Well-tempered variation on 15th-century tuning

Hello, there, everyone, and some recent discussions of 15th-century 
tuning systems based on a kind of modified Pythagorean tuning with a Wolf 
fifth among _musica recta_ notes (e.g. D-A or A-E), and with certain 
schismatic thirds (Pythagorean diminished fourths or augmented seconds) 
simplified to pure ratios of 5:4 and 6:5, has prompted me to post a 
12-note well-temperament which _might_ have been derived from such a 
tuning system.

The example here is derived from one 12-note tuning set in the manuscript 
_Volens facere clavicordium_, as ably explicated by Roland Hutchinson, 
with one convenient choice: the system apparently has C#/Db at either 
135/128 or 16/15 above C depending on the octave, and here I've used 
135/128.

The basic idea is to adjust the two notes of the Wolf fifth, D-A, a 
syntonic comma (81:80, ~21.51 cents) narrow in the original scheme, so 
that the lower note D is moved down 1/3 syntonic comma, and the upper note 
A moved up by the same amount, leaving this fifth impure by 1/3-comma 
(~7.17 cents). This amount of tempering, applied to three fifths in the 
tuning, is identical to that in the 1/3-comma meantone of Salinas (1577), 
earlier discussed and described as "languid" by Zarlino (1571); and almost 
identical to that in the 19-tone equal temperament advocated by Costeley 
(1570) with 19 equal thirdtones to an octave.

Interestingly, as Mark Lindley has noted, the kind of temperament produced 
by such a procedure is very similar to the system of Charles, Earl of 
Stanhope (1806), a well-temperament where some fifths are tempered by up 
to 1/3 sytonic comma.

Please let me caution that while this procedure produces something than 
can indeed be called a "well-temperament" under a conventional 
18th-century definition, it isn't necessarily ideal for English keyboard 
music around 1600, the topic of a lot of these discussions about 
15th-century tunings. For example E-G#-B will have a major third over 19 
cents wide of 5:4, and only a schisma (~1.95 cents) narrow of Pythagorean.

Here's a Scala file for the tuning showing steps other than the 
tempered ones as integer ratios. Values in cents are shown for the 
tempered steps -- for values in cents for all steps and intervals in the 
tuning, see the data further below.

---------------- Scala file starts on next line of text --------------

! anovolwt.scl
!
Well-tempered variation on  _Volens facere clavicordium_ (15th c.)
 12
!
 135/128
 196.74124
 32/27
 5/4
 4/3
 45/32
 3/2
 128/81
 891.52748
 16/9
 15/8
 2/1

---------- Scala file ended on newline after previous line of text ------

Here is some data from Scala on the locations of steps and sizes of 
intervals in cents, etc.:

