Comparison of mode notation systems: Difference between revisions

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==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**==  

To find a [[MODMOS Scales|MODMOS]] 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. //[This may change]// 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.

Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:

Melodic minor modes:

==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**==  

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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Naming Rank-2 Scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; &lt;/h1&gt;

  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;&lt;span style="font-size: 1.3em; line-height: 1.5;"&gt;Proposed method of naming all possible rank-2 scales&lt;/span&gt;&lt;/u&gt;&lt;/h2&gt;

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Meantone [7,+3,-6] means that the 3rd note in the &lt;u&gt;compacted&lt;/u&gt; genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. The alterations are the exact opposite of the alterations needed to close the gaps in the uncompacted genchain. &amp;quot;+&amp;quot; and &amp;quot;-&amp;quot; are preferred over &amp;quot;#&amp;quot; and &amp;quot;b&amp;quot; because in the case of a chroma-negative generator, &amp;quot;+&amp;quot; makes the note flatter, as in the last example:&lt;br /&gt;

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Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:&lt;br /&gt;

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Melodic minor modes:&lt;br /&gt;

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  Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:&lt;br /&gt;

Eb -- Bb -- F --- C --- G&lt;br /&gt;

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  Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples:&lt;br /&gt;

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Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.&lt;br /&gt;

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.&lt;br /&gt;

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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:&lt;br /&gt;

&amp;quot;A foolish consistency is the hobgoblin of little minds&amp;quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike &lt;u&gt;wise&lt;/u&gt; consistencies, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.&lt;br /&gt;

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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.)&lt;br /&gt;

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See below.&lt;br /&gt;

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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.&lt;br /&gt;

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 &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow"&gt;zero-based counting&lt;/a&gt; and Mode Numbers notation uses the more intuitive one-based counting.&lt;br /&gt;

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  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&lt;br /&gt;

(1) as little knowledge needed as possible, to help the less-sophisticated user,&lt;br /&gt;

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[7] gives me the diatonic major scale LLsLLLs, or CDEFGABC.&lt;br /&gt;

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  &lt;span style="line-height: 1.5;"&gt;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. &lt;/span&gt;&lt;br /&gt;

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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.&lt;br /&gt;

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