Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 580965555 - Original comment: ** |
Wikispaces>TallKite **Imported revision 580965825 - 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>2016-04-23 02: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-23 02:51:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>580965825</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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**__Why not number the modes in the order they occur in the scale?__** | **__Why not number the modes in the order they occur in the scale?__** | ||
This would order the modes Ionian, Dorian, Phrygian, etc. | This would order the modes Ionian, Dorian, Phrygian, etc. The advantage of scale-based numbering is that the modes are easier to find on one's instrument. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. | ||
__**Why make an exception for 3/2 vs 4/3 as the generator?**__ | __**Why make an exception for 3/2 vs 4/3 as the generator?**__ | ||
Because of centuries of established thought | Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show [emphasis mine]: | ||
"Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio **<span class="mw-redirect">3:2</span>**." [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]] | "Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio __**<span class="mw-redirect">3:2</span>**__." [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]] | ||
"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect **fifth**." [[https://en.wikipedia.org/wiki/Syntonic_temperament|en.wikipedia.org/wiki/Syntonic_temperament]] | "The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect __**fifth**__." [[https://en.wikipedia.org/wiki/Syntonic_temperament|en.wikipedia.org/wiki/Syntonic_temperament]] | ||
"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect **fifths**" | "Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect __**fifths**__" | ||
[[https://en.wikipedia.org/wiki/Meantone_temperament|en.wikipedia.org/wiki/Meantone_temperament]] | [[https://en.wikipedia.org/wiki/Meantone_temperament|en.wikipedia.org/wiki/Meantone_temperament]] | ||
"In this system the perfect **fifth** is flattened by one quarter of a syntonic comma" [[https://en.wikipedia.org/wiki/Quarter-comma_meantone|en.wikipedia.org/wiki/Quarter-comma_meantone | "In this system the perfect __**fifth**__ is flattened by one quarter of a syntonic comma" [[https://en.wikipedia.org/wiki/Quarter-comma_meantone|en.wikipedia.org/wiki/Quarter-comma_meantone]] | ||
__**Then why not choose the larger of the two generators?**__ | __**Then why not choose the larger of the two generators?**__ | ||
Because | Because the interval arithmetic is easier with smaller intervals. It's easier to compute the sum of stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain often is identical to the scale, simplifying mode numbering. | ||
__**Why not just use UDP notation?**__ | |||
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. | One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. | ||
|| scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || | || scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule:8:&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:8 --><strong><u>MODMOS scales</u></strong></h2> | ||
As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.<br /> | As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:9:&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-Fractional-octave periods&quot; title=&quot;Anchor: How to name rank-2 scales-Fractional-octave periods&quot;/&gt; --><a name="How to name rank-2 scales-Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:9 --><strong><u>Fractional-octave periods</u></strong></h2> | ||
Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br /> | Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br /> | ||
Eb -- Bb -- F --- C --- G<br /> | Eb -- Bb -- F --- C --- G<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:10:&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-Non-MOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-Non-MOS scales&quot;/&gt; --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:10 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | ||
Compact the genchain to remove any gaps via chromatic alterations. The mode number is derived from the compacted genchain. Examples:<br /> | Compact the genchain to remove any gaps via chromatic alterations. The mode number is derived from the compacted genchain. Examples:<br /> | ||
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<strong><u>Why not number the modes in the order they occur in the scale?</u></strong><br /> | <strong><u>Why not number the modes in the order they occur in the scale?</u></strong><br /> | ||
This would order the modes Ionian, Dorian, Phrygian, etc. | This would order the modes Ionian, Dorian, Phrygian, etc. The advantage of scale-based numbering is that the modes are easier to find on one's instrument. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. <br /> | ||
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<u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u><br /> | <u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u><br /> | ||
Because of centuries of established thought | Because of 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 /> | ||
&quot;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <strong><span class="mw-redirect">3:2</span></strong>.&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br /> | &quot;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u>.&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br /> | ||
<br /> | <br /> | ||
&quot;The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <strong>fifth</strong>.&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow">en.wikipedia.org/wiki/Syntonic_temperament</a><br /> | &quot;The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <u><strong>fifth</strong></u>.&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow">en.wikipedia.org/wiki/Syntonic_temperament</a><br /> | ||
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&quot;Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <strong>fifths</strong>&quot;<br /> | &quot;Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <u><strong>fifths</strong></u>&quot;<br /> | ||
<a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow">en.wikipedia.org/wiki/Meantone_temperament</a><br /> | <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow">en.wikipedia.org/wiki/Meantone_temperament</a><br /> | ||
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&quot;In this system the perfect <strong>fifth</strong> is flattened by one quarter of a syntonic comma&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow">en.wikipedia.org/wiki/Quarter-comma_meantone | &quot;In this system the perfect <u><strong>fifth</strong></u> is flattened by one quarter of a syntonic comma&quot; <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow">en.wikipedia.org/wiki/Quarter-comma_meantone</a><br /> | ||
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<u><strong>Then why not choose the larger of the two generators?</strong></u><br /> | <u><strong>Then why not choose the larger of the two generators?</strong></u><br /> | ||
Because | Because the interval arithmetic is easier with smaller intervals. It's easier to compute the sum of stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain often is identical to the scale, simplifying mode numbering.<br /> | ||
<br /> | |||
<br /> | <br /> | ||
< | <u><strong>Why not just use UDP notation?</strong></u><br /> | ||
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.<br /> | |||
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Admins: Please delete the blank page at <!-- ws:start:WikiTextUrlRule: | Admins: Please delete the blank page at <!-- ws:start:WikiTextUrlRule:1383:http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position --><a href="http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position">http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position</a><!-- ws:end:WikiTextUrlRule:1383 --></body></html></pre></div> | ||