|
Well-tempered variation on  _Volens facere clavicordium_ (15th c.)
  0:          1/1            0.000000 unison, perfect prime
  1:        135/128          92.17876 major limma, large chroma
  2:        196.741 cents    196.7413
  3:         32/27           294.1351 Pythagorean minor third
  4:          5/4            386.3139 major third
  5:          4/3            498.0452 perfect fourth
  6:         45/32           590.2239 tritone
  7:          3/2            701.9553 perfect fifth
  8:        128/81           792.1803 Pythagorean minor sixth
  9:        891.527 cents    891.5278
 10:         16/9            996.0905 Pythagorean minor seventh
 11:         15/8            1088.269 classic major seventh
 12:          2/1            1200.000 octave
|
Temperings of       5/4           3/2
  0: 0.000:         0.0000        0.0000
  1: 92.179:        19.5526      -1.9537
  2: 196.741:       7.1688       -7.1688
  3: 294.135:       21.5063       0.0000
  4: 386.314:       19.5526       0.0000
  5: 498.045:       7.1688        0.0000
  6: 590.224:       19.5526       0.0000
  7: 701.955:       0.0000       -7.1688
  8: 792.180:       21.5063       0.0000
  9: 891.527:       14.3375      -7.1688
 10: 996.090:       14.3375       0.0000
 11: 1088.269:      19.5526       0.0000
 12: 1200.000:      0.0000        0.0000
|
Temperings of       6/5           3/2
  0: 0.000:        -21.5063       0.0000
  1: 92.179:       -21.5063      -1.9537
  2: 196.741:      -14.3375      -7.1688
  3: 294.135:      -19.5526       0.0000
  4: 386.314:       0.0000        0.0000
  5: 498.045:      -21.5063       0.0000
  6: 590.224:      -14.3375       0.0000
  7: 701.955:      -21.5063      -7.1688
  8: 792.180:      -19.5526       0.0000
  9: 891.527:      -7.1688       -7.1688
 10: 996.090:      -19.5526       0.0000
 11: 1088.269:     -7.1688        0.0000
 12: 1200.000:     -21.5063       0.0000
|
Temperings of       19/16         3/2
  0: 0.000:        -3.3780        0.0000
  1: 92.179:       -3.3780       -1.9537
  2: 196.741:       3.7907       -7.1688
  3: 294.135:      -1.4243        0.0000
  4: 386.314:       18.1283       0.0000
  5: 498.045:      -3.3780        0.0000
  6: 590.224:       3.7907        0.0000
  7: 701.955:      -3.3780       -7.1688
  8: 792.180:      -1.4243        0.0000
  9: 891.527:       10.9595      -7.1688
 10: 996.090:      -1.4243        0.0000
 11: 1088.269:      10.9595       0.0000
 12: 1200.000:     -3.3780        0.0000
|
Interval class, Number of incidences, Size:
  1:  1  256/243           90.225 cents   Pythagorean limma
  1:  4  135/128           92.179 cents   major limma, large chroma
  1:  1  97.394 cents
  1:  1  99.347 cents
  1:  2  104.563 cents
  1:  3  16/15             111.731 cents  minor diatonic semitone
  2:  2  189.572 cents
  2:  2  196.741 cents
  2:  2  4096/3645         201.956 cents 
  2:  6  9/8               203.910 cents  major whole tone
  3:  4  32/27             294.135 cents  Pythagorean minor third
  3:  3  1215/1024         296.089 cents 
  3:  2  301.304 cents
  3:  2  308.473 cents
  3:  1  6/5               315.641 cents  minor third
  4:  2  5/4               386.314 cents  major third
  4:  2  393.482 cents
  4:  2  400.651 cents
  4:  4  512/405           405.866 cents  narrow diminished fourth
  4:  2  81/64             407.820 cents  Pythagorean major third
  5:  8  4/3               498.045 cents  perfect fourth
  5:  1  10935/8192        499.999 cents  fourth + schisma, 5-limit approximation to 
ET fourth
  5:  3  505.214 cents
  6:  4  45/32             590.224 cents  tritone
  6:  1  595.439 cents
  6:  1  597.392 cents
  6:  1  602.608 cents
  6:  1  604.561 cents
  6:  4  64/45             609.776 cents  2nd tritone
  7:  3  694.786 cents
  7:  1  16384/10935       700.001 cents  fifth - schisma, 5-limit approximation to ET 
fifth
  7:  8  3/2               701.955 cents  perfect fifth
  8:  2  128/81            792.180 cents  Pythagorean minor sixth
  8:  4  405/256           794.134 cents  wide augmented fifth
  8:  2  799.349 cents
  8:  2  806.518 cents
  8:  2  8/5               813.686 cents  minor sixth
  9:  1  5/3               884.359 cents  major sixth, BP sixth
  9:  2  891.527 cents
  9:  2  898.696 cents
  9:  3  2048/1215         903.911 cents 
  9:  4  27/16             905.865 cents  Pythagorean major sixth
 10:  6  16/9              996.090 cents  Pythagorean minor seventh
 10:  2  3645/2048         998.044 cents 
 10:  2  1003.259 cents
 10:  2  1010.428 cents
 11:  3  15/8              1088.269 cents classic major seventh
 11:  2  1095.437 cents
 11:  1  1100.653 cents
 11:  1  1102.606 cents
 11:  4  256/135           1107.821 cents octave - major limma
 11:  1  243/128           1109.775 cents Pythagorean major seventh
|
Number of notes                      : 12
Smallest interval                    : 256/243, 90.2250 cents
Average interval (divided octave)    : 100.000 cents
Average / Smallest interval          : 1.108340
Largest interval of one step         : 16/15, 111.7313 cents
Largest / Average interval           : 1.117313
Largest / Smallest interval          : 1.238363
Least squares average interval       : 99.3102 cents
Median interval                      : 98.371 cents
Prime limit                          : 5
Odd number limit                     : 135
Scale is strictly proper
Scale is a well-temperament. It has 3 different sizes of fifth
Scale has monotonic third-sizes over circle of fifths
Number of different intervals        : 52, relative 4.72727
Limited transpositions with margin 14.3375 cents :
 2 5 7 9 10
Average distance from equal tempered   : 6.9319 cents,  0.069319 steps
Standard deviation from equal tempered : 2.3751 cents,  0.023750 steps
Maximum distance from equal tempered   : 13.6863 cents,  0.136862 steps
Geometric average of pitches  0..n: 594.435 cents
Arithmetic average of pitches 0..n: 634.647 cents
Harmonic average of pitches   0..n: 554.224 cents
Geometric average of pitches  1..n: 643.972 cents
Arithmetic average of pitches 1..n: 678.367 cents
Harmonic average of pitches   1..n: 609.576 cents
Geometric average of pitches  1..n-1: 593.424 cents
Arithmetic average of pitches 1..n-1: 622.123 cents
Harmonic average of pitches   1..n-1: 564.724 cents

By the way, the tempering data on a sonority of 16:19:24 might be of 
interest if one is playing music where a minor third above the lowest note 
is apt to occur in conclusive sonorities, something that seems to happen 
around 1500 more often than in later Renaissance/Manneristic music (where 
closes including a third tend to call for a major third above the lowest 
voice). This tuning of the division of the fifth into minor third below 
and major third above can seem more conclusive than the simpler 10:12:15, 
likely because the lowest note of 16:19:24 is an octave of the acoustical 
"fundamental" (that is, an even power of 2, here 16).

With Elizabethan or Jacobean music, the desire for pure or near-pure major 
thirds (around 5:4) is likely a more important consideration, with minor 
thirds at or near 6:5 stylistically appropriate, their "tentativeness" 
part of the intonational setting. However, I mention that question of 
16:19:24 since a tuning might be used in many settings, some of which 
could favor this kind of sonority.

Most appreciatively,

Margo Schulter
[EMAIL PROTECTED]