Kite's thoughts on pergens: Difference between revisions

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>2018-03-29 01:32:30 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-29 01:43:22 UTC</tt>.<br>

: The original revision id was <tt>~~628120301~~</tt>.<br>

: The original revision id was <tt>628120387</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 657:

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

~~Not~~ all Blackwood-like pergens are of the form (P8/m, ^1~~). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2~~).

It is not yet known if all Blackwood-like pergens are of the form (P8/m, ^1).

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:

Line 663:

||= large quintuple blue ||= 2.3.7 ||= (-14,0,0,5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= D E^^ Gv A^ Cvv D ||= C G D A E... ||= 49/48 ||

||= small quintuple red ||= 2.3.7 ||= (22,-5,0,-5) ||= " ||= " ||= " ||= " ||= " ||= 64/63 ||

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section).

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A Blackwood-like pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

Line 4,061:

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.<br />

<br />

~~Not~~ all Blackwood-like pergens are of the form (P8/m, ^1~~). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2~~).<br />

It is not yet known if all Blackwood-like pergens are of the form (P8/m, ^1).<br />

<br />

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:<br />

Line 4,129:

</table>

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &quot;missing notes&quot; (see next section).<br />

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &quot;missing notes&quot; (see next section). A Blackwood-like pergen should never have an uninflected genchain.<br />

<br />

<h2 id="toc18"><a name="Further Discussion-Notating non-8ve and no-5ths pergens"></a>Notating non-8ve and no-5ths pergens</h2>

@@ 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>2018-03-29 01:32:30 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-29 01:43:22 UTC</tt>.<br>
-: The original revision id was <tt>628120301</tt>.<br>
+: The original revision id was <tt>628120387</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 657: / Line 657: @@
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.
-Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2).
+It is not yet known if all Blackwood-like pergens are of the form (P8/m, ^1).
 It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:
@@ Line 663: / Line 663: @@
 ||= large quintuple blue ||= 2.3.7 ||= (-14,0,0,5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= D E^^ Gv A^ Cvv D ||= C G D A E... ||= 49/48 ||
 ||= small quintuple red ||= 2.3.7 ||= (22,-5,0,-5) ||= " ||= " ||= " ||= " ||= " ||= 64/63 ||
-Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section).
+Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A Blackwood-like pergen should never have an uninflected genchain.
 ==Notating non-8ve and no-5ths pergens==
@@ Line 4,061: / Line 4,061: @@
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.&lt;br /&gt;
 &lt;br /&gt;
-Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2).&lt;br /&gt;
+It is not yet known if all Blackwood-like pergens are of the form (P8/m, ^1).&lt;br /&gt;
 &lt;br /&gt;
 It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:&lt;br /&gt;
@@ Line 4,129: / Line 4,129: @@
 &lt;/table&gt;
-Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &amp;quot;missing notes&amp;quot; (see next section).&lt;br /&gt;
+Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &amp;quot;missing notes&amp;quot; (see next section). A Blackwood-like pergen should never have an uninflected genchain.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&gt;Notating non-8ve and no-5ths pergens&lt;/h2&gt;