Kite's Genchain mode numbering: Difference between revisions
Wikispaces>TallKite **Imported revision 593196876 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593196904 - 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>2016-09-24 03: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-24 03:53:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593196904</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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|| Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | || Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | ||
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). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see [[Naming Rank-2 Scales using Mode Numbers#Why | 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). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see [[Naming Rank-2 Scales using Mode Numbers#Why%20not%20just%20use%20UDP%20notation?|Why not just use UDP notation?]] below. | ||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
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==[[#Explanation / Rationale]]__Explanation / Rationale__== | ==[[#Explanation / Rationale]]__Explanation / Rationale__== | ||
**__Why not number the modes in the order they occur in the scale?__** | |||
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. | ||
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The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain. | The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain. | ||
__**Why make an exception for 3/2 vs 4/3 as the generator?**__ | |||
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): | 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): | ||
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"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. | "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. | ||
__**Then why not always choose the larger of the two generators?**__ | |||
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.) | 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.) | ||
**__Why not always choose the chroma-positive generator?__** | |||
See below. | See below. | ||
__**Why not just use UDP notation?**__ | |||
One problem with [[Modal UDP Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging. | One problem with [[Modal UDP Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging. | ||
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<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>Naming Rank-2 Scales using Mode Numbers</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1> | <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>Naming Rank-2 Scales using Mode Numbers</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 2em;"><a href="#x-MODMOS scales">MODMOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 2em;"><a href="#x-Fractional-octave periods">Fractional-octave periods</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 2em;"><a href="#x-Rank-2 scales that are neither MOS nor MODMOS">Rank-2 scales that are neither MOS nor MODMOS</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#x-Numbering considerations">Numbering considerations</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#x-Explanation / Rationale">Explanation / Rationale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div> | ||
<!-- ws:end:WikiTextTocRule:19 --><strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br /> | |||
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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). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are several disadvantages of only using chroma-positive generators, see <a class="wiki_link" href="/Naming%20Rank-2%20Scales%20using%20Mode%20Numbers#Why | 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). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are several disadvantages of only using chroma-positive generators, see <a class="wiki_link" href="/Naming%20Rank-2%20Scales%20using%20Mode%20Numbers#Why%20not%20just%20use%20UDP%20notation?">Why not just use UDP notation?</a> below.<br /> | ||
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Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:20:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-MODMOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-MODMOS scales&quot;/&gt; --><a name="How to name rank-2 scales-MODMOS scales"></a><!-- ws:end:WikiTextAnchorRule:20 --><strong><u>MODMOS scales</u></strong></h2> | ||
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<a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The &quot;#&quot; symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G.<br /> | <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The &quot;#&quot; symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:21:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Fractional-octave periods&quot; title=&quot;Anchor: Fractional-octave periods&quot;/&gt; --><a name="Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:21 --><strong><u>Fractional-octave periods</u></strong></h2> | ||
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Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br /> | Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:22:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank-2 scales that are neither MOS nor MODMOS&quot; title=&quot;Anchor: Rank-2 scales that are neither MOS nor MODMOS&quot;/&gt; --><a name="Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextAnchorRule:22 --><strong><u>Rank-2 scales that are neither MOS nor MODMOS</u></strong></h2> | ||
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Some scales have too many or too few notes to be MOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.<br /> | Some scales have too many or too few notes to be MOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Numbering considerations"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Numbering considerations"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:23:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Numbering considerations&quot; title=&quot;Anchor: Numbering considerations&quot;/&gt; --><a name="Numbering considerations"></a><!-- ws:end:WikiTextAnchorRule:23 --><u>Numbering considerations</u></h2> | ||
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As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what &quot;#4&quot; means in a pentatonic or hexatonic context.<br /> | As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what &quot;#4&quot; means in a pentatonic or hexatonic context.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:24:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Explanation / Rationale&quot; title=&quot;Anchor: Explanation / Rationale&quot;/&gt; --><a name="Explanation / Rationale"></a><!-- ws:end:WikiTextAnchorRule:24 --><u>Explanation / Rationale</u></h2> | ||
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<strong><u>Why not number the modes in the order they occur in the scale?</u></strong><br /> | |||
<br /> | |||
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | ||
<br /> | <br /> | ||
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The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br /> | The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br /> | ||
<br /> | <br /> | ||
<u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u><br /> | |||
<br /> | |||
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):<br /> | 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):<br /> | ||
<br /> | <br /> | ||
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&quot;A foolish consistency is the hobgoblin of little minds&quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | &quot;A foolish consistency is the hobgoblin of little minds&quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | ||
<br /> | <br /> | ||
<u><strong>Then why not always choose the larger of the two generators?</strong></u><br /> | |||
<br /> | |||
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.)<br /> | 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.)<br /> | ||
<br /> | <br /> | ||
< | <strong><u>Why not always choose the chroma-positive generator?</u></strong><br /> | ||
<br /> | |||
See below.<br /> | See below.<br /> | ||
<br /> | <br /> | ||
<u><strong>Why not just use UDP notation?</strong></u><br /> | |||
<br /> | |||
One problem with <a class="wiki_link" href="/Modal%20UDP%20Notation">UDP</a> is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.<br /> | One problem with <a class="wiki_link" href="/Modal%20UDP%20Notation">UDP</a> is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.<br /> | ||