Re: Gesualdo (Part I of II)

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Gesualdo, intonation, and instruments:
Some comments on a diverse thread
Part I of II
---------------------------------------------

In response to the varied issues raised in the Gesualdo discussion, I
would like to focus on a few points which may reveal not only some
evidence provided by Gesualdo's music and his contemporaries on
questions of intonation, but also some pet distinctions and issues of
some of us who revel in Renaissance/Manneristic techniques.

Thanks especially to Paul Erlich, who called my attention to the
excellent examples in Easley Blackwood's book[1]. Now I can explain
more clearly my own statement that I have not yet seen any examples of
_direct_ "enharmonic" or fifthtone progressions a la Nicola Vicentino
in Gesualdo, although Blackwood provides splendid illustrations of
diesis distinctions between notes in very close proximity, and I shall
add two more examples here.

In Section 1, Paul, I clarify my rather narrow and possibly fussy
definitions of concepts such as "direct chromaticism" or "direct
enharmonicism," and confirm that Blackwood, you, and I seem much in
accord as to the fine fit between Gesualdo's music and the 31-note
meantone system described and realized by his early contemporary
Nicola Vicentino and late contemporaries Scipione Stella and Fabio
Colonna, among others.

In Section 2, I consider the complexity of Gesualdo's music from the
viewpoints of modality, altered melodic intervals, and vertical
sonance, urging that it be approached as much as possible in its own
terms, and in the context of various 16th-century chromatic and other
techniques which Gesualdo cultivates and further develops.

In Section 3, I argue that a system of 31 meantone pitch classes (of
which Gesualdo uses 20 in his madrigals) is a felicitous intonational
model, and that meantone is a tuning system at once sweet for the
stable concords of this music and strikingly assertive for the
augmented or diminished intervals and diesis contrasts of a type
described by Blackwood.

In Section 4, distinguishing between the later historical concept of
"standard pitch" and the 16th-century problem of pitch drift within an
ensemble, I cite Giovanni Battista Benedetti's argument around 1563
that just intonation following the syntonic diatonic is impractical
for performers because it would result in such drift by multiple
commas. Interestingly, as discussed by Claude Palisca, Benedetti may
have been one of the first known European theorists to use frequency
ratios in comparing degrees of concord between pure intervals.

In Section 5, surveying Gesualdo's musical neighborhood, I note his
connection with the composer and keyboard designer Scipione Stella,
whose 31-note meantone instrument with some notes replicated for
easier fingering evidently served as a basis for Fabio Colonna's
_Sambuca Lincea_, with some dispute between Stella and Colonna as to
exactly who invented what first.

In Section 6, I consider specific and general indications that
Gesualdo's madrigals were not infrequently performed by instrumental
ensembles, or by mixed voices and instruments, although _a capella_
renditions (voices alone) are certainly one possibility; and also a
keyboard canzona by Gesualdo likely written for an instrument with 19
or more pitch classes available, such as the Neapolitan _cembalo
chromatico_ or a full 31-note instrument of the Stella/Colonna
variety.

In Section 7, finally, I consider a theme suggested by some of the
posts of Paul Erlich: the imprecision of musical perception, which may
not observe nuances such as the small inflections involved in
meantone-based adaptive JI (on Vicentino's keyboard in his second
tuning, or by singers or players of flexible-pitch instruments), or
the mathematical distinction between 1/4-comma meantone and 31-tET.
This imprecision leaves open some tantalizing if uncertain inferences
from remarks of theorists at the time.

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1. Gesualdo's "near-direct" diesis distinctions and Vicentino
-------------------------------------------------------------

As some readers may be not be too surprised to see demonstrated, the
essence of a musicological mindframe may often seem to be a love of
rather fussy distinctions. The following discussion of Vicentino
(1511-1576) and Don Carlo Gesualdo, Prince of Venosa (1560-1613) may
illustrate this trait, while confirming my agreement with and
indebtedness to both Easley Blackwood and Paul Erlich for some superb
examples of what I might term "almost-direct" diesis contrasts in
Gesualdo's madrigals.

Earlier in this thread, I offered the remark that I had not seen any
examples in Gesualdo of Vicentino's _direct_ "enharmonic" or fifthtone
progressions, for example C-C*-C#, where an ASCII asterisk (*) stands
for Vicentino's dot above a note raising it by a diesis equal to about
1/5-tone or 1/31 octave.

