Comparison of mode notation systems: Difference between revisions

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These [[MOSScales|MOS scales]] are formed from a segment of the [[periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as "fourth meantone heptatonic" or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".

Sensi[8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)

To find the 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. For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. "+" and "-" are used to indicate the altered notes. For example, Meantone[7,+3,-6] means that the 3rd note in the compacted genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. "+" and "-" are preferred over "#" and "b" because in the case of a chroma-negative generator, "+" makes the note flatter.

|| Harmonic minor || A B C D E F G# A || F C * D __**A**__ E B * * G# || F C G D __**A**__ E B || 5th Meantone[7,+3] ||

|| Ascending melodic minor || A B C D E F# G# A || C * D __**A**__ E B F# * G# || F C G D __**A**__ E B || 5th Meantone[7,+1,+3] ||

||= " ||= " ||= " || C G D __**A**__ E B F# || 4th Meantone[7,+2] ||

||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-6] ||

|| Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || F C G D __**A**__ E B || 5th Meantone[7,+3,+4] ||

||= " ||= " ||= " || __**A**__ E B F# C# G# D# || 1st Meantone[7,-4,-5] ||

|| Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || Bb F C G D __**A**__ E || 6th Meantone[7,+3,+4] ||

||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-4,-5] ||

|| &lt;span class="mw-redirect"&gt;Hungarian gypsy &lt;/span&gt;minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || F C G D __**A**__ E B || 5th Meantone[7,+4] ||

|| Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone[7,+3] ||

|| Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5,-4,-5] ||

|| (a mode of the above) || F A B C E F || __**F**__ C * * A E B || A E B __**F#**__ C# || 4th Meantone[5,-4,-5] ||

In the Japanese pentatonic scale, "[5,-4,-5]" moves the notes 7 steps in the genchain, not 5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written "b5".

 

The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.

C D E F F# G A B C has a genchain F __**C**__ G D A E B F#, and is named C 2nd Meantone[8].

The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The disadvantage of genchain-based numbering is that the mode numbers are harder to find in the scale. However this is arguably an advantage, because in the course of

Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show [emphasis mine]:

"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike __wise__ consistencies, it doesn't reduce memorization, because everyone already knows that the generator is historically 3/2.

A D G C ||= 3/2 || C G D A E B F# C# G#  

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

D# A# E# B# Fx Cx Gx  

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 N-edo interval to slightly sharp of it.

These &lt;a class="wiki_link" href="/MOSScales"&gt;MOS scales&lt;/a&gt; are formed from a segment of the &lt;a class="wiki_link" href="/periods%20and%20generators"&gt;generator-chain&lt;/a&gt;, or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as &amp;quot;fourth meantone heptatonic&amp;quot; or possibly &amp;quot;fourth meantone seven&amp;quot;. If in D, as above, it would be &amp;quot;D fourth meantone heptatonic&amp;quot;.&lt;br /&gt;

Sensi[8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)&lt;br /&gt;

  To find the 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 &lt;u&gt;compacted&lt;/u&gt; genchain. For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. &amp;quot;+&amp;quot; and &amp;quot;-&amp;quot; are used to indicate the altered notes. For example, Meantone[7,+3,-6] means that the 3rd note in the compacted genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. &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.&lt;br /&gt;

         &lt;td&gt;5th Meantone[7,+3]&lt;br /&gt;

         &lt;td&gt;5th Meantone[7,+1,+3]&lt;br /&gt;

         &lt;td&gt;4th Meantone[7,+2]&lt;br /&gt;

         &lt;td&gt;2nd Meantone[7,-6]&lt;br /&gt;

         &lt;td&gt;5th Meantone[7,+3,+4]&lt;br /&gt;

         &lt;td&gt;1st Meantone[7,-4,-5]&lt;br /&gt;

         &lt;td&gt;6th Meantone[7,+3,+4]&lt;br /&gt;

         &lt;td&gt;2nd Meantone[7,-4,-5]&lt;br /&gt;

         &lt;td&gt;5th Meantone[7,+4]&lt;br /&gt;

         &lt;td&gt;6th Meantone[7,+3]&lt;br /&gt;

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

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

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

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

         &lt;td&gt;1st Meantone[5,-4,-5]&lt;br /&gt;

         &lt;td&gt;(a mode of the above)&lt;br /&gt;

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

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

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

         &lt;td&gt;4th Meantone[5,-4,-5]&lt;br /&gt;

In the Japanese pentatonic scale, &amp;quot;[5,-4,-5]&amp;quot; moves the notes 7 steps in the genchain, not 5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written &amp;quot;b5&amp;quot;.&lt;br /&gt;

&lt;em&gt;A point to consider:&lt;/em&gt;&lt;br /&gt;

&lt;br /&gt;

The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.&lt;br /&gt;

C D E F F# G A B C has a genchain F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F#, and is named C 2nd Meantone[8].&lt;br /&gt;

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

Genchain-based: genchain = 1 2 3 4 5 6 7, Lydian scale = 1 3 5 7 2 4 6 1, major scale = 2 4 6 1 3 5 7 2&lt;br /&gt;

Scale-based:&lt;br /&gt;

The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The disadvantage of genchain-based numbering is that the mode numbers are harder to find in the scale. However this is arguably an advantage, because in the course of&lt;br /&gt;

  Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show [emphasis mine]:&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 doesn't reduce memorization, because everyone already knows that the generator is historically 3/2.&lt;br /&gt;

  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;

  See below.&lt;br /&gt;

  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;

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

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

D# A# E# B# Fx Cx Gx &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 N-edo interval to slightly sharp of it.&lt;br /&gt;

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