Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 19:29:18 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 22:13:05 UTC</tt>.

: The original revision id was <tt>~~624808841~~</tt>.

: The original revision id was <tt>624810837</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< ||

||||||~ Octotonic MOS scales ||~ ||

||= 1L 7s ||= (P8, P) [8] ||= octotonic ||< ||

||= 1L 7s ||= (P8, P4/4) [8] ||= quarter-4th octotonic ||< fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th ||

||= 2L 6s ||= (P8, P) [8] ||= octotonic ||< ||

||= 2L 6s ||= (P8/2, P5) [8] ||= half-octave octotonic ||< ||

||= 3L 5s ||= (P8, P) [8] ||= octotonic ||< ||

||= 3L 5s ||= (P8, P11/4) [8] ||= quarter-11th octotonic ||< seventh-WW4th, seventh-WW5th ||

||= 4L 4s ||= (P8, P) [8] ||= octotonic ||< ||

||= 4L 4s ||= (P8/4, P5) [8] ||= quarter-8ve octotonic ||< ||

||= 5L 3s ||= (P8, P) [8] ||= octotonic ||< ||

||= 5L 3s ||= (P8, P12/4) [8] ||= quarter-12th octotonic ||< (very lopsided, unless 5th is quite flat) ||

||= 6L 2s ||= (P8, P) [8] ||= octotonic ||< ||

||= 6L 2s ||= (P8/2, P4/3) [8] ||= half-8ve third-4th octotonic ||< ||

||= 7L 1s ||= (P8, P) [8] ||= octotonic ||< ||

||= 7L 1s ||= (P8, P4/3) [8] ||= third-4th octotonic ||< ||

||||||~ Nonatonic MOS scales ||~ ||

||= 1L 8s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 1L 8s ||= (P8, P4/4) [9] ||= quarter-4th nonatonic ||< fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th ||

||= 2L 7s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 2L 7s ||= (P8, W3P5/8) [9] ||= eighth-W35th nonatonic ||< third-11th, fifth-WW4th ||

||= 3L 6s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 3L 6s ||= (P8/3, P5) [9] ||= third-8ve nonatonic ||< third-8ve half-5th ||

||= 4L 5s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 4L 5s ||= (P8, P12/7) [9] ||= seventh-12th nonatonic ||< sixth-11th ||

||= 5L 4s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 5L 4s ||= (P8, P4/2) [9] ||= half-4th nonatonic ||< (lopsided unless 4th is sharp), seventh-11th ||

||= 6L 3s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 6L 3s ||= (P8/3, P4/2) [9] ||= third-8ve half-4th nonatonic ||< ||

||= 7L 2s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 7L 2s ||= (P8, WWP5/6) [9] ||= sixth-WW5th nonatonic ||< (lopsided unless 5th is sharp) ||

||= 8L 1s ||= (P8, P) [9] ||= nonatonic ||< ||

||= 8L 1s ||= (P8, P5/5) [9] ||= fifth-5th nonatonic ||< ||

||||||~ Decatonic MOS scales ||~ ||

||= 1L 9s ||= (P8, P) [10] ||= decatonic ||< ||

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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 tipping-point edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢~~. In practice, the lower limit of this range is ~646¢, for 13b-edo~~. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

||||~ bare enharmonic interval ||~ 3-exponent ||~ tipping

||||~ bare enharmonic

interval ||~ 3-exponent ||~ tipping

point edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 5th is just ||

||= M2 ||= C - D ||= 2 ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||

Line 1,357:

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<td style="text-align: center;">1L 7s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8, P4/4) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">quarter-4th octotonic

</td>

<td style="text-align: left;">fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th

</td>

</tr>

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<td style="text-align: center;">2L 6s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8/2, P5) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">half-octave octotonic

</td>

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<td style="text-align: center;">3L 5s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8, P11/4) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">quarter-11th octotonic

</td>

<td style="text-align: left;">seventh-WW4th, seventh-WW5th

</td>

</tr>

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<td style="text-align: center;">4L 4s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8/4, P5) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">quarter-8ve octotonic

</td>

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<td style="text-align: center;">5L 3s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8, P12/4) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">quarter-12th octotonic

</td>

<td style="text-align: left;">(very lopsided, unless 5th is quite flat)

</td>

</tr>

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<td style="text-align: center;">6L 2s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8/2, P4/3) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">half-8ve third-4th octotonic