At the same time, I added that Gesualdo uses notes an enharmonic
diesis apart in rather close proximity, for example B# and C. Thanks
to you, Paul, and to Blackwood, I now can joyfully affirm that this
was rather an understatement, since melodic figures such as C-B-B#[2]
or Eb-D-D# (see below) do occur with only a single note intervening
between the pair a fifthtone apart. Another example below shows an
equally close juxtaposition of sonorities with A# and Bb, although
here not in the same voice.

We now encounter the musicological quibble of which the reader has
been warned: a tendency to define _direct_ "chromaticism" or
"enharmonicism" quite narrowly to mean the writing of an outright
melodic interval of a chromatic semitone (e.g. G-G#) or enharmonic
diesis (e.g. B#-C or D#-Eb). This very narrow definition may still
provide a distinction between Gesualdo and Vicentino -- at least
unless and until someone cites a direct enharmonic progression in
Gesualdo.[3]

However, I enthusiastically agree with Blackwood and Paul that figures
such as C-B-B#, like the "direct fifthtone progressions" C-B#-B or
B-B#-C, very neatly illustrate the use of diesis contrasts available
in a 1/4-comma meantone (or "adaptive meantone-based JI") system, or
the almost identical 31-tone equal temperament (31-tET).

While Blackwood, to my recollection, does not discuss the direct
enharmonicism of Vicentino, his eloquent words about Gesualdo might
nicely convey the kinship between these practices[4]:

"The availability of more than twelve notes makes possible
the placement in close proximity of two notes that differ
by a diesis (41.059 cents). This unusual and expressive
melodic device is strikingly used in _Merce grido
piangendo_.

In bars 1 and 2, the soprano makes the succession C-B-B#
and a listener hears at once that B# is very substantially
lower than C.... Subjectively, this device is perceived as
an expressive inflection of a sort totally impossible
within the confines of conventional 12-note equal tuning."

Recognizing this kinship, we might still point to two features of
Vicentino's enharmonic technique which may distinguish it from
Gesualdo's. The first is the use of direct melodic fifthtones, whether
spelled using conventional notation (e.g. C#-Db) or Vicentino's diesis
signs (e.g. Ab-Ab*-A). The second is what Vicentino terms "enharmonic
cadences" or related progressions where the "major semitone" is
avoided (the usual diatonic semitone of 3/5-tone) while the minor or
chromatic semitone (2/5-tone) is favored.[5]

To quote an example from Vicentino's treatise, using a MIDI-style
notation where C4 is middle C and "r" shows a rest, in 2/2 meter (with
the minim or half-note as a moderate beat):

1 & 2 | 1 2

r A4 Ab4 A4
E4 E*4 E4
C4 B*3 C2 C#2
A2 E*3 A2

This example illustrates at once the direct fifthtone steps E4-E*4-E4
in the next-to-highest part; the altered melodic fifths A2-E*3-A2 in
the bass (equivalent to meantone diminished sixths, ~49:32); and the
cadential progression between the outer voices of major third or tenth
to octave (E*3-Ab4 to A2-A4) with the upper voice ascending by a
chromatic semitone (Ab-A) of 2/5-tone. The concluding ornamental
figure of C2-C#2 in the next-to-lowest voice, shifting from the minor
third to the more conclusive major third above the bass, has an
affinity to somewhat more conventional 16th-century chromatic styles.

Here it will be observed that Vicentino's enharmonic nuances leave the
usual vertical consonances of third-plus-fifth-or-sixth above the bass
unaltered; for example, E*3-B*3-E*4-Ab4 could be described as the
usual cadential sonority E3-B3-E4-G#4 with each note raised by a
diesis (one could alternatively write E*3-B*3-E*4-G#*4).

From this perspective, let us return to the "almost-direct" diesisism
of Gesualdo. In addition to Blackwood's example from Gesualdo's
madrigal _Merce, grido piagendo_ (Book V, 1611 and 1613), I have found
an instance at the end of this phrase from _Tu piangi, o Filli mia_
(Book VI, 1611 and 1613), p. 21, bars 24-27 of the same edition used
by Blackwood[6], here read in 4/2 meter, with a Bb signature.

Here I'll use Bn (for B-natural) to confirm the presence of this
inflection, with each note understood to be sustained until a rest or
the conclusion of the excerpt, thus generally avoiding the problem of
ties (the dash before the first note of the middle part shows that it
is a suspension from the previous measure). I do use dots to show the
prolongation of a semibreve or whole note by half of its usual value
(a total duration of three beats rather than two), supplying such dots
apparently omitted by the modern editor at bar 27 in the two highest
sounding voices:

25
1 2 3 4 | 1 2 3 4 & | 1 & 2 3 & 4 | 1 2 3 4 |
Eb5 D5 r r
r Bn4 Bb4 A4 A4 A4 G4 F4. F#4
_C5 Bn4 r G#4 G4 F4 F4 F4. Eb4 D4. D#4
G4 r r Db4 C4 Bb3 Bb3 Bb3 Bb3 A3 A3 Bn3
G3 r E3 F3 F3 G3 G3 G3 D3 D3 Bn2

Here we have lots of conventional late 16th-century dissonances and
idioms, and characteristic traits such as frequent motions of the bass
by a third up or down, as well as the striking figure Eb4-D4-D#4 at
the conclusion of the middle voice.

