Caspar Johannes Walter

Meantone Circles (2016)

Models for systematical structures of meantone and other tunings

I - The divisions of octave, allocated to the primary meantone tunings 19, 31 and 50

Macro structure:

The octave consists of two semitones (a) and five full tones (b) of the same size

a and b follow consecutive pairs of numbers of the Fibonacci-sequence.

[12-tone circle; circle of fifths from A♭ to C#]; {values for a and b: 1,2}
19-tone circle; circle of fifths from G♭ to B#; corresponds to the $\frac{1}{3}$ -comma tuning; {2,3}; approaches quasi perfectly the pure minor third (ratio $\frac{6}{5}$ )
31-tone circle; circle of fifths from G♭♭ to A##; corresponds to the $\frac{1}{4}$ -comma tuning; {3,5};approaches quasi perfectly the pure major third (ratio $\frac{5}{4}$ )
50-tone circle; circle of fifths from F♭♭♭ to F####; corresponds to the $\frac{2}{7}$ -comma tuning; {5,8}; approaches quasi perfectly the chromatic semitone (ratio $\frac{25}{24}$ ) and provides a good balance between the for the primary meantone tuning most important intervals of the major and the minor third.

II - The divisions of octave, allocated to the secondary meantone tunings 43 and 55

Macro structure:

The octave consists of two semitones (a) and five full tones (b) of the same size

a and b follow first consecutive pairs of numbers of the Fibonacci-sequence (circles 12, 19, 31). Later always 12 is added the the actual final number of the sequence of circles. So, the value of the fifth is always improved, but the values of the the thirds are lossing quality.

[12-tone circle; circle of fifths from A♭ to C#]; {values for a and b: 1,2}
[19-tone circle; circle of fifths from G♭ to B#]; corresponds to the $\frac{1}{3}$ -comma tuning; {2,3}; approaches quasi perfectly the pure minor third (ratio $\frac{6}{5}$ )
[31-tone circle; circle of fifths from G♭♭ to A##]; corresponds to the $\frac{1}{4}$ -comma tuning; {3,5};approaches quasi perfectly the pure major third (ratio $\frac{5}{4}$ )
43-tone circle; circle of fifths from A♭♭♭ to A###; corresponds to the $\frac{1}{5}$ -comma tuning; {4,7}; approaches quasi perfectly the diatonic major seventh (ratio $\frac{15}{8}$ ) and provides a good balance between the for the primary meantone tuning most important intervals of the major and the minor third.
55-tone circle; circle of fifths from A♭♭♭♭ bis F####; corresponds to the $\frac{1}{6}$ -comma tuning; {5,9}; approaches quasi perfectly the diatonic tritone (ratio $\frac{45}{32}$ ).

III - The divisions of octave, with the best approxiamtion to the perfect fifths

The octave consists of two semitones (a) and five full tones (b) of the same size.

First a circle of 12 is set (values for a and b: 1,2). Later the formula $b = 2a + 1$ is used to enlarge the tempered fith with the aime of approaching perfect fith (ratio $\frac{3}{2}$ ). For the circles of 17, 29 and 43 the fiths are still too large, for 53 it is fairly perfect.

[12-tone circle; circle of fifths from A♭ to C#]; {values for a and b: 1,2}
[17-tone circle; circle of fifths from A♭ bis B#]; {1,3}
[29-tone circle; circle of fifths from A♭♭ bis A##]; {2,5}
41-tone circle; circle of fifths from B♭♭♭ bis A###; {3,7}
53-tone circle; circle of fifths from A♭♭♭♭ to G####; {4,9}; tuning of fifths with quasi syntonic comma as smallest step

comment

Another formula leading to the 53-circle is $2a + 5b + 3$

This is a formula truly expressing the idea of a limit-5 just intonation tuning, where the octave consists of two (diatonic) semitones (ratio $\frac{15}{8}$ ), three large full tones (ratio $\frac{9}{8}$ ) and two small full tones (ratio $\frac{10}{9}$ ). When pairs of Fibonacci-numpers are pushed into the formula, circles of 15, 22, 34 and 53 are appearing. With the 53-circle the smallest step is close to a syntonic comma ( $\frac{81}{80}$ , the differenence between $\frac{9}{8}$ and $\frac{10}{9}$ ).

