Kite's Genchain mode numbering: Difference between revisions
Wikispaces>TallKite **Imported revision 593196824 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593196876 - 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 03:49:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593196876</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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<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;white-space: pre-wrap ! important" class="old-revision-html">= = | <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;white-space: pre-wrap ! important" class="old-revision-html">= = | ||
[[toc]] | [[toc]] | ||
**Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] 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: | **Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] 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: | ||
|| old scale name || new scale name || sL pattern || example on white keys || genchain || | || old scale name || new scale name || sL pattern || example on white keys || genchain || | ||
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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 [[ | 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 not just use UDP notation?|Why not just use UDP notation?]] below. | ||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
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See below. | See below. | ||
===__**Why not just use UDP notation?**__=== | ===[#Why not just use UDP notation?]]__**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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Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C. | Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C. | ||
Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. | Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.</pre></div> | ||
</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>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> | ||
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<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-Then why not always choose the larger of the two generators?">Then why not always choose the larger of the two generators?</a></div> | <!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-Then why not always choose the larger of the two generators?">Then why not always choose the larger of the two generators?</a></div> | ||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-Why not always choose the chroma-positive generator?">Why not always choose the chroma-positive generator?</a></div> | <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-Why not always choose the chroma-positive generator?">Why not always choose the chroma-positive generator?</a></div> | ||
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-Why not just use UDP notation?">Why not just use UDP notation?</a></div> | <!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 3em;"><a href="#x-Explanation / Rationale-[#Why not just use UDP notation?]]Why not just use UDP notation?">[#Why not just use UDP notation?]]Why not just use UDP notation?</a></div> | ||
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --></div> | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --></div> | ||
<!-- ws:end:WikiTextTocRule:34 --> | <!-- ws:end:WikiTextTocRule:34 --><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 /> | ||
<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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<br /> | <br /> | ||
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=" | 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 not just use UDP notation?">Why not just use UDP notation?</a> below.<br /> | ||
<br /> | <br /> | ||
Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
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See below.<br /> | See below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>Why not just use UDP notation?</strong></u></h3> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Explanation / Rationale-[#Why not just use UDP notation?]]Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:20 -->[#Why not just use UDP notation?]]<u><strong>Why not just use UDP notation?</strong></u></h3> | ||
<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 /> | ||