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Circles of fifths for Fokker blocks, and merged blocks

🔗Gene Ward Smith <gwsmith@svpal.org>

8/11/2004 4:36:37 PM

Here are circles of approximate fifths, giving the deviation from a
pure fifth, for various 12-note, 5-limit Fokker blocks. These are in a
standardized order which makes the notional "wolf", the approximate
fifth connecting the end of the block in meantone order with the
beginning, the last one listed.

By a "merged" block I mean one where two of the commas have merged,
giving one more pure fifth than the other blocks with the given
commas. These clearly seem worth singling out, and they can serve as
boundries between two different kinds of Fokker blocks, and example
being syndia6 and schisdia6.

Using the circle of fifths also allows us to put these in a logical
order, which I do below.

Syndies

Merged
# Third Syndie 81/80 128/125 scale ~ duodene.scl = efg33355.scl
syndie3c := [1, 1, 1, 80/81, 1, 1, 1, 80/81, 1, 1, 1, 2048/2025];

# First Syndie 81/80 128/125 scale ~ sauveur_ji.scl

syndie1c := [1, 1, 80/81, 1, 1, 1, 80/81, 1, 1, 1, 80/81, 128/125];

# Fourth Syndie 81/80 128/125 scale = marpurg1.scl

syndie4c := [1, 80/81, 1, 1, 1, 80/81, 1, 1, 1, 80/81, 1, 128/125];

# Second Syndie 81/80 128/125 scale = fogliano1.scl

syndie2c := [80/81, 1, 1, 1, 80/81, 1, 1, 1, 80/81, 1, 1, 128/125];

Syndias

Merged
# First 81/80 2048/2025 Fokker block = ramis.scl

syndia1c := [1, 1, 1, 1, 1, 80/81, 1, 1, 1, 1, 1, 32768/32805];

# Fifth 81/80 2048/2025 Fokker block = pipedum_12.scl

syndia5c := [1, 1, 1, 1, 80/81, 1, 1, 1, 1, 1, 80/81, 2048/2025];

# Third 81/80 2048/2025 Fokker block

syndia3c := [1, 1, 1, 80/81, 1, 1, 1, 1, 1, 80/81, 1, 2048/2025];

# Second 81/80 2048/2025 Fokker block

syndia2c := [1, 1, 80/81, 1, 1, 1, 1, 1, 80/81, 1, 1, 2048/2025];

# Fourth 81/80 2048/2025 Fokker block

syndia4c := [1, 80/81, 1, 1, 1, 1, 1, 80/81, 1, 1, 1, 2048/2025];

# Sixth 81/80 2048/2025 Fokker block

syndia6c := [80/81, 1, 1, 1, 1, 1, 80/81, 1, 1, 1, 1, 2048/2025];

Diadies

# First Diadie 2048/2025 128/125 scale = lumma5r.scl

diadie1c:=[1, 125/128, 1, 2048/2025, 1, 1, 1, 80/81,
1, 1, 1, 2048/2025];

# Second Diadie 2048/2025 128/125 scale ~ pipedum_12a.scl

diadie2c := [125/128, 1, 1, 2048/2025, 1, 1, 125/128,
2048/2025, 1, 1, 1, 2048/2025];

Schisdias

Merged
# Sixth schisdia 32805/32768 2048/2025 scale ~ ramis tamil_vi syndia1

schisdia6c := [1, 1, 1, 1, 1, 80/81,
1, 1, 1, 1, 1, 32768/32805];

# First schisdia 32805/32768 2048/2025 scale

schisdia1c := [1, 1, 1, 1, 2025/2048, 32768/32805,
1, 1, 1, 1, 1, 32768/32805];

# Fourth schisdia 32805/32768 2048/2025 scale

schisdia4c := [1, 1, 1, 2025/2048, 1, 32768/32805,
1, 1, 1, 1, 1, 32768/32805];

# Fifth schisdia 32805/32768 2048/2025 scale

schisdia5c := [1, 1, 2025/2048, 1, 1, 32768/32805,
1, 1, 1, 1, 1, 32768/32805];

# Third schisdia 32805/32768 2048/2025 scale

schisdia3c := [1, 2025/2048, 1, 1, 1, 32768/32805,
1, 1, 1, 1, 1, 32768/32805];

# Second schisdia 32805/32768 2048/2025 scale

schisdia2c := [2025/2048, 1, 1, 1, 1, 32768/32805,
1, 1, 1, 1, 1, 32768/32805];

Majsyn

Merged
# Third Majsyn 648/625 81/80 scale ~ duodene_skew.scl

majsyn3c := [1, 1, 80/81, 1, 1, 80/81, 1, 1,
80/81, 1, 1, 128/125];

# Second Majsyn 648/625 81/80 scale

majsyn2c := [1, 80/81, 1, 1, 80/81, 1, 1, 80/81,
1, 1, 80/81, 648/625];

# First Majsyn 648/625 81/80 scale

majsyn1c := [80/81, 1, 1, 80/81, 1, 1, 80/81,
1, 1, 80/81, 1, 648/625];