Comparison of mode notation systems: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-22 03:28:02 UTC</tt>.<br>

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In order to be a MOS scale, the genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.

There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5, which is necessary to avoid duplicate note names.

Examples:

An even larger problem is that 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. In other words the notation is overly tuning-dependent. 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 note in N-edo to slightly sharp of it.

In order to be a MOS scale, the genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.&lt;br /&gt;

There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5, which is necessary to avoid duplicate note names.&lt;br /&gt;

  Examples:&lt;br /&gt;

An even larger problem is that 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. In other words the notation is overly tuning-dependent. 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 note in N-edo to slightly sharp of it.&lt;br /&gt;