schismatic tuning

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 "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 i put them 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 of this tuning:




   

i have colored the 8 distinct notes in green,
the schismatically-related pairs in orange, and
the two notes which were half of a schismatic pair
but were not used, in grey.

here is 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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Updated:

2003.03.23 -- page created

by Joe Monzo

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(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

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