Comparison of mode notation systems: Difference between revisions
Wikispaces>jdfreivald **Imported revision 581131771 - 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: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 11:37:31 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>581131851</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;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">=__**Kite** Giedraitis method__= | ||
[[toc|flat]] | |||
==__**Proposed method of naming all possible rank-2 scales**__== | ==__**Proposed method of naming all possible rank-2 scales**__== | ||
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What if the algorithm were something like this: | What if the algorithm were something like this: | ||
Start with whatever you have more of, L or s. Put the smallest cluster of those you can at the beginning. Then alternate, using the smallest clusters of steps you can until it's done. | Start with whatever you have more of, L or s. Put the smallest cluster of those you can at the beginning. Then alternate, using the smallest clusters of steps you can until it's done. | ||
Some examples: | Some examples: | ||
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For 4L+5s, which is sLsLsLsLs, it seems a little more complicated -- but not much. 5>4, so I have to start with s. I see one place where there are two s's together, so that has to go before the last L: sLsLsLssL | For 4L+5s, which is sLsLsLsLs, it seems a little more complicated -- but not much. 5>4, so I have to start with s. I see one place where there are two s's together, so that has to go before the last L: sLsLsLssL | ||
For 9L+4s, which is sLLsLLsLLsLLL, we start with L. The longest string of L's I have is three, so mode 1 is LLsLLsLLsLLLs. | For 9L+4s, which is sLLsLLsLLsLLL, we start with L. The longest string of L's I have is three, so mode 1 is LLsLLsLLsLLLs. | ||
For 7L+8s, which is sLsLsLsLsLsLsLs, we start with s and put the string ss just before the last L: mode 1 for this structure (e.g., for porcupine[15]) is sLsLsLsLsLsLssL. | For 7L+8s, which is sLsLsLsLsLsLsLs, we start with s and put the string ss just before the last L: mode 1 for this structure (e.g., for porcupine[15]) is sLsLsLsLsLsLssL. | ||
For an MOS like 3L+3s, make it as much "like meantone[7] major" as you can: L to start, and a small leading tone: LsLsLs. | For an MOS like 3L+3s, make it as much "like meantone[7] major" as you can: L to start, and a small leading tone: LsLsLs. | ||
Note the things I *don't* need to know to do this: I don't have to know what a generator is, what mappings are, what utonality or otonality is, or a host of other things. I won't get confused by seeing intervals that don't map to JI well (e.g., phi). This is, in fact, just basic string manipulation. | Note the things I *don't* need to know to do this: I don't have to know what a generator is, what mappings are, what utonality or otonality is, or a host of other things. I won't get confused by seeing intervals that don't map to JI well (e.g., phi). This is, in fact, just basic string manipulation. | ||
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. | 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. | ||
==Extending to non-MOS== | ==Extending to non-MOS== | ||
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(Note that the word "scale" is ambiguous in some of what follows. I don't think we'll get away from that in real life, so I'm just pushing on, even where it sounds weird.) | (Note that the word "scale" is ambiguous in some of what follows. I don't think we'll get away from that in real life, so I'm just pushing on, even where it sounds weird.) | ||
I'm going to start with some of the scales Kite has already used on the wiki page he created. | I'm going to start with some of the scales Kite has already used on the wiki page he created. | ||
The harmonic minor scale is already structured this way: A B C D E F G# A is MsMMsLs. There are the same number of M and s steps, so M goes first. The one-step cluster of M goes first, and the two-M cluster goes second. The harmonic minor scale is mode 1 of this scale. The phyrigian dominant scale, which features in a lot of world music (I think of it as the "Hava Nagila scale"), is mode 5 of this scale. | The harmonic minor scale is already structured this way: A B C D E F G# A is MsMMsLs. There are the same number of M and s steps, so M goes first. The one-step cluster of M goes first, and the two-M cluster goes second. The harmonic minor scale is mode 1 of this scale. The phyrigian dominant scale, which features in a lot of world music (I think of it as the "Hava Nagila scale"), is mode 5 of this scale. | ||
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Double harmonic minor (never heard of this -- I learn something new every day on this list) is A B C D# E F G# A, or MsLssLs. Mode 1 will start with a small step and have the largest cluster of s's last in the scale: sMsLssL. Double harmonic minor is thus mode 2 of this scale. | Double harmonic minor (never heard of this -- I learn something new every day on this list) is A B C D# E F G# A, or MsLssLs. Mode 1 will start with a small step and have the largest cluster of s's last in the scale: sMsLssL. Double harmonic minor is thus mode 2 of this scale. | ||