Like C-B-B# in Blackwood's example, we have diatonic semitone
(3/5-tone) down immediately followed by a chromatic semitone
(2/5-tone) up, with an enharmonic diesis between the first note and
the last.

Here's an example of a "diagonal" near-direct diesis relationship from
_Ancor che per amarti_, at p. 94, mm. 25-26, of the same modern
edition of the Sixth Book of Madrigals[7], here a piece without
signature, so that I show the two forms of the step B/Bb simply as B
and Bb. This passage sets the text _Poi che vil fango anchor_:

25
| 1 2 3 & 4 | 1 & 2 ...
r
r A#4 A#4 A#4 B4 D5 C#5
r C#4 C#4 C#4 B3 G4 A4
r F#4 F#4 F#4 E4 D4 E4
r F#3 F#3 F#3 G3 Bb3 A3

We have an almost-direct diesis contrast between A# in the highest
sounding voice of the repeated opening sonority and Bb at the
beginning of the second measure, with only a single sonority
intervening.

The expressive tritone suspension Bb3-E4 opening the second measure
leads to a conventional cadence of the remissive variety (with
descending semitone Bb3-A3) to A3-E4-A4-C#5, concluding on Zarlino's
harmonic division of the fifth (major third above bass, in contrast to
the arithmetic division with the minor third above the bass). This
cadence features the traditional progressions expanding from major
third to fifth (Bb3-D4 to A3-E4) and from major sixth to octave
(Bb3-G4 to A3-A4).

Gesualdo's use of A# here makes possible the more euphonious harmonic
division of the fifth F#3-C#4-F#4-A#4, while also setting the stage
for the near-direct diesis contrast with Bb. The melodic diminished
fourth outlined in the bass, F#3-G3-Bb3, is also noteworthy, with
other composers such as Giaches de Wert and Claudio Monteverdi using
this interval expressively.

In sum, I heartily agree with Blackwood that diesis contrasts are a
vital part of Gesualdo's music, and with you, Paul, that someone
encountering melodic progressions such as C-B-B# or Eb-D-D# might well
speak of "quartertones" -- which I would amend in a friendly manner to
the more precise "fifthtones." Such figures are distinguished by only
one intervening note from the direct fifthtone music of Vicentino or
Colonna based more expressly on a full 31-note cycle.

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2. Complexity and sophistication in Gesualdo
--------------------------------------------

On a philosophical note, I would caution against an excessive reliance
on evolutionary concepts of history both older and newer which tend to
propose a more or less progressive development from the simple to the
complex. Gesualdo's art is an immensely complex and sophisticated one,
and the challenges as well as opportunities for ensembles seeking
felicitous intonations are at once nontrivial and exhilarating.

This music is based on a system of 12 modes adorned with various
chromatic and nearly-direct enharmonic figures, as well as altered
intervals both melodic and vertical of a kind partially described by
Vicentino (1555) and explored by composers such as Wert.

This Manneristic agenda colors the use of an extended meantone system,
distinguishing it from either a possible 18th-century agenda based on
what might be described as in some ways a much simplified major/minor
key system, or a possible 21st-century agenda based on using
approximations of such ratios as 7:9:12 in neo-Gothic progressions.

Vicentino, in exploring the new as well as traditional intervals of
his 31-note system, shows that then as now, tastes may vary. He finds
the "minimal third" of 7/5-tone (~7:6) and "proximate major sixth" of
24/5-tone (~12:7) rather dissonant, and the minimal seventh of
25/5-tone (~7:4) somewhat more so, although noting that any of these
intervals might be used to express an appropriate text.[8]

In contrast, he finds the "proximate minor third" of 9/5-tone, which
he describes as having an approximate ratio of 11:9 and of leaning
toward the outstanding concord of the usual major third at 5:4, to be
rather concordant, and uses it as a cadential embellishment in some of
his examples.

Thus "n-limit" concepts may not capture the subtle distinctions
of Joe Monzo's "sonance" between intervals which may be treated as
unstable or inconclusive but nevertheless recognized to have a degree
of "concord" or "compatibility" in a given style.

