ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3</span> = D ||= C F^ Cv F ||= ||= " || | ||
||= {P8/3, P4/2} ||= ||= ||= ||= ||= ||= 15, 18b*, 24, 30 || | ||= {P8/3, P4/2} ||= ||= ||= ||= ||= ||= 15, 18b*, 24, 30 || | ||
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P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= ^<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> ||= 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 ||= ^<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> ||= 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* || | ||
||= " ||= P8/2 = vA4 = ^d5 | ||= " ||= P8/2 = vA4 = ^d5 | ||
P5/3 = /M2 = \\m3 ||= ^^d2, | P5/3 = /M2 = \\m3 ||= ^^d2, | ||
\\\m2 ||= C^^ = B# | \\\m2 ||= C^^ = B# | ||
C/// = Db ||= C - F#v=Gb^ - C | C/// = Db ||= C - F#v=Gb^ - C | ||
C - /D - \F - G ||= ||= " || | C - /D - \F - G ||= ||= " || | ||
||= {P8/2, P11/3} ||= ||= ||= ||= ||= ||= | ||= {P8/2, P11/3} ||= ||= ||= ||= ||= ||= || | ||
||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= 15, 21, 30* || | ||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= 15, 21, 30* || | ||
||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/4, P5} ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 | ||= {P8/4, P5} ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= v<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> = Dbb ||= C Ebv Gbvv A^ C ||= diminished, | ||
^1 = 81/80 ||= 12, 16, 20, 24*, 28 || | ^1 = 81/80 ||= 12, 16, 20, 24*, 28 || | ||
||= {P8, P4/4} ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> = B## ||= C Db^ Ebb^^ Ev F ||= ||= || | ||= {P8, P4/4} ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> = B## ||= C Db^ Ebb^^ Ev F ||= ||= || | ||
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||= {P8/2, P4/4} ||= ||= ||= ||= ||= ||= || | ||= {P8/2, P4/4} ||= ||= ||= ||= ||= ||= || | ||
||= {P8/2, P5/4} ||= ||= ||= ||= ||= ||= || | ||= {P8/2, P5/4} ||= ||= ||= ||= ||= ||= || | ||
||= etc. ||= ||= ||= ||= ||= ||= || | ||= etc. ||= ||= ||= ||= ||= ||= || | ||
||= ||= ||= ||= ||= ||= ||= | ||~ fifths ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= | ||= {P8/5, P5} ||= P8/5 = ||= vm2 ||= ||= ||= ||= || | ||
||= {P8, P5/5} || P5/5 = ^m2 = v<span style="vertical-align: super;">4</span>A<span style="vertical-align: super;">3</span>1 ||= ^<span style="vertical-align: super;">5</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">5</span> = B#<span style="vertical-align: super;">3</span> ||= C - Db^ - Ebb^^= | |||
D<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span> E#vv F#v G ||= ||= || | |||
Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. If the rank-2 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 make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and If it's descending, upped. Thus __**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 previous table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens are upped or downed as if the 5th were just. | |||
Heptatonic 5th-based notation is only possible if the 5th ranges between 600¢ and 720¢. For every enharmonic interval, the following table says in what parts of that range the interval should be upped or downed. | |||
||~ enharmonic | |||
interval ||~ implied edo ||~ size of the | |||
edo's 5th ||~ upping range ||~ downing range || | |||
||= M2 ||= 2-edo ||= 600¢ ||= none ||= all || | |||
||= m2 ||= 5-edo ||= 720¢ ||= none ||= all || | |||
||= A1 ||= 7-edo ||= 686¢ ||= 600-686¢ ||= 686¢-720¢ || | |||
||= d2 ||= 12-edo ||= 700¢ ||= 700-720¢ ||= 600-700¢ || | |||
||= dd3 ||= 17-edo ||= 706¢ ||= 706-720¢ ||= 600-706¢ || | |||
||= dd2 ||= 19-edo ||= 695¢ ||= 695-720¢ ||= 600-695¢ || | |||
||= d<span style="vertical-align: super;">3</span>2 ||= 26-edo ||= 692¢ ||= 692-720¢ ||= 600-692¢ || | |||
||= d<span style="vertical-align: super;">4</span>4 ||= 29-edo ||= 703¢ ||= 703-720¢ ||= 600-703¢ || | |||
||= d<span style="vertical-align: super;">4</span>3 ||= 31-edo ||= 697¢ ||= 697-720¢ ||= 600-697¢ || | |||
(to be continued)</pre></div> | (to be continued)</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:WikiTextTocRule: | <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:WikiTextTocRule:32:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Derivation">Derivation</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Applications">Applications</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#x^/P4"> ^/P4 </a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#B# ||"> B# ||</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#C# ||"> C# ||</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Db ||"> Db ||</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#B## ||"> B## ||</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --> | <a href="#D ||"> D ||</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#B#3 ||"> B#3 ||</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | <a href="#Dbb ||"> Dbb ||</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#B## ||"> B## ||</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#C# ||"> C# ||</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Eb3 ||"> Eb3 ||</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --> | <a href="#Db ||"> Db ||</a><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#B#3 ||"> B#3 ||</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 --><u><strong>Definition</strong></u></h1> | ||
