Comparison of mode notation systems: Difference between revisions
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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">==__**Proposed method of naming all possible rank-2 scales**__== | <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__= | ||
==__**Proposed method of naming all possible rank-2 scales**__== | |||
**This page is a work in progress...** | **This page is a work in progress...** | ||
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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 [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. | 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 [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. | ||
=Jake Freivald method= | |||
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 "right" answer for meantone[7], and (4) extensibility of the method beyond MOS. | |||
I've created a method that uses step sizes, generally identified as s for small, M for medium, and L for large. (That would need to be extended for scales with more than three sizes of steps, of course, but the principle remains the same.) | |||
We'll start with MOS scales. For MOS scales, we'll only have s and L steps sizes. | |||
Inversion seems important for intuition. If 5L+2s is LLsLLLs, 5s+2L should be ssLsssL. | |||
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. | |||
Some examples: | |||
For 5L+2s, which is some rotation of LLsLLsL, I know I have to start with L. The smallest cluster I have is LL, so I get LLsLLLs. | |||
For 1L+ys where y>1, I know I start with s, which means all of them go at the front and L goes at the end: s..sL. xL+1s is L..Ls. | |||
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 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. | |||
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. | |||
< | ==Extending to non-MOS== | ||
<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> | |||
</ | <span style="line-height: 1.5;">If the scale is made of xL+ys+zM, and any two of x, y, or z are the same, and they're larger than the third number, then the scale starts with the larger of the steps -- just like meantone major starts with larger steps. In other words, if it's 3L+3s+2M, we start with L. If it's 2L+3s+3M, we start with M.</span> | ||
</ | (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. | ||
< | 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. | ||
</ | Melodic minor ascending is also already structured that way: A B C D E F# G# A is LsLLLLs. | ||
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. | ||
</ | 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. | ||
</ | What Kite calls "a pentatonic scale" on the wiki page he just made is C D E G A#, or LLssL. That would be LLLss, and Kite's pentatonic would be mode 2 of that scale. | ||
</ | 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.) | ||
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.) | |||
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. | |||
* http://x31eq.com/cgi-bin/scala.cgi?ets=12_31_22&limit=11&tuning=po | |||
Admins: Please delete the blank page at http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position | |||
<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"><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> | |||
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<strong>This page is a work in progress...</strong><br /> | |||
Question: number MODMOS modes by position in compacted or uncompacted genchain?<br /> | |||
Question: resolve the ambiguity for MODMOS mode numbers?<br /> | |||
<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 /> | |||
<table class="wiki_table"> | |||
<tr> | <tr> | ||
<td> | <td>old scale name<br /> | ||
</td> | </td> | ||
<td> | <td>new scale name<br /> | ||
</td> | </td> | ||
<td> | <td>Ls pattern<br /> | ||
</td> | </td> | ||
<td>C D E F G A B C<br /> | <td>example on white keys<br /> | ||
</td> | |||
<td>genchain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Lydian<br /> | |||
</td> | |||
<td>1st Meantone [7]<br /> | |||
</td> | |||
<td>LLLs LLs<br /> | |||
</td> | |||
<td>F G A B C D E F<br /> | |||
</td> | |||
<td><u><strong>F</strong></u> C G D A E B<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Ionian (major)<br /> | |||
</td> | |||
<td>2nd Meantone [7]<br /> | |||
</td> | |||
<td>LLsL LLs<br /> | |||
</td> | |||
<td>C D E F G A B C<br /> | |||
</td> | </td> | ||
<td>F <u><strong>C</strong></u> G D A E B<br /> | <td>F <u><strong>C</strong></u> G D A E B<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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:26:&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:26 --><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:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:27:&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:27 --><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: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:28:&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:28 --><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:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:29:&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:29 --><u>Explanation / Rationale</u></h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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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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:14:&lt;h3&gt; --><h3 id="toc7"><a name="Kite Giedraitis method-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:14 --><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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Kite Giedraitis method-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | ||
<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:18 --><u>Why not always choose the chroma-positive generator?</u></h3> | ||
<br /> | <br /> | ||
See below.<br /> | See below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>Why not just use UDP notation?</strong></u></h3> | ||
<br /> | <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 /> | 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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<tr> | <tr> | ||
<td>Meantone [11]<br /> | <td>Meantone [11]<br /> | ||
</td> | </td> | ||
<td>???<br /> | <td>???<br /> | ||
</td> | </td> | ||
<td>C G D A E B F# C# G# D# A#<br /> | <td>C G D A E B F# C# G# D# A#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Meantone [12] if generator &lt; 700¢<br /> | <td>Meantone [12] if generator &lt; 700¢<br /> | ||
</td> | </td> | ||
<td>E# A# D# G# C# F# B E A D G C<br /> | <td>E# A# D# G# C# F# B E A D G C<br /> | ||
</td> | </td> | ||
<td>C G D A E B F# C# G# D# A# E#<br /> | <td>C G D A E B F# C# G# D# A# E#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: left;">Meantone [12] if generator &gt; 700¢<br /> | <td style="text-align: left;">Meantone [12] if generator &gt; 700¢<br /> | ||
</td> | </td> | ||
<td>C G D A E B F# C# G# D# A# E#<br /> | <td>C G D A E B F# C# G# D# A# E#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C G D A E B F# C# G# D# A# E#<br /> | <td style="text-align: center;">C G D A E B F# C# G# D# A# E#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
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An even larger problem is that the notation is overly tuning-dependent. Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth[12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant[12] ambiguous. For example &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant[12]&quot; or &quot;9th Dominant[12]&quot;. Something similar happens with Meantone[19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone[19] ambiguous. Another example is Dicot[7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore[5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<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 /> | |||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Jake Freivald method"></a><!-- ws:end:WikiTextHeadingRule:22 -->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 /> | |||
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I've created a method that uses step sizes, generally identified as s for small, M for medium, and L for large. (That would need to be extended for scales with more than three sizes of steps, of course, but the principle remains the same.)<br /> | |||
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We'll start with MOS scales. For MOS scales, we'll only have s and L steps sizes.<br /> | |||
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Inversion seems important for intuition. If 5L+2s is LLsLLLs, 5s+2L should be ssLsssL.<br /> | |||
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What if the algorithm were something like this:<br /> | |||
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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 /> | |||
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For 5L+2s, which is some rotation of LLsLLsL, I know I have to start with L. The smallest cluster I have is LL, so I get LLsLLLs.<br /> | |||
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For 1L+ys where y&gt;1, I know I start with s, which means all of them go at the front and L goes at the end: s..sL. xL+1s is L..Ls.<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 /> | |||
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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 /> | |||
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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 /> | |||
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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 /> | |||
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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 /> | |||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Jake Freivald method-Extending to non-MOS"></a><!-- ws:end:WikiTextHeadingRule:24 -->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 /> | |||
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<span style="line-height: 1.5;">If the scale is made of xL+ys+zM, and any two of x, y, or z are the same, and they're larger than the third number, then the scale starts with the larger of the steps -- just like meantone major starts with larger steps. In other words, if it's 3L+3s+2M, we start with L. If it's 2L+3s+3M, we start with M.</span><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 /> | |||
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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 /> | |||
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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 /> | |||
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Melodic minor ascending is also already structured that way: A B C D E F# G# A is LsLLLLs.<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 /> | |||
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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 /> | |||
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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 /> | |||
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What Kite calls &quot;a pentatonic scale&quot; on the wiki page he just made is C D E G A#, or LLssL. That would be LLLss, and Kite's pentatonic would be mode 2 of that 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 /> | |||
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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 /> | |||
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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 /> | |||
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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 /> | |||
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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 /> | |||
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