Kite's Genchain mode numbering: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-25 07:57:27 UTC</tt>.<br>

: The original revision id was <tt>593233858</tt>.<br>

For example, here are all the modes of Meantone [7], using ~3/2 as the generator:

The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in "Explanation"). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see the critique of UDP at the bottom of this page.

Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, Srutal [10] might look like this:

The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.

There are only two Blackwood [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":

=[[#Explanation / Rationale]]__Explanation__=  

There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):

"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a __wise__ consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.

Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)

&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Explanation"&gt;Explanation&lt;/a&gt;&lt;/div&gt;

For example, here are all the modes of Meantone [7], using ~3/2 as the generator:&lt;br /&gt;

The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in &amp;quot;Explanation&amp;quot;). &lt;strong&gt;&lt;u&gt;Unlike modal UDP notation, the generator isn't always chroma-positive&lt;/u&gt;.&lt;/strong&gt; There are several disadvantages of only using chroma-positive generators, see the critique of UDP at the bottom of this page.&lt;br /&gt;

Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation"&gt;ups and downs notation&lt;/a&gt;. Because the generator is a 2nd, the genchain resembles the scale.&lt;br /&gt;

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, Srutal [10] might look like this:&lt;br /&gt;

The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.&lt;br /&gt;

There are only two Blackwood [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &amp;quot;height&amp;quot;:&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Explanation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:24:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Explanation / Rationale&amp;quot; title=&amp;quot;Anchor: Explanation / Rationale&amp;quot;/&amp;gt; --&gt;&lt;a name="Explanation / Rationale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:24 --&gt;&lt;u&gt;Explanation&lt;/u&gt;&lt;/h1&gt;

There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):&lt;br /&gt;

&amp;quot;A foolish consistency is the hobgoblin of little minds&amp;quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a &lt;u&gt;wise&lt;/u&gt; consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.&lt;br /&gt;

Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)&lt;br /&gt;