In Vicentino or Gesualdo, the most complex stable ratios are defined
ideally by factors of 5 (5:4, 6:5, 5:3, 8:5), but following and
expanding earlier precedents they deploy a great variety of altered
intervals as well as more or less conventionally treated dissonances.

Whether we are focusing on the altered intervals of these composers,
or on the subtle adjustments singers and players may have made to
obtain purer vertical consonances, Vicentino's words of advice in
introducing his examples of enharmonic cadences are worthy of our
attention.

Noting that the enharmonic genus with its fifthtone steps "permits the
creation of steps and leaps beyond the rational," from which reason
"such a division is called an irrational ratio," he continues [9]:

"A pupil must learn such disproportioned steps and
leaps in order to become a perfect musician and
perfect singer. Also, [s/]he should know how to
match and accompany with harmony all sorts of
disproportioned and irrational intervals, and
also how to sing them so as to show the world
that [s/]he is exceptional and can accomplish
with artifice that which cannot be done by reason."
(Inclusive English pronouns mine -- M.S.)

In Gesualdo, devices of this nature range from the "commatic" figure
described as early as 1357 by Johannes Boen with two successive
descending (or sometimes ascending) diatonic semitones, e.g. Ab-G-F#,
to unusual leaps of a diminished fifth or diminished fourth or minor
ninth.

The striking vertical dissonances favored by Gesualdo -- like
Monteverdi and other contemporaries around 1600 -- also present subtle
questions of tuning, with a 31-note meantone model providing an
attractive starting point. While modern writers ranging from Blackwood
to many more specialized historians often lean toward an 18th-19th
century terminology in describing these sonorities, a more period-
inclined approach may emphasize the plurality of forms and
combinations in this fluid modal setting.

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3. Meantone: assertive as well as sweet
---------------------------------------

While singers and players of flexible-pitch instruments would
certainly not be bound by the fixed tuning of 19-note or 31-note
keyboard, even such a keyboard can provide a reasonably faithful and
enthralling realization of Manneristic music with its chromatic and
sometimes enharmonic idioms.

At the outset, it might be well to note Blackwood's less than precise
language when he comments that Gesualdo's vocal compositions fit the
conventions of meantone tuning "with one exception -- they frequently
use more than twelve notes. It should not be surprising that a
composer of vocal music should find it unnecessary to heed a
restriction associated solely with the limitations of keyboard
instruments."[10]

This might easily be amended: "Gesualdo's compositions frequently use
more than 12 notes, thus exceeding the range of a standard meantone
keyboard, but nicely according with a musical environment where
harpsichords or organs offering 19 notes or even a full 31-note cycle
inspired their own repertory of keyboard music by composers such as
Trabaci and, in at least one apparent instance, Gesualdo himself."

In fairness to Blackwood, I would add that his discussion of meantone
emphasizes the many chromatic and other special effects available
_even_ within a usual 12-note compass, and then presents Gesualdo's
music to show the yet expanded possibilities such as diesis contrasts
which a larger gamut offers.

In my view, it is unnecessary to turn to Vicentino or Gesualdo in
order to experience the assertive qualities of 16th-century meantone,
although their music exemplifies this quality _par excellence_. The
diminished fourths or augmented fifths of Spanish vocal and keyboard
music, including the beautiful pieces of Antonio de Cabezon for organ
or clavier (harpsichord or clavichord), provide one prime example.

This is not to mention the audacious enharmonic sonorities of an early
17th-century musician such as Fabio Colonna, who includes ratios such
as 17:12 in his theoretical system and favors fifthtone "slides"
producing vertical intervals such as a fourth a diesis wider than the
usual interval (not too far from 11:8).

Of course, meantone also expresses itself in the tranquil flow of
smoothly concordant sonorities, punctuated here and there by an
ornamental figure or suspension dissonance, typical of so much
16th-century music.

The genius of the tuning is that it can at once approximate the ideal
of a seamless musical fabric woven largely of Renaissance JI concords,
and realize the chromatic or enharmonic drama of a Lasso, a Vicentino,
or a Gesualdo.

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4. Just intonation and pitch drift: a 16th-century view
-------------------------------------------------------

While absolute pitch in the 16th century apparently varied widely,
with each locale or even ensemble having its own standards[11], the
issue of pitch drift within a given ensemble during the performance of
a given piece was both known and asserted as a problem of just
intonation based on Ptolemy's syntonic diatonic as described by
Ludovico Fogliano (1529) and Zarlino (1558).

