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schismic tuning / schismatic tuning

[Joe Monzo]

According to Mark Lindley (Lindley/Turner-Smith 1993), from the evidence of musical usage in manuscripts of compositions, during the late 1300s and early 1400s composers began writing for keyboards tuned in a pythagorean chain in which the notes tuned as Gb:Db:Ab:Eb were actually used often as F#:C#:G#:D#.

This tuning, called "Quasi-Pythagorean" by Lindley and dubbed "schismic tuning" or "schismatic tuning" by other modern tuning theorists, assumes the equivalence of notes tuned a skhisma apart. Since this is only ~2 cents and therefore below the threshold of normal manual tuning error, these schismatic pairs of notes may indeed be assumed to be equivalent for all intents and purposes.

here is the scale as it was tuned, in a pythagorean chain:

  note    3^x    ~cents

(Gb)/"F#"  -6   588.2699948
(Db)/"C#"  -5    90.22499567
  Ab/"G#"  -4   792.1799965
  Eb/"D#"  -3   294.1349974
   Bb      -2   996.0899983
   F       -1   498.0449991
   C        0     0
   G        1   701.9550009
   D        2   203.9100017
   A        3   905.8650026
   E        4   407.8200035
   B        5  1109.775004

the scale was tuned 3^(-6 ... +5), but Lindley says
that Gb and Db were not in use at this time, so that's
why they appear in parentheses.


here it is in descending order of pitch, as a chromatic scale:


  note    3^x    ~cents

   B        5  1109.775
   Bb      -2   996.090
   A        3   905.865
  Ab/"G#"  -4   792.180
   G        1   701.955
(Gb)/"F#"  -6   588.270
   F       -1   498.045
   E        4   407.820
  Eb/"D#"  -3   294.135
   D        2   203.910
(Db)/"C#"  -5    90.225
   C        0     0.000
		

Below is a lattice-diagram of this tuning:

schismic tuning: lattice diagram

On this diagram, the 8 distinct notes are colored green, the schismatically-related pairs are orange, and the two notes which were half of a schismatic pair but were not used, are grey.

Below is a table listing the complete set of intervals between all pairs of notes in this tuning:

~cents instances
1110 F:E, Ab/"G#":G, Eb/"D#":D, Bb:A, "F#":F, "C#":C, C:B
1086 D:"C#", B:Bb, G:"F#", E:Eb/"D#", A:Ab/"G#"
1020 "F#":E, "C#":B
996.1 Eb/"D#":"C#", Ab/"G#":"F#", B:A, F:Eb/"D#", A:G, D:C, E:D, G:F, Bb:Ab/"G#", C:Bb
905.9 F:D, G:E, Ab/"G#":F, Bb:G, Eb/"D#":C, "F#":Eb/"D#", C:A, "C#":Bb, D:B,
882.4 E:"C#", A:"F#", B:Ab/"G#"
815.6 Ab/"G#":E, "F#":D, "C#":A, Eb/"D#":B
792.2 F:"C#", B:G, Bb:"F#", G:Eb/"D#", A:F, E:C, C:Ab/"G#", D:Bb
702 "F#":"C#", Ab/"G#":Eb/"D#", A:E, F:C, G:D, Bb:F, C:G, D:A, Eb/"D#":Bb, "C#":Ab/"G#", E:B
678.5 B:"F#"
611.7 Ab/"G#":D, Bb:E, "F#":C, Eb/"D#":A, "C#":G, F:B
588.3 G:"C#", B:F, A:Eb/"D#", C:"F#", D:Ab/"G#", E:Bb
521.5 "F#":B
498 Ab/"G#":"C#", B:E, A:D, Bb:Eb/"D#", G:C, C:F, D:G, F:Bb, "C#":"F#", Eb/"D#":Ab/"G#", E:A
407.8 Ab/"G#":C, Bb:D, C:E, Eb/"D#":G, F:A, "F#":Bb, "C#":F, G:B
384.4 A:"C#", B:Eb/"D#", D:"F#", E:Ab/"G#"
317.6 "C#":E, "F#":A, Ab/"G#":B
294.1 Bb:"C#", B:D, A:C, C:Eb/"D#", Eb/"D#":"F#", D:F, E:G, F:Ab/"G#", G:Bb
203.9 Bb:C, C:D, D:E, F:G, Ab/"G#":Bb, Eb/"D#":F, G:A, "C#":Eb/"D#", "F#":Ab/"G#", A:B
180.4 B:"C#", E:"F#"
113.7 Eb/"D#":E, Ab/"G#":A, "F#":G, "C#":D, Bb:B
90.2 B:C, C:"C#", F:"F#", D:Eb/"D#", A:Bb, E:F, G:Ab/"G#"
0 C:C, "C#":"C#", etc.
. . . . . . . . .

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