Double harmonic major (never heard of this either) is A Bb C# D E F G# A, or sLsMssLs. Start with the smallest cluster of small steps, and this has to be sMssLssL. Double harmonic major is mode 7 of this scale. | Double harmonic major (never heard of this either) is A Bb C# D E F G# A, or sLsMssLs. Start with the smallest cluster of small steps, and this has to be sMssLssL. Double harmonic major is mode 7 of this scale. | ||
Hungarian gypsy minor is A B C D# E F G A, or MsLssMM. We have the same number of s's and M's, so we start the mode with M. After that decision, there are no more to be made: It's MMMsLss. Hungarian gypsy minor is mode 3 of this scale. | Hungarian gypsy minor is A B C D# E F G A, or MsLssMM. We have the same number of s's and M's, so we start the mode with M. After that decision, there are no more to be made: It's MMMsLss. Hungarian gypsy minor is mode 3 of this scale. | ||
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None of these scales have had a problem that I'm about to address and resolve. To wit: | None of these scales have had a problem that I'm about to address and resolve. To wit: | ||
Let's pick a rank-3 scale: 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1. (Note that this is not a temperament at all.) That's LMsLMLs. With three L's and only two M's and s's, L has to go first. But there are no larger or smaller clusters of L's! They only come one at a time. So let's pick the mode that has the longest string of non-L's to go first. (That pushes the last L out as far as it will go.) | Let's pick a rank-3 scale: 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1. (Note that this is not a temperament at all.) That's LMsLMLs. With three L's and only two M's and s's, L has to go first. But there are no larger or smaller clusters of L's! They only come one at a time. So let's pick the mode that has the longest string of non-L's to go first. (That pushes the last L out as far as it will go.) | ||
This also works for Kite's question about meantone[8], which is LMsMLLML. There are more L's than any other step size, so the scale has to start with L. The L's are in equal clusters of two, so there's no obvious way to pick which one goes first: mode 1 must start with LL. So let's pick the mode that has the longest string of non-L's to go after that first LL: Mode 1 of meantone[8] is LLMsMLLM. (That's C - D - E - F - F# - G - A - B - C, or something like it.) | This also works for Kite's question about meantone[8], which is LMsMLLML. There are more L's than any other step size, so the scale has to start with L. The L's are in equal clusters of two, so there's no obvious way to pick which one goes first: mode 1 must start with LL. So let's pick the mode that has the longest string of non-L's to go after that first LL: Mode 1 of meantone[8] is LLMsMLLM. (That's C - D - E - F - F# - G - A - B - C, or something like it.) | ||
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Let's try something harder: the rank-3 scale minerva[12], which I found through Graham's temperament finder.* Since there are four step sizes, I'm going to label them LMms, where the capital M is larger than the small m. With steps of 113, 113, 87, 113, 87, 99, 113, 87, 113, 87, 113, and 73, that's LLmLmMLmLmLs, 6L+1M+4m+1s. L has to go first. There's a cluster of two L's in the string, and I push that as close to the end as possible: LmMLmLmLsLLm. Graham is showing mode 10 of this scale in his temperament finder. | Let's try something harder: the rank-3 scale minerva[12], which I found through Graham's temperament finder.* Since there are four step sizes, I'm going to label them LMms, where the capital M is larger than the small m. With steps of 113, 113, 87, 113, 87, 99, 113, 87, 113, 87, 113, and 73, that's LLmLmMLmLmLs, 6L+1M+4m+1s. L has to go first. There's a cluster of two L's in the string, and I push that as close to the end as possible: LmMLmLmLsLLm. Graham is showing mode 10 of this scale in his temperament finder. | ||
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. | 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. | ||
* http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&limit=11&tuning=po | * http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&limit=11&tuning=po | ||
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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</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Kite Giedraitis method"></a><!-- ws:end:WikiTextHeadingRule:0 --><u><strong>Kite</strong> Giedraitis method</u></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</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Kite Giedraitis method"></a><!-- ws:end:WikiTextHeadingRule:0 --><u><strong>Kite</strong> Giedraitis method</u></h1> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Proposed method of naming all possible rank-2 scales</strong></u></h2> | <!