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13edo: Difference between revisions

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13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.
13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.


Enharmonic intervals: v⁴A1, ^m2
Enharmonic unisons: v⁴A1, ^m2


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{| class="wikitable center-all right-2"
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The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".
The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".


The first approach has enharmonic intervals of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.
The first approach has Enharmonic unisons of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.


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{| class="wikitable center-all right-2"
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Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)
Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)


Enharmonic interval: dds3
Enharmonic unison: dds3
{| class="wikitable"
{| class="wikitable"
|+notes/intervals in melodic order (s = sub-, d = -oid)
|+notes/intervals in melodic order (s = sub-, d = -oid)
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Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)  
Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)  


Enharmonic interval: d2
Enharmonic unison: d2
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{| class="wikitable"
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|+notes/intervals in melodic order
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Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)
Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)


Enharmonic interval: dd2
Enharmonic unison: dd2
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{| class="wikitable"
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|+notes/intervals in melodic order
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Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)
Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)


Enharmonic intervals: vvA1, vm2
Enharmonic unisons: vvA1, vm2
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{| class="wikitable"
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|+notes/intervals in melodic order
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Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)
Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)


Enharmonic interval: ddd2
Enharmonic unison: ddd2
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{| class="wikitable"
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|+notes/intervals in melodic order
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A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.
A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.


[[File:Archaeotonic.png|alt=Archaeotonic.png|Archaeotonic.png]]
[[File:Archaeotonic.png|Archaeotonic.png|link=Special:FilePath/Archaeotonic.png]]


Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
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