</td>

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<td style="text-align: center;">7L 1s

</td>

<td style="text-align: center;">(P8, P) [8]

<td style="text-align: center;">(P8, P4/3) [8]

</td>

<td style="text-align: center;">octotonic

<td style="text-align: center;">third-4th octotonic

</td>

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<td style="text-align: center;">1L 8s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, P4/4) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">quarter-4th nonatonic

</td>

<td style="text-align: left;">fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th

</td>

</tr>

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<td style="text-align: center;">2L 7s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, W3P5/8) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">eighth-W35th nonatonic

</td>

<td style="text-align: left;">third-11th, fifth-WW4th

</td>

</tr>

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<td style="text-align: center;">3L 6s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8/3, P5) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">third-8ve nonatonic

</td>

<td style="text-align: left;">third-8ve half-5th

</td>

</tr>

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<td style="text-align: center;">4L 5s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, P12/7) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">seventh-12th nonatonic

</td>

<td style="text-align: left;">sixth-11th

</td>

</tr>

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<td style="text-align: center;">5L 4s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, P4/2) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">half-4th nonatonic

</td>

<td style="text-align: left;">(lopsided unless 4th is sharp), seventh-11th

</td>

</tr>

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<td style="text-align: center;">6L 3s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8/3, P4/2) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">third-8ve half-4th nonatonic

</td>

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<td style="text-align: center;">7L 2s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, WWP5/6) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">sixth-WW5th nonatonic

</td>

<td style="text-align: left;">(lopsided unless 5th is sharp)

</td>

</tr>

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<td style="text-align: center;">8L 1s

</td>

<td style="text-align: center;">(P8, P) [9]

<td style="text-align: center;">(P8, P5/5) [9]

</td>

<td style="text-align: center;">nonatonic

<td style="text-align: center;">fifth-5th nonatonic

</td>

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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 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 &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.

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 tipping-point edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢~~. In practice, the lower limit of this range is ~646¢, for 13b-edo~~. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