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Kite Giedraitis method">Kite Giedraitis method</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Jake Freivald method">Jake Freivald method</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Proposed method of naming all possible rank-2 scales</strong></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:4:&lt;h2&gt; --><h2 id="toc2"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:41:&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:41 --><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:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:42:&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:42 --><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:8:&lt;h2&gt; --><h2 id="toc4"><a name="Kite Giedraitis method-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Kite Giedraitis method-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:43:&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:43 --><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:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- 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-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:44 --><u>Explanation / Rationale</u></h2> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Kite Giedraitis method-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:12 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Kite Giedraitis method-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:12 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | ||
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What if the algorithm were something like this:<br /> | What if the algorithm were something like this:<br /> | ||
<br /> | <br /> | ||
Start with whatever you have more of, L or s. Put the smallest cluster of those you can at the beginning. Then alternate, using the smallest clusters of steps you can until it's done. <br /> | Start with whatever you have more of, L or s. Put the smallest cluster of those you can at the beginning. Then alternate, using the smallest clusters of steps you can until it's done.<br /> | ||
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Some examples:<br /> | Some examples:<br /> | ||
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For 4L+5s, which is sLsLsLsLs, it seems a little more complicated -- but not much. 5&gt;4, so I have to start with s. I see one place where there are two s's together, so that has to go before the last L: sLsLsLssL<br /> | For 4L+5s, which is sLsLsLsLs, it seems a little more complicated -- but not much. 5&gt;4, so I have to start with s. I see one place where there are two s's together, so that has to go before the last L: sLsLsLssL<br /> | ||
<br /> | <br /> | ||
For 9L+4s, which is sLLsLLsLLsLLL, we start with L. The longest string of L's I have is three, so mode 1 is LLsLLsLLsLLLs. <br /> | For 9L+4s, which is sLLsLLsLLsLLL, we start with L. The longest string of L's I have is three, so mode 1 is LLsLLsLLsLLLs.<br /> | ||
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For 7L+8s, which is sLsLsLsLsLsLsLs, we start with s and put the string ss just before the last L: mode 1 for this structure (e.g., for porcupine[15]) is sLsLsLsLsLsLssL. <br /> | For 7L+8s, which is sLsLsLsLsLsLsLs, we start with s and put the string ss just before the last L: mode 1 for this structure (e.g., for porcupine[15]) is sLsLsLsLsLsLssL.<br /> | ||
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For an MOS like 3L+3s, make it as much &quot;like meantone[7] major&quot; as you can: L to start, and a small leading tone: LsLsLs.<br /> | For an MOS like 3L+3s, make it as much &quot;like meantone[7] major&quot; as you can: L to start, and a small leading tone: LsLsLs.<br /> | ||
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Note the things I *don't* need to know to do this: I don't have to know what a generator is, what mappings are, what utonality or otonality is, or a host of other things. I won't get confused by seeing intervals that don't map to JI well (e.g., phi). This is, in fact, just basic string manipulation. <br /> | Note the things I *don't* need to know to do this: I don't have to know what a generator is, what mappings are, what utonality or otonality is, or a host of other things. I won't get confused by seeing intervals that don't map to JI well (e.g., phi). This is, in fact, just basic string manipulation.<br /> | ||
<br /> | <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 /> | 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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(Note that the word &quot;scale&quot; is ambiguous in some of what follows. I don't think we'll get away from that in real life, so I'm just pushing on, even where it sounds weird.)<br /> | (Note that the word &quot;scale&quot; is ambiguous in some of what follows. I don't think we'll get away from that in real life, so I'm just pushing on, even where it sounds weird.)<br /> | ||
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I'm going to start with some of the scales Kite has already used on the wiki page he created. <br /> | I'm going to start with some of the scales Kite has already used on the wiki page he created.<br /> | ||
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The harmonic minor scale is already structured this way: A B C D E F G# A is MsMMsLs. There are the same number of M and s steps, so M goes first. The one-step cluster of M goes first, and the two-M cluster goes second. The harmonic minor scale is mode 1 of this scale. The phyrigian dominant scale, which features in a lot of world music (I think of it as the &quot;Hava Nagila scale&quot;), is mode 5 of this scale.<br /> | The harmonic minor scale is already structured this way: A B C D E F G# A is MsMMsLs. There are the same number of M and s steps, so M goes first. The one-step cluster of M goes first, and the two-M cluster goes second. The harmonic minor scale is mode 1 of this scale. The phyrigian dominant scale, which features in a lot of world music (I think of it as the &quot;Hava Nagila scale&quot;), is mode 5 of this scale.<br /> | ||
