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Vicentino's adaptive-JI of 1555

[Joe Monzo]

Vicentino's "2nd tuning of 1555" is composed of two chains of 1/4-comma meantone, the first one a 19-tone chain from Gb to B#, which can be closely approximated by a 19-tone subset of 31edo, and the second a 17-tone chain from Gb to A#, 1/4-comma higher than the first chain.

here's a rectangular lattice-diagram of Vicentino's "2nd tuning of 1555"; the 3-axis is horizontal, the 5-axis is vertical and the pitches indicate exponents which increase by increments of 1/4. notes a "5th" apart (3:2 ratio) which have integer exponents of 5 are connected by dashed lines. Approximate cents-values are given after the note-names.

B#  269
E#  773-----
A#   76
D#  579     A#   81
G# 1083     D#  585
C#  386-----G# 1088
F#  890     C#  392
B   193     F#  895
E   697     B   199
A     0-----E   702
D   503     A     5
G  1007     D   509
C   310     G  1012
F   814-----C   316
Bb  117     F   819
Eb  621     Bb  122
Ab 1124     Eb  626
Db  427-----Ab 1129
Gb  931     Db  433
            Gb  936

the 19-tone "first chain" is the column on the left-hand side of this lattice.

the generator "5th" in 1/4-comma meantone can be factored as 5^(1/4), thus 4 steps of the generator (i.e., 4 "5ths") is exactly 5^1, which represents the 5:4 ratio; thus, 4 "5ths" = a "major-3rd", the basic equation of meantone, which is only approximately true in other variants belonging to the meantone family of temperaments, but is exact here in 1/4-comma MT. thus, on the lattice, a chain of 1/4-comma meantone goes straight up and down the 5-axis.

Vicentino's second chain, 1/4-comma higher than the first chain, thus ends up being in effect a 3:2 higher than the first chain, as the lattice shows in the right-hand column. Thus, it is an adaptive-JI which provides a great number of vertical sonorities which are in exact 5-limit just-intonation, but without the problems of commatic drift, since it is based on meantone.

here's an attempt at redrawing the same lattice in triangular format:

                                                      B# 269
                                                   E# 773 ----------
                                                A# 76 '.
                                             D# 579     '.        A# 81
                                          G# 1083         '.   D# 585
                                       C# 386 ------------- G# 1088
                                    F# 890 '.            C# 392
                                 B 193       '.       F# 895
                              E 697            '.  B 199
                           A 0 ---------------- E 702
                        D 503 '.             A 5
                     G 1007     '.        D 509
                  C 310           '.   G 1012
               F 814 -------------- C 316
            Bb 117 '.            F 819
         Eb 621      '.       Bb 122
      Ab 1124          '.  Eb 626
   Db 427 ------------- Ab 1129
Gb 931               Db 433
                  Gb 936

the lattices show that the following triads are exactly in tune to low-integer proportions:

Major = 4:5:6
Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#

minor = 1/(6:5:4)
eb, bb, f, c, g, d, a, e, b, f#, c#, g#, d#

here is the whole scale. i'd have to do some reading to find out Vicentino's notation, so i'll just use a plus sign to designate the 1/4-comma-higher note of each like-named pair.

