Comparison of mode notation systems: Difference between revisions

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For example, Meantone [5] is generated by 3/2, not 4/3. Because 5 fifths take one down a semitone, not up, the generator is chroma-negative, and the modes proceed from flatter to sharper. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C.

Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.

[[MODMOS Scales|MODMOS]] scales are named as chromatic alterations of a MOS scale, similar to UDP notation. "#" means raised by L-s. For //some-temperament-name//[N], "#" means moved N steps on the genchain, forwards if the generator is chroma-positive, otherwise backwards.

A pentatonic scale like C D E G A# is written 1st Meantone [5] #6. Scale degrees are heptatonic not pentatonic (#6 not #5) because while the scale is pentatonic, the notation uses 7 letters and is inherently heptatonic. If the scale were written H J K L #M, one would use #5.

==[[#How to name rank-2 scales-Non-MOS scales]]**__Rank-2 scales that are neither MOS nor MODMOS__**==  

One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they could be named Meantone [6], Meantone [8], etc. However chromatic modifications create genchains with gaps that are very difficult to name. Instead, these scales are named as altered MOS scales, using "add" and "no" to add or subtract notes, analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale/mode.  

Reserving the //name//[//number//] format for only MOS and MODMOS scales has the advantage of identifying what scale sizes can be MOS in unfamiliar temperaments. For example, Porcupine [8] is MOS but Porcupine [9] isn't. Writing Porcupine [9] as an altered Porcupine [8] indicates this.

The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be "F 1st Meantone [5] no5 add b6" and "A 3rd Meantone [5] no4 no7 add #5, #2".

G# D# A# E# B#  

FxCx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G#

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 "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". 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.

&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kite Giedraitis method-Rank-2 scales that are neither MOS nor MODMOS"&gt;Rank-2 scales that are neither MOS nor MODMOS&lt;/a&gt;&lt;/div&gt;

For example, Meantone [5] is generated by 3/2, not 4/3. Because 5 fifths take one down a semitone, not up, the generator is chroma-negative, and the modes proceed from flatter to sharper. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C.&lt;br /&gt;

Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation"&gt;ups and downs notation&lt;/a&gt;. Because the generator is a 2nd, the genchain resembles the scale.&lt;br /&gt;

&lt;a class="wiki_link" href="/MODMOS%20Scales"&gt;MODMOS&lt;/a&gt; scales are named as chromatic alterations of a MOS scale, similar to UDP notation. &amp;quot;#&amp;quot; means raised by L-s. For &lt;em&gt;some-temperament-name&lt;/em&gt;[N], &amp;quot;#&amp;quot; means moved N steps on the genchain, forwards if the generator is chroma-positive, otherwise backwards.&lt;br /&gt;

A pentatonic scale like C D E G A# is written 1st Meantone [5] #6. Scale degrees are heptatonic not pentatonic (#6 not #5) because while the scale is pentatonic, the notation uses 7 letters and is inherently heptatonic. If the scale were written H J K L #M, one would use #5.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Kite Giedraitis method-Rank-2 scales that are neither MOS nor MODMOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:49:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&amp;quot; title=&amp;quot;Anchor: How to name rank-2 scales-Non-MOS scales&amp;quot;/&amp;gt; --&gt;&lt;a name="How to name rank-2 scales-Non-MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:49 --&gt;&lt;strong&gt;&lt;u&gt;Rank-2 scales that are neither MOS nor MODMOS&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;

One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they could be named Meantone [6], Meantone [8], etc. However chromatic modifications create genchains with gaps that are very difficult to name. Instead, these scales are named as altered MOS scales, using &amp;quot;add&amp;quot; and &amp;quot;no&amp;quot; to add or subtract notes, analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale/mode. &lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

         &lt;td&gt;A B C D D# E F G# A&lt;br /&gt;

         &lt;td&gt;F C * D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * G# D#&lt;br /&gt;

         &lt;td&gt;A 5th Meantone [7] #7 add #4&lt;br /&gt;

         &lt;td&gt;A B C D D# E G# A&lt;br /&gt;

         &lt;td&gt;C * D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * G# D#&lt;br /&gt;

         &lt;td&gt;A 5th Meantone [7] #7 add #4 no6&lt;br /&gt;

         &lt;td&gt;nonotonic:&lt;br /&gt;

         &lt;td&gt;A B C# D D# E F# G G# A&lt;br /&gt;

         &lt;td&gt;F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * G# D#&lt;br /&gt;

         &lt;td&gt;F 1st Meantone [7] no4&lt;br /&gt;

         &lt;td&gt;G A C D E F# G&lt;br /&gt;

        &lt;td&gt;C &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E * F#&lt;br /&gt;

        &lt;td&gt;G 2nd Meantone [7] no3&lt;br /&gt;

    &lt;/tr&gt;

    &lt;tr&gt;

        &lt;td&gt;pentatonic:&lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

         &lt;td&gt;F G A C E F&lt;br /&gt;

         &lt;td&gt;&lt;u&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;/u&gt; C G * A E&lt;br /&gt;

         &lt;td&gt;F 2nd Meantone [7] no4 no6&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;

         &lt;td&gt;F 1st Meantone [7] no4 no6&lt;br /&gt;

G# D# A# E# B# &lt;br /&gt;

FxCx Gx Dx Ax Ex&lt;br /&gt;

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 &amp;quot;Dominant 8|3&amp;quot; could mean either &amp;quot;4th Dominant [12]&amp;quot; or &amp;quot;9th Dominant [12]&amp;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.&lt;br /&gt;

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