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Double harmonic minor (never heard of this -- I learn something new every day on this list) is A B C D# E F G# A, or MsLssLs. Mode 1 will start with a small step and have the largest cluster of s's last in the scale: sMsLssL. Double harmonic minor is thus mode 2 of this scale.<br /> | Double harmonic minor (never heard of this -- I learn something new every day on this list) is A B C D# E F G# A, or MsLssLs. Mode 1 will start with a small step and have the largest cluster of s's last in the scale: sMsLssL. Double harmonic minor is thus mode 2 of this scale.<br /> | ||
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Double harmonic major (never heard of this either) is A Bb C# D E F G# A, or sLsMssLs. Start with the smallest cluster of small steps, and this has to be sMssLssL. Double harmonic major is mode 7 of this scale. <br /> | Double harmonic major (never heard of this either) is A Bb C# D E F G# A, or sLsMssLs. Start with the smallest cluster of small steps, and this has to be sMssLssL. Double harmonic major is mode 7 of this scale.<br /> | ||
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Hungarian gypsy minor is A B C D# E F G A, or MsLssMM. We have the same number of s's and M's, so we start the mode with M. After that decision, there are no more to be made: It's MMMsLss. Hungarian gypsy minor is mode 3 of this scale.<br /> | Hungarian gypsy minor is A B C D# E F G A, or MsLssMM. We have the same number of s's and M's, so we start the mode with M. After that decision, there are no more to be made: It's MMMsLss. Hungarian gypsy minor is mode 3 of this scale.<br /> | ||
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None of these scales have had a problem that I'm about to address and resolve. To wit:<br /> | None of these scales have had a problem that I'm about to address and resolve. To wit:<br /> | ||
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Let's pick a rank-3 scale: 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1. (Note that this is not a temperament at all.) That's LMsLMLs. With three L's and only two M's and s's, L has to go first. But there are no larger or smaller clusters of L's! They only come one at a time. So let's pick the mode that has the longest string of non-L's to go first. (That pushes the last L out as far as it will go.) <br /> | Let's pick a rank-3 scale: 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1. (Note that this is not a temperament at all.) That's LMsLMLs. With three L's and only two M's and s's, L has to go first. But there are no larger or smaller clusters of L's! They only come one at a time. So let's pick the mode that has the longest string of non-L's to go first. (That pushes the last L out as far as it will go.)<br /> | ||
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This also works for Kite's question about meantone[8], which is LMsMLLML. There are more L's than any other step size, so the scale has to start with L. The L's are in equal clusters of two, so there's no obvious way to pick which one goes first: mode 1 must start with LL. So let's pick the mode that has the longest string of non-L's to go after that first LL: Mode 1 of meantone[8] is LLMsMLLM. (That's C - D - E - F - F# - G - A - B - C, or something like it.)<br /> | This also works for Kite's question about meantone[8], which is LMsMLLML. There are more L's than any other step size, so the scale has to start with L. The L's are in equal clusters of two, so there's no obvious way to pick which one goes first: mode 1 must start with LL. So let's pick the mode that has the longest string of non-L's to go after that first LL: Mode 1 of meantone[8] is LLMsMLLM. (That's C - D - E - F - F# - G - A - B - C, or something like it.)<br /> | ||
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Let's try something harder: the rank-3 scale minerva[12], which I found through Graham's temperament finder.* Since there are four step sizes, I'm going to label them LMms, where the capital M is larger than the small m. With steps of 113, 113, 87, 113, 87, 99, 113, 87, 113, 87, 113, and 73, that's LLmLmMLmLmLs, 6L+1M+4m+1s. L has to go first. There's a cluster of two L's in the string, and I push that as close to the end as possible: LmMLmLmLsLLm. Graham is showing mode 10 of this scale in his temperament finder.<br /> | Let's try something harder: the rank-3 scale minerva[12], which I found through Graham's temperament finder.* Since there are four step sizes, I'm going to label them LMms, where the capital M is larger than the small m. With steps of 113, 113, 87, 113, 87, 99, 113, 87, 113, 87, 113, and 73, that's LLmLmMLmLmLs, 6L+1M+4m+1s. L has to go first. There's a cluster of two L's in the string, and I push that as close to the end as possible: LmMLmLmLsLLm. Graham is showing mode 10 of this scale in his temperament finder.<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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