note ~cents {3,5}-vector 12edo note w/ cawapus

 Ab+  1129   [1  -8/4]   G#/Ab +1201
 Ab   1124   [0  -7/4]   G#/Ab  +981
 G#+  1088   [1   4/4]   G#/Ab  -481
 G#   1083   [0   5/4]   G#/Ab  -701
 G+   1012   [1  -3/4]     G    +501
 G    1007   [0  -2/4]     G    +280
 Gb+   936   [1 -10/4]   F#/Gb +1482
 Gb    931   [0  -9/4]   F#/Gb +1261
 F#+   895   [1   2/4]   F#/Gb  -200
 F#    890   [0   3/4]   F#/Gb  -420
 F+    819   [1  -5/4]     F    +781
 F     814   [0  -4/4]     F    +561
 E#    773   [0   8/4]     F   -1121
 E+    702   [1   0/4]     E     +80
 E     697   [0   1/4]     E    -140
 Eb+   626   [1  -7/4]   D#/Eb +1061
 Eb    621   [0  -6/4]   D#/Eb  +841
 D#+   585   [1   5/4]   D#/Eb  -621
 D#    579   [0   6/4]   D#/Eb  -841
 D+    509   [1  -2/4]     D    +360
 D     503   [0  -1/4]     D    +140
 Db+   433   [1  -9/4]   C#/Db +1341
 Db    427   [0  -8/4]   C#/Db +1121
 C#+   392   [1   3/4]   C#/Db  -340
 C#    386   [0   4/4]   C#/Db  -561
 C+    316   [1  -4/4]     C    +641
 C     310   [0  -3/4]     C    +420
 B#    269   [0   9/4]     C   -1261
 B+    199   [1   1/4]     B     -60
 B     193   [0   2/4]     B    -280
 Bb+   122   [1  -6/4]   A#/Bb  +921
 Bb    117   [0  -5/4]   A#/Bb  +701
 A#+    81   [1   6/4]   A#/Bb  -761
 A#     76   [0   7/4]   A#/Bb  -981
 A+      5   [1  -1/4]     A    +220
 A       0   [0   0/4]     A       0

below is an interval matrix of Vicentino's adaptive-JI:

just out of curiosity i analyzed what intervals might approximate the 7:4 ratio (~969 cents). here's the list:

~971 cents:

 Gb : E+
  F : D#+
 Eb : C#+
 Db : B+
  C : A#+
 Bb : G#+
 Ab : F#+


~966 cents:

  G : E#
Gb+ : E+
 Gb : E
 F+ : D#+
  F : D#
Eb+ : C#+
 Eb : C#
  D : B#
Db+ : B+
 Db : B
 C+ : A#+
  C : A#
 Bb : G#
Bb+ : G#+
Ab+ : F#+
 Ab : F#

The graphic below shows the 36-out-of-217edo approximation of Vicentino's tuning. With a maximum error of ~2 cents, which is nearly half the difference between Vicentino's two chains, 217edo was chosen less because of its accuracy than because of the fact that, as 7 bike-chains of 31edo, each higher than the last by 1 degree of 217edo (which is close to 1/4 of a comma), its structure resembles that of Vicentino's tuning so well; therefore, a 36-tone subset of 217edo (19 tones from the first 31edo bike-chain, plus 17 tones from the next bike-chain above it) can work on the same principles as those of Vicentino's actual tuning.

217edo provides, among other things, an excellent set of pitches to achieve fixed-pitch adaptive-JI. A subset of 217edo which consists of 2 bike-chains of 31edo, approximately 1/4-comma apart, provides a good approximation to Vicentino's tuning. (See below, on Margo Schulter's and Paul Erlich's 62-tone proposals.)


Below is an example by Dave Keenan of a I-VI-II-V-I "comma-pump" chord progression, written in the "sagittal" notation developed by he and George Secor in 2002, and tuned by me in 217edo (click on the graphic to hear the MIDI-file).


Below are two musical illustrations of the first 5 measures of Orlando di Lasso's motet Ave regina coelorum, tuned in Vicentino's adaptive-JI tuning. This example was chosen because it is used as an illustration by Easley Blackwood of the unsuitability of strict-JI for "common-practice" repertoire (see Blackwood 1985, p 133-135). The first illustration uses Pythagorean tuning as the basis for the letter-names and accidentals, and shows the deviation of Vicentino's tuning from Pythagorean. The second illustration uses 1/4-comma meantone as the notational basis and shows the deviation of Vicentino's tuning from that. [click on the illustration to hear a MIDI-file of it]


REFERENCEs


Vicentino, Nicola. 1555.
L'antica musica ridotta alla moderna prattica
["ancient music restored to modern practice"]
Antonio Barre, Rome. republished 1557.
English translation:

Blackwood, Easley. 1985.
The Structure of Recognizable Diatonic Tunings
Princeton University Press, Princeton, NJ.
ISBN 0-691-09129-3


Paul Erlich commented, in Yahoo tuning list message 40414 (Thu Oct 31, 2002 4:37 am):


Margo Schulter commented, in Yahoo tuning list message 40451 (Thu Oct 31, 2002 4:57 pm):