Comparison of mode notation systems: Difference between revisions
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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:jdfreivald|jdfreivald]] and made on <tt>2016-04-25 11: | : This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2016-04-25 11:39:40 UTC</tt>.<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">=__**Kite** Giedraitis method__= | <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]] | |||
==__**Proposed method of naming all possible rank-2 scales**__== | =__**Kite** Giedraitis method__= | ||
==__<span style="font-size: 1.3em; line-height: 1.5;">****Proposed method of naming all possible rank-2 scales****</span>__== | |||
**This page is a work in progress...** | **This page is a work in progress...** | ||
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<span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 4019.5px; width: 1px;"><span style="background-color: #f6f7f8; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 12px; line-height: 16.08px;">**<span style="color: #3b5998;">[[https://www.facebook.com/kite.giedraitis?fref=ufi|Giedraitis]]</span>** <span class="UFICommentBody">Been</span></span></span></pre></div> | <span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 4019.5px; width: 1px;"><span style="background-color: #f6f7f8; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 12px; line-height: 16.08px;">**<span style="color: #3b5998;">[[https://www.facebook.com/kite.giedraitis?fref=ufi|Giedraitis]]</span>** <span class="UFICommentBody">Been</span></span></span></pre></div> | ||
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Naming Rank-2 Scales</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0" | <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</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:28:&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:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Kite Giedraitis method">Kite Giedraitis method</a></div> | ||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-**Proposed method of naming all possible rank-2 scales**">**Proposed method of naming all possible rank-2 scales**</a></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-MODMOS scales">MODMOS scales</a></div> | |||
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Fractional-octave periods">Fractional-octave periods</a></div> | |||
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Non-MOS non-MODMOS scales">Non-MOS non-MODMOS scales</a></div> | |||
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Explanation / Rationale">Explanation / Rationale</a></div> | |||
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-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?</a></div> | |||
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-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?</a></div> | |||
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-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:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?">Why not always choose the chroma-positive generator?</a></div> | |||
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?">Why not just use UDP notation?</a></div> | |||
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Jake Freivald method">Jake Freivald method</a></div> | |||
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Jake Freivald method-Extending to non-MOS">Extending to non-MOS</a></div> | |||
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --></div> | |||
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Kite Giedraitis method"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Kite</strong> Giedraitis method</u></h1> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Kite Giedraitis method-**Proposed method of naming all possible rank-2 scales**"></a><!-- ws:end:WikiTextHeadingRule:4 --><u><span style="font-size: 1.3em; line-height: 1.5;">Proposed method of naming all possible rank-2 scales</span></u></h2> | |||
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<strong>This page is a work in progress...</strong><br /> | <strong>This page is a work in progress...</strong><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:44:&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:44 --><strong><u>MODMOS scales</u></strong></h2> | ||
To find a <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> 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 <u>compacted</u> genchain. <em>[This may change]</em> For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. This is so that two notes an aug or dim fifth apart will have adjacent mode numbers. Just like A and E are adjacent, Ab and E are too. In other words, determining the mode number from the scale degree remains fifth-based.<br /> | To find a <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> 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 <u>compacted</u> genchain. <em>[This may change]</em> For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. This is so that two notes an aug or dim fifth apart will have adjacent mode numbers. Just like A and E are adjacent, Ab and E are too. In other words, determining the mode number from the scale degree remains fifth-based.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:45:&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:45 --><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: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:46:&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:46 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | ||
Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples:<br /> | Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:47:&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:47 --><u>Explanation / Rationale</u></h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Kite Giedraitis method-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:14 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | ||
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Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Kite Giedraitis method-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3> | ||
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Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show:<br /> | Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show:<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 <u>wise</u> consistencies, 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 <u>wise</u> consistencies, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Kite Giedraitis method-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:18 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:20 --><u>Why not always choose the chroma-positive generator?</u></h3> | ||
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See below.<br /> | See below.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:22 --><u><strong>Why not just use UDP notation?</strong></u></h3> | ||
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One problem with 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.<br /> | One problem with 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.<br /> | ||
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A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, 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. Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and Mode Numbers notation uses the more intuitive one-based counting.<br /> | A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, 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. Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and Mode Numbers notation uses the more intuitive one-based counting.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Jake Freivald method"></a><!-- ws:end:WikiTextHeadingRule:24 -->Jake Freivald method</h1> | ||
My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include (1) as little knowledge needed as possible, to help the less-sophisticated user, (2) reasonably intuitive if possible, (3) easy to remember and check your own work, and therefore (3a) biased toward major being the &quot;right&quot; answer for meantone[7], and (4) extensibility of the method beyond MOS.<br /> | My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include (1) as little knowledge needed as possible, to help the less-sophisticated user, (2) reasonably intuitive if possible, (3) easy to remember and check your own work, and therefore (3a) biased toward major being the &quot;right&quot; answer for meantone[7], and (4) extensibility of the method beyond MOS.<br /> | ||
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I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone[5 or 7] gives me pentatonic major ssLsL, or CDEGAC, and diatonic major LLsLLLs, or CDEFGABC.<br /> | I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone[5 or 7] gives me pentatonic major ssLsL, or CDEGAC, and diatonic major LLsLLLs, or CDEFGABC.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Jake Freivald method-Extending to non-MOS"></a><!-- ws:end:WikiTextHeadingRule:26 -->Extending to non-MOS</h2> | ||
<span style="line-height: 1.5;">My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible. </span><br /> | <span style="line-height: 1.5;">My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible. </span><br /> | ||
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NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.<br /> | NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.<br /> | ||
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<ul><li><!-- ws:start:WikiTextUrlRule: | <ul><li><!-- ws:start:WikiTextUrlRule:1753:http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&amp;limit=11&amp;tuning=po --><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&amp;limit=11&amp;tuning=po" rel="nofollow">http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&amp;limit=11&amp;tuning=po</a><!-- ws:end:WikiTextUrlRule:1753 --></li></ul><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:1754: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:1754 --><br /> | ||
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