<tr>

<th colspan="2">bare enharmonic interval

<th colspan="2">bare enharmonic

interval

</th>

<th>3-exponent

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<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 19:29:18 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 22:13:05 UTC</tt>.<br>
-: The original revision id was <tt>624808841</tt>.<br>
+: The original revision id was <tt>624810837</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>
@@ Line 142: / Line 142: @@
 ||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||&lt;   ||
 ||||||~ Octotonic MOS scales ||~   ||
-||= 1L 7s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 1L 7s ||= (P8, P4/4) [8] ||= quarter-4th octotonic ||&lt; fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th ||
-||= 2L 6s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 2L 6s ||= (P8/2, P5) [8] ||= half-octave octotonic ||&lt;   ||
-||= 3L 5s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 3L 5s ||= (P8, P11/4) [8] ||= quarter-11th octotonic ||&lt; seventh-WW4th, seventh-WW5th ||
-||= 4L 4s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 4L 4s ||= (P8/4, P5) [8] ||= quarter-8ve octotonic ||&lt;   ||
-||= 5L 3s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 5L 3s ||= (P8, P12/4) [8] ||= quarter-12th octotonic ||&lt; (very lopsided, unless 5th is quite flat) ||
-||= 6L 2s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 6L 2s ||= (P8/2, P4/3) [8] ||= half-8ve third-4th octotonic ||&lt;   ||
-||= 7L 1s ||= (P8, P) [8] ||= octotonic ||&lt;   ||
+||= 7L 1s ||= (P8, P4/3) [8] ||= third-4th octotonic ||&lt;   ||
 ||||||~ Nonatonic MOS scales ||~   ||
-||= 1L 8s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 1L 8s ||= (P8, P4/4) [9] ||= quarter-4th nonatonic ||&lt; fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th ||
-||= 2L 7s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 2L 7s ||= (P8, W&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;P5/8) [9] ||= eighth-W&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5th nonatonic ||&lt; third-11th, fifth-WW4th ||
-||= 3L 6s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 3L 6s ||= (P8/3, P5) [9] ||= third-8ve nonatonic ||&lt; third-8ve half-5th ||
-||= 4L 5s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 4L 5s ||= (P8, P12/7) [9] ||= seventh-12th nonatonic ||&lt; sixth-11th ||
-||= 5L 4s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 5L 4s ||= (P8, P4/2) [9] ||= half-4th nonatonic ||&lt; (lopsided unless 4th is sharp), seventh-11th ||
-||= 6L 3s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 6L 3s ||= (P8/3, P4/2) [9] ||= third-8ve half-4th nonatonic ||&lt;   ||
-||= 7L 2s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 7L 2s ||= (P8, WWP5/6) [9] ||= sixth-WW5th nonatonic ||&lt; (lopsided unless 5th is sharp) ||
-||= 8L 1s ||= (P8, P) [9] ||= nonatonic ||&lt;   ||
+||= 8L 1s ||= (P8, P5/5) [9] ||= fifth-5th nonatonic ||&lt;   ||
 ||||||~ Decatonic MOS scales ||~   ||
 ||= 1L 9s ||= (P8, P) [10] ||= decatonic ||&lt;   ||
@@ Line 311: / Line 311: @@
 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 tipping-point edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.
-||||~ bare enharmonic interval ||~ 3-exponent ||~ tipping
+||||~ bare enharmonic
+interval ||~ 3-exponent ||~ tipping
 point edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 5th is just ||
 ||= M2 ||= C - D ||= 2 ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||
@@ Line 1,357: / Line 1,358: @@
          &lt;td style="text-align: center;"&gt;1L 7s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/4) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quarter-4th octotonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,367: / Line 1,368: @@
          &lt;td style="text-align: center;"&gt;2L 6s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-octave octotonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 1,377: / Line 1,378: @@
          &lt;td style="text-align: center;"&gt;3L 5s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P11/4) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quarter-11th octotonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;seventh-WW4th, seventh-WW5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,387: / Line 1,388: @@
          &lt;td style="text-align: center;"&gt;4L 4s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/4, P5) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quarter-8ve octotonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 1,397: / Line 1,398: @@
          &lt;td style="text-align: center;"&gt;5L 3s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P12/4) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quarter-12th octotonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;(very lopsided, unless 5th is quite flat)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,407: / Line 1,408: @@
          &lt;td style="text-align: center;"&gt;6L 2s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P4/3) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-8ve third-4th octotonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 1,417: / Line 1,418: @@
          &lt;td style="text-align: center;"&gt;7L 1s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [8]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/3) [8]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;octotonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;third-4th octotonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 1,433: / Line 1,434: @@
          &lt;td style="text-align: center;"&gt;1L 8s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/4) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quarter-4th nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,443: / Line 1,444: @@
          &lt;td style="text-align: center;"&gt;2L 7s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, W&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;P5/8) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;eighth-W&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5th nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;third-11th, fifth-WW4th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,453: / Line 1,454: @@
          &lt;td style="text-align: center;"&gt;3L 6s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P5) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;third-8ve nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;third-8ve half-5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,463: / Line 1,464: @@
          &lt;td style="text-align: center;"&gt;4L 5s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P12/7) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;seventh-12th nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;sixth-11th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,473: / Line 1,474: @@
          &lt;td style="text-align: center;"&gt;5L 4s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/2) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-4th nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;(lopsided unless 4th is sharp), seventh-11th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,483: / Line 1,484: @@
          &lt;td style="text-align: center;"&gt;6L 3s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P4/2) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;third-8ve half-4th nonatonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 1,493: / Line 1,494: @@
          &lt;td style="text-align: center;"&gt;7L 2s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, WWP5/6) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;sixth-WW5th nonatonic&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;(lopsided unless 5th is sharp)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,503: / Line 1,504: @@
          &lt;td style="text-align: center;"&gt;8L 1s&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P) [9]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5/5) [9]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;nonatonic&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;fifth-5th nonatonic&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 2,328: / Line 2,329: @@
 Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;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 &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;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 &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
-The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the &amp;quot;rungspan&amp;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.&lt;br /&gt;
+The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the &amp;quot;rungspan&amp;quot;) is larger. If the sweet spot contains the tipping point, and the 5th equals the tipping-point edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.&lt;br /&gt;
 &lt;br /&gt;
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.&lt;br /&gt;
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.&lt;br /&gt;
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;bare enharmonic interval&lt;br /&gt;
+         &lt;th colspan="2"&gt;bare enharmonic&lt;br /&gt;
+interval&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;3-exponent&lt;br /&gt;
@@ Line 3,010: / Line 3,012: @@
 &lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Misc notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Misc notes&lt;/h2&gt;
   &lt;br /&gt;
-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3751:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3751 --&gt;&lt;br /&gt;
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3756:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3756 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:&lt;br /&gt;