IV - comparison of the circles 50, 53 and 55

V - Appendix 1: note-names of the circles, sorted by chains of fifths

The primary meantone tunings 12, 19, 31 and 50 (sorted chromatically)

12-tone circle; circle of fifths from A♭ to C#

["C♮", "C#", "D♮", "E♭", "E♮", "F♮", "F#", "G♮", "A♭", "A♮", "B♭", "B♮"]

19-tone circle; circle of fifths from G♭ to B#

["C♮", "C#", "D♭", "D♮", "D#", "E♭", "E♮", "E#", "F♮", "F#", "G♭", "G♮", "G#", "A♭", "A♮", "A#", "B♭", "B♮", "B#"]

31-tone circle; circle of fifths from G♭♭ to A##

["C♮", "D♭♭", "C#", "D♭", "C##", "D♮", "E♭♭", "D#", "E♭", "D##", "E♮", "F♭", "E#", "F♮", "G♭♭", "F#", "G♭", "F##", "G♮", "A♭♭", "G#", "A♭", "G##", "A♮", "B♭♭", "A#", "B♭", "A##", "B♮", "C♭", "B#"]

50-tone circle; circle of fifths from F♭♭♭ to F####

["C♮", "B##", "D♭♭", "C#", "E♭♭♭♭", "D♭", "C##", "E♭♭♭", "D♮", "C###", "E♭♭", "D#", "F♭♭♭", "E♭", "D##", "F♭♭", "E♮", "D###", "F♭", "E#", "G♭♭♭", "F♮", "E##", "G♭♭", "F#", "E###", "G♭", "F##", "A♭♭♭", "G♮", "F###", "A♭♭", "G#", "B♭♭♭♭", "A♭", "G##", "B♭♭♭", "A♮", "G###", "B♭♭", "A#", "C♭♭♭", "B♭", "A##", "C♭♭", "B♮", "A###", "C♭", "B#", "D♭♭♭"]

The primary meantone tunings 12, 19, 31 and 50 (sorted by fifths)

12-tone circle; circle of fifths from A♭ to C#

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "A♭", "E♭", "B♭", "F♮"]

19-tone circle; circle of fifths from G♭ to B#

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

31-tone circle; circle of fifths from G♭♭ to A##

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

50-tone circle; circle of fifths from F♭♭♭ to F####

The secondary meantone tunings 43, 55 and 67 (sorted chromatically)

43-tone circle; circle of fifths from C♭♭♭ to C###

["C♮", "D♭♭", "B##", "C#", "D♭", "E♭♭♭", "C##", "D♮", "E♭♭", "C###", "D#", "E♭", "F♭♭", "D##", "E♮", "F♭", "G♭♭♭", "E#", "F♮", "G♭♭", "E##", "F#", "G♭", "A♭♭♭", "F##", "G♮", "A♭♭", "F###", "G#", "A♭", "B♭♭♭", "G##", "A♮", "B♭♭", "C♭♭♭", "A#", "B♭", "C♭♭", "A##", "B♮", "C♭", "D♭♭♭", "B#"]

55-tone circle; circle of fifths from G♭♭♭♭ to F####

["C♮", "D♭♭", "E♭♭♭♭", "B##", "C#", "D♭", "E♭♭♭", "B###", "C##", "D♮", "E♭♭", "F♭♭♭", "C###", "D#", "E♭", "F♭♭", "G♭♭♭♭", "D##", "E♮", "F♭", "G♭♭♭", "D###", "E#", "F♮", "G♭♭", "A♭♭♭♭", "E##", "F#", "G♭", "A♭♭♭", "E###", "F##", "G♮", "A♭♭", "B♭♭♭♭", "F###", "G#", "A♭", "B♭♭♭", "F####", "G##", "A♮", "B♭♭", "C♭♭♭", "G###", "A#", "B♭", "C♭♭", "D♭♭♭♭", "A##", "B♮", "C♭", "D♭♭♭", "A###", "B#"]

67-tone circle; circle of fifths from B♭♭♭♭♭♭ to G####; [not yet in the right order]

["C♮", "B##", "E♭♭♭♭♭", "D♭♭", "C#", "B###", "E♭♭♭♭", "D♭", "C##", "F♭♭♭♭♭", "E♭♭♭", "D♮", "C###", "F♭♭♭♭", "E♭♭", "D#", "C####", "F♭♭♭", "E♭", "D##", "G♭♭♭♭♭", "F♭♭", "E♮", "D###", "G♭♭♭♭", "F♭", "E#", "G♭♭♭", "F♮", "E##", "A♭♭♭♭♭", "G♭♭", "F#", "E###", "A♭♭♭♭", "G♭", "F##", "B♭♭♭♭♭♭", "A♭♭♭", "G♮", "F###", "B♭♭♭♭♭", "A♭♭", "G#", "F####", "B♭♭♭♭", "A♭", "G##", "C♭♭♭♭♭", "B♭♭♭", "A♮", "G###", "C♭♭♭♭", "B♭♭", "A#", "G####", "C♭♭♭", "B♭", "A##", "D♭♭♭♭♭", "C♭♭", "B♮", "A###", "D♭♭♭♭", "C♭", "B#", "D♭♭♭"]

The secondary meantone tunings 43, 55 and 67 (sorted by fifths)