<br /> | <br /> | ||
A <strong>pergen</strong> set (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br /> | A <strong>pergen</strong> set (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br /> | ||
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<td style="text-align: center;">none<br /> | <td style="text-align: center;">none<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">none<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">none<br /> | <td style="text-align: center;">none<br /> | ||
| Line 806: | Line 823: | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
(if 5th &gt; 700¢)<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">^^d2<br /> | <td style="text-align: center;">^^d2<br /> | ||
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^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">12, 20*, 22, 24*, 30*<br /> | <td style="text-align: center;">12, 14, 16, 18b, 20*,<br /> | ||
22, 24*, 26, 28*, 30*<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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^1 = 64/63<br /> | ^1 = 64/63<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">semaphore<br /> | <td style="text-align: center;">semaphore<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">14, 15*, 18b*, 19, 20*,<br /> | ||
23, 24, 25*, 28*, 29,<br /> | |||
30*<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">{P8/2, P4/2}<br /> | <td style="text-align: center;">{P8/2, P4/2}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2 = /M2 = \m3 | <td style="text-align: center;">P4/2 = /M2 = \m3<br /> | ||
P5/2 = ^m3 = vM3 | P5/2 = ^m3 = vM3<br /> | ||
P8/2 = v/A4 = ^\d5<br /> | P8/2 = v/A4 = ^\d5<br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x^/P4"></a><!-- ws:end:WikiTextHeadingRule:6 --> ^/P4 </h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x^/P4"></a><!-- ws:end:WikiTextHeadingRule:6 --> ^/P4 </h1> | ||
| Line 935: | Line 938: | ||
</td> | </td> | ||
<td style="text-align: center;">C<em> = Db,<br /> | <td style="text-align: center;">C<em> = Db,<br /> | ||
C^^ = C# | C^^ = C#<br /> | ||
C^^</em> = D<br /> | C^^</em> = D<br /> | ||
</td> | </td> | ||
| Line 944: | Line 947: | ||
</td> | </td> | ||
<td style="text-align: center;">bb&amp;aaT<br /> | <td style="text-align: center;">bb&amp;aaT<br /> | ||
^1 = 33/32 | ^1 = 33/32<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
</td> | </td> | ||
| Line 954: | Line 957: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5 | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
P4/2 = /M2 = \m3 | P4/2 = /M2 = \m3<br /> | ||
P5/2 = ^/m3 = v\M3<br /> | P5/2 = ^/m3 = v\M3<br /> | ||
</td> | </td> | ||
| Line 962: | Line 965: | ||
vv\\A1<br /> | vv\\A1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^^ = B# | <td style="text-align: center;">C^^ = B#<br /> | ||
C<em> = Db,<br /> | C<em> = Db,<br /> | ||
C^^</em> = C#<br /> | C^^</em> = C#<br /> | ||
| Line 971: | Line 974: | ||
</td> | </td> | ||
<td style="text-align: center;">sgg&amp;bbT<br /> | <td style="text-align: center;">sgg&amp;bbT<br /> | ||
^1 = 81/80 | ^1 = 81/80<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
</td> | </td> | ||
| Line 980: | Line 983: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5 | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
P5/2 = /m3 = \M3 | P5/2 = /m3 = \M3<br /> | ||
P4/2 =v/M2 = ^\m3<br /> | P4/2 =v/M2 = ^\m3<br /> | ||
</td> | </td> | ||
| Line 988: | Line 991: | ||
^^\\m2<br /> | ^^\\m2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^^ = B# | <td style="text-align: center;">C^^ = B#<br /> | ||
C<em> = C#<br /> | C<em> = C#<br /> | ||
C^^</em> = B##<br /> | C^^</em> = B##<br /> | ||
| Line 997: | Line 1,000: | ||
</td> | </td> | ||
<td style="text-align: center;">sgg&amp;aaT<br /> | <td style="text-align: center;">sgg&amp;aaT<br /> | ||
^1 = 81/80 | ^1 = 81/80<br /> | ||
/1 = 33/32<br /> | /1 = 33/32<br /> | ||
</td> | </td> | ||
| Line 1,023: | Line 1,026: | ||
</td> | </td> | ||
<td style="text-align: center;">P8/3 = vM3 = ^^d4<br /> | <td style="text-align: center;">P8/3 = vM3 = ^^d4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br /> | <td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br /> | ||
| Line 1,033: | Line 1,035: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">12, 15, 18b*, 21 | <td style="text-align: center;">12, 15, 18b*, 21,<br /> | ||
27, 30*<br /> | 24*, 27, 30*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,084: | Line 1,086: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,172: | Line 1,174: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = vA4 = ^d5 | <td style="text-align: center;">P8/2 = vA4 = ^d5<br /> | ||
P5/3 = /M2 = \\m3<br /> | P5/3 = /M2 = \\m3<br /> | ||
</td> | </td> | ||
| Line 1,178: | Line 1,180: | ||
\\\m2<br /> | \\\m2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^^ = B# | <td style="text-align: center;">C^^ = B#<br /> | ||
C/// = Db<br /> | C/// = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C - F#v=Gb^ - C | <td style="text-align: center;">C - F#v=Gb^ - C<br /> | ||
C - /D - \F - G<br /> | C - /D - \F - G<br /> | ||
</td> | </td> | ||
| Line 1,202: | Line 1,204: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,241: | Line 1,243: | ||
