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:
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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