43-tone circle; circle of fifths from C♭♭♭ to C###

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "E##", "B##", "F###", "C###", "C♭♭♭", "G♭♭♭", "D♭♭♭", "A♭♭♭", "E♭♭♭", "B♭♭♭", "F♭♭", "C♭♭", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

55-tone circle; circle of fifths from G♭♭♭♭ to F####

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "E##", "B##", "F###", "C###", "G###", "D###", "A###", "E###", "B###", "F####", "G♭♭♭♭", "D♭♭♭♭", "A♭♭♭♭", "E♭♭♭♭", "B♭♭♭♭", "F♭♭♭", "C♭♭♭", "G♭♭♭", "D♭♭♭", "A♭♭♭", "E♭♭♭", "B♭♭♭", "F♭♭", "C♭♭", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

67-tone circle; circle of fifths from B♭♭♭♭♭♭ to G####; [not yet corrected]

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "E##", "B##", "F###", "C###", "G###", "D###", "A###", "E###", "B###", "F####", "C####", "G####", "B♭♭♭♭♭♭", "F♭♭♭♭♭", "C♭♭♭♭♭", "G♭♭♭♭♭", "D♭♭♭♭♭", "A♭♭♭♭♭", "E♭♭♭♭♭", "B♭♭♭♭♭", "F♭♭♭♭", "C♭♭♭♭", "G♭♭♭♭", "D♭♭♭♭", "A♭♭♭♭", "E♭♭♭♭", "B♭♭♭♭", "F♭♭♭", "C♭♭♭", "G♭♭♭", "D♭♭♭", "A♭♭♭", "E♭♭♭", "B♭♭♭", "F♭♭", "C♭♭", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

The tunings 17, 29, 41 and 53 (sorted chromatically)

17-tone circle; circle of fifths from G♭ to A#

["C♮", "D♭", "C#", "D♮", "E♭", "D#", "E♮", "F♮", "G♭", "F#", "G♮", "A♭", "G#", "A♮", "B♭", "A#", "B♮"]

29-tone circle; circle of fifths from A♭♭ to A##

["C♮", "B#", "D♭", "C#", "E♭♭", "D♮", "C##", "E♭", "D#", "F♭", "E♮", "G♭♭", "F♮", "E#", "G♭", "F#", "A♭♭", "G♮", "F##", "A♭", "G#", "B♭♭", "A♮", "G##", "B♭", "A#", "C♭", "B♮", "A##"]

41-tone circle; circle of fifths from A♭♭♭ to G###

["C♮", "B#", "E♭♭♭", "D♭", "C#", "B##", "E♭♭", "D♮", "C##", "F♭♭", "E♭", "D#", "C###", "F♭", "E♮", "D##", "G♭♭", "F♮", "E#", "A♭♭♭", "G♭", "F#", "E##", "A♭♭", "G♮", "F##", "B♭♭♭", "A♭", "G#", "F###", "B♭♭", "A♮", "G##", "C♭♭", "B♭", "A#", "G###", "C♭", "B♮", "A##", "D♭♭"]

53-tone circle; circle of fifths from D♭♭♭♭ bis B###

["C♮", "B#", "A###", "E♭♭♭", "D♭", "C#", "B##", "F♭♭♭", "E♭♭", "D♮", "C##", "B###", "F♭♭", "E♭", "D#", "C###", "G♭♭♭", "F♭", "E♮", "D##", "A♭♭♭♭", "G♭♭", "F♮", "E#", "D###", "A♭♭♭", "G♭", "F#", "E##", "B♭♭♭♭", "A♭♭", "G♮", "F##", "E###", "B♭♭♭", "A♭", "G#", "F###", "C♭♭♭", "B♭♭", "A♮", "G##", "D♭♭♭♭", "C♭♭", "B♭", "A#", "G###", "D♭♭♭", "C♭", "B♮", "A##", "E♭♭♭♭", "D♭♭"]

The tunings 17, 29, 41 and 53 (sorted by fifths)

17-tone circle; circle of fifths from G♭ to A#

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

29-tone circle; circle of fifths from A♭♭ to A##

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

41-tone circle; circle of fifths from A♭♭♭ to G###

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "C##", "G##", "D##", "A##", "E##", "B##", "F###", "C###", "G###", "A♭♭♭", "E♭♭♭", "B♭♭♭", "F♭♭", "C♭♭", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

53-tone circle; circle of fifths from D♭♭♭♭ bis B###

53-tone circles with schismatic changes

53-tone circle, 1 Mercator-comma + 2 Schismae: 13, 1, 12 // 12, 1, 13

c - fisis (13 fifths); fisis - D+1 (1 schismatic diminished fifth);

d+1- cisis+1 (12 fifths); cisis+1 - ceses-1 (Mercator-comma);

ceses-1 – B♭-1 (12 fifths); B♭-1 - geses ( 1 schismatic diminished fifth));

geses - c (13 fifths)

sorted by fifths:

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "A#", "E#", "B#", "F##", "D♮"+1, "A♮+1", "E♮+1", "B♮+1", "F#+1", "C#+1", "G#+1", "D#+1", "A#+1", "E#+1", "B#+1", "F##+1", "C##+1", "C♭♭-1", "G♭♭-1", "D♭♭-1", "A♭♭-1", "E♭♭-1", "B♭♭-1", "F♭-1", "C♭-1", "G♭-1", "D♭-1", "A♭-1", "E♭-1", "B♭-1", "G♭♭", "D♭♭", "A♭♭", "E♭♭", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

53-tone circle, 1 Mercator-comma + 2 Schismae: 9, 1, 17 // 17, 1, 9

c – dis; b+1 – aisis+1 //eseses-1 - d-1; heses – c

sorted by fifths:

["C♮", "G♮", "D♮", "A♮", "E♮", "B♮", "F#", "C#", "G#", "D#", "B♭+1", "F♮+1", "C♮+1", "G♮+1", "D♮"+1, "A♮+1", "E♮+1", "B♮+1", "F#+1", "C#+1", "G#+1", "D#+1", "A#+1", "E#+1", "B#+1", "F##+1", "C##+1", "G##+1", "D##+1", "A##+1", "E♭♭♭-1", "B♭♭♭-1", "F♭♭-1", "C♭♭-1", "C♭♭-1", "G♭♭-1", "D♭♭-1", "A♭♭-1", "E♭♭-1", "B♭♭-1", "F♭-1", "C♭-1", "G♭-1", "D♭-1", "A♭-1", "E♭-1", "B♭-1", "F♮-1", "C♮-1", "G♮-1", "D♮-1", "B♭♭", "F♭", "C♭", "G♭", "D♭", "A♭", "E♭", "B♭", "F♮"]

53-tone circle, 1 Mercator-comma + 4 Schismae

c – fis; des+1 – cis+1; as+2 – d+2 //b-2 – e-2; ces-1 - h-1; ges – c

sorted by chromatically:

["C♮", "C♮+1", "C♮+2", "D♭-1", "D♭", "D♭+1", "C#+1", "D♮-2", "D♮-1", "D♮", "D♮+1", "D♮+2", "E♭-1", "E♭", "E♭+1", "E♭+2", "E♮-2", "E♮-1", "E♮", "E♮+1", "F♮-2", "F♮-1", "F♮", "F♮+1", "F♮+2", "G♭-1", "G♭", "F#", "F#+1", "G♮-2", "G♮-1", "G♮", "G♮+1", "G♮+2", "A♭-1", "A♭", "A♭+1", "A♭+2", "A♮-2", "A♮-1", "A♮", "A♮+1", "B♭-2", "B♭-1", "B♭", "B♭+1", "B♭+1", "C♭-1", "B♮-1", "B♮", "B♮+1", "C♮-2", "C♮-1"]

5 Schismae: 10, 11, 5+1+5, 11, 10

c – h; ges+1 – e+1; ces+2 – d+2 // b-2 – cis-2; as-1 - fis-1; des – c

6 Schismae: 4,9,9,4+1+4,9,9,4

c – e; ces+1 – g+1; eses+2 – b+2; geses+3 – es+3 // ais-3 – fisis-3; d-2 – ais-2; f-1 - cis-1; as – c

5,8,9,4+1+4,9,8,5

c – e; ces+1 – g+1; eses+2 – es+2; ceses+3 – es+3 // ais-3 – cisis-3; a-2 – ais-2; f-1 - cis-1; as – c

sorted by fifths:

["C♮", "G♮", "D♮", "A♮", "E♮", "C♭+1", "G♭+1", "D♭+1", "A♭+1", "E♭+1", "B♭+1", "F♮+1", "C♮+1", "G♮+1", "E♭♭+2", "B♭♭+2", "F♭+2", "C♭+2", "G♭+2", "D♭+2", "A♭+2", "E♭+2", "C♭♭+3", "G♭♭+3", "D♭♭+3", "A♭♭+3", "E♭♭+3", "A#-3", "E#-3", "B#-3", "F#-3", "C#-3", "A♮-2", "E♮-2", "B♮-2", "F#-2", "C#-2", "G#-2", "D#-2", "A#-2", "F♮-1", "C♮-1", "G♮-1", "D♮-1", "A♮-1", "E♮-1", "B♮-1", "F#-1", "C#-1", "A♭", "E♭", "B♭", "F♮"]

This circle could be compared with the $\frac{1}{4}$ -comma circle.

VI - Appendix 2: circles, sorted by fifths

The primary meantone tunings 12, 19, 31 and 50

The secondary meantone tunings [12], [19], [31], 43 and 55

The tunings [12], 17, 29, 41 and 53

53-tone circles with 6, 4, 2, 0 schismatic changes