</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 /> | ||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>d2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>d2<br /> | ||
| Line 1,368: | Line 1,369: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">etc.<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,384: | Line 1,385: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <th>fifths<br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/5, P5}<br /> | |||
</td> | |||
<td style="text-align: center;">P8/5 =<br /> | |||
</td> | |||
<td style="text-align: center;">vm2<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,393: | Line 1,414: | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8, P5/5}<br /> | |||
</td> | |||
<td>P5/5 = ^m2 = v<span style="vertical-align: super;">4</span>A<span style="vertical-align: super;">3</span>1<br /> | |||
</td> | |||
<td style="text-align: center;">^<span style="vertical-align: super;">5</span>d<span style="vertical-align: super;">3</span>2<br /> | |||
</td> | |||
<td style="text-align: center;">C^<span style="vertical-align: super;">5</span> <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="B#3 ||"></a><!-- ws:end:WikiTextHeadingRule:30 --> B#<span style="vertical-align: super;">3</span><br /> | |||
</td> | |||
<td></h1> | |||
C - Db^ - Ebb^^=D<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span> E#vv F#v G<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. If the rank-2 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 make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and If it's descending, upped. Thus <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 previous table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens are upped or downed as if the 5th were just.<br /> | |||
<br /> | |||
Heptatonic 5th-based notation is only possible if the 5th ranges between 600¢ and 720¢. For every enharmonic interval, the following table says in what parts of that range the interval should be upped or downed.<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>enharmonic<br /> | |||
interval<br /> | |||
</th> | |||
<th>implied edo<br /> | |||
</th> | |||
<th>size of the<br /> | |||
edo's 5th<br /> | |||
</th> | |||
<th>upping range<br /> | |||
</th> | |||
<th>downing range<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">M2<br /> | |||
</td> | |||
<td style="text-align: center;">2-edo<br /> | |||
</td> | |||
<td style="text-align: center;">600¢<br /> | |||
</td> | |||
<td style="text-align: center;">none<br /> | |||
</td> | |||
<td style="text-align: center;">all<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">m2<br /> | |||
</td> | |||
<td style="text-align: center;">5-edo<br /> | |||
</td> | |||
<td style="text-align: center;">720¢<br /> | |||
</td> | |||
<td style="text-align: center;">none<br /> | |||
</td> | |||
<td style="text-align: center;">all<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">A1<br /> | |||
</td> | |||
<td style="text-align: center;">7-edo<br /> | |||
</td> | |||
<td style="text-align: center;">686¢<br /> | |||
</td> | |||
<td style="text-align: center;">600-686¢<br /> | |||
</td> | |||
<td style="text-align: center;">686¢-720¢<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">d2<br /> | |||
</td> | |||
<td style="text-align: center;">12-edo<br /> | |||
</td> | |||
<td style="text-align: center;">700¢<br /> | |||
</td> | |||
<td style="text-align: center;">700-720¢<br /> | |||
</td> | |||
<td style="text-align: center;">600-700¢<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">dd3<br /> | ||
</td> | |||
<td style="text-align: center;">17-edo<br /> | |||
</td> | |||
<td style="text-align: center;">706¢<br /> | |||
</td> | |||
<td style="text-align: center;">706-720¢<br /> | |||
</td> | |||
<td style="text-align: center;">600-706¢<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">dd2<br /> | |||
</td> | |||
<td style="text-align: center;">19-edo<br /> | |||
</td> | |||
<td style="text-align: center;">695¢<br /> | |||
</td> | |||
<td style="text-align: center;">695-720¢<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">600-695¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | </tr> | ||
<tr> | |||
<td style="text-align: center;">d<span style="vertical-align: super;">3</span>2<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">26-edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">692¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">692-720¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">600-692¢<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">d<span style="vertical-align: super;">4</span>4<br /> | ||
</td> | |||
<td style="text-align: center;">29-edo<br /> | |||
</td> | |||
<td style="text-align: center;">703¢<br /> | |||
</td> | |||
<td style="text-align: center;">703-720¢<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">600-703¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | </tr> | ||
<tr> | |||
<td style="text-align: center;">d<span style="vertical-align: super;">4</span>3<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">31-edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">697¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">697-720¢<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">600-697¢<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
(to be continued)</body></html></pre></div> | (to be continued)</body></html></pre></div> | ||