My purpose here is not to resolve the open question debated among
modern scholars of Renaissance polyphony such as Margaret Bent and
Roger Wibberley, as well as advocates of various alternative tuning
systems, as to whether such drift is good or bad, only to document
that the issue was a 16th-century concern, and for at least one
theorist a serious objection to classic JI as a practical ensemble
tuning.

Around 1563, as chronicled by Claude Palisca, the philosopher,
mathematician, and musician Giovanni Battista Benedetti corresponded
with the great composer Cipriano de Rore (1516-1565) on the question
of interval ratios and intonation.[12]

While traditional medieval and Renaissance theory defined intervals
in terms of string ratios, Benedetti took some passages from Aristotle
as the basis for what might be termed a frequency-based approach,
focusing on the _intervallum tremoris_ or "period of vibration" for a
string, which he states is inversely proportional to the length of the
string.[13]

For example, in the case of a 2:1 octave, "the larger portion of the
string will complete one period of vibration (_intervallum tremoris_)
during the time it takes the shorter to complete two." Likewise for a
3:2 fifth, "the longer portion of the string completing two periods of
vibration during the time the lesser portion completes three."[14]

"Benedetti then arrives at the law which states that
that the product of the number representing the string
length and the number of periods of the longer portion
of the string will equal the product of the number
representing the string length of the shorter portion
and the number of periods of this portion....

He proceeds to calculate the products for each of the
consonances recognized by Fogliano, which are:
diapason [2:1] 2, diapente [3:2] 6, diatessaron [4:3] 12,
major sixth [5:3] 15, ditone [5:4] 20, semiditone [6:5] 20,
semiditone [6:5] 30, and minor sixth [8:5] 40. He notes
that these numbers agree among themselves with a wonderful
reasonableness (_mirabili analogia_).
(Ratios in brackets added for convenience -- M.S.)

Benedetti thus surveys the standard 16th-century consonances in terms
of "the order of concurrence of the terminations of percussions of
waves of the air through which sounds are generated" (_ordinem
concursus percussionum terminorum, seu undarum aeris, vnde sonus
generatur_).[15] The smaller the product of the two terms of a musical
ratio, the more frequently their periods of vibration concur.

As Palisca remarks, this approach to consonance theory seems to fit
neatly with the just intonation models of Fogliano and Zarlino. For
Benedetti, however, a complication precluded the practical use of the
syntonic diatonic by voices or instruments: the problem of pitch
drift.

Taking some musical excerpts, including an example from a chanson of
Rore himself, Benedetti proceeds in his first of two letters to Rore
to analyze the melodic whole-tone and semitone sizes of the syntonic
diatonic required to maintain just vertical concords.

Only in the second letter, however, does he come to the point of this
exercise: "Bendetti declares that if these different sizes of
semitones and whole tones are used, as they must be if the consonances
are tuned justly, a vocal composition will not end on the same pitch
as it began but either higher or lower."[16]

He proceeds to give examples of what might now be termed two "comma
pump" progressions, each not surprisingly involving the maintenance of
pure fifths at both G-D and D-A. These patterns, causing a rise or
descent by a syntonic comma (81:80, ~21.51 cents), are typical of such
modes as G Mixolydian or D Dorian[17]:

1 2 | 1 2 | 1 ... 1 & 2 & | 1 & 2 & | 1 ...
G4 A4 G4 r D4 C#4 D4 E4 G4
D4 E4 D4 r G3 G3 A3 D3 G3 r
G3 D4 C4 G3 G3 E3 A3 r G3

(a) Rise of 81:80 (b) Fall of 81:80
Upper voice: 9:8 - 10:9 Upper voice: 27:25 - 16:15
G4 A4 G4 D4 C#4 D4

While Benedetti concludes that temperament of the vertical concords is
a practical necessity for performers, if one wishes to avoid such
drifting of pitch then another solution is the adaptive JI tuning
described by Vicentino (1555, 1561) as one alternative for his
archicembalo and arciorgano with "perfect fifths and perfect thirds."

Again, my purpose here is not to argue the issue of whether comma
drift _should_ be considered a musical vice and/or virtue, only to
show that the issue was fully recognized by a 16th-century theorist
and composer keenly attuned to acoustical and mathematical questions
of consonance.

-----
Notes
-----

1. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_
(Princeton: Princeton University Press, 1985); on Gesualdo and
meantone tuning, see pp. 185-187 and musical examples.

2. Ibid., p. 187, example from _Merc`e grido piangendo_, bars 1-2 of
Blackwood's excerpt in highest voice (D5-C5-B4-B#4).

3. There is a common musicological distinction between "accidentalism"
or the use of accidentals (routinely mandated by 14th-16th century
guidelines such as "closest approach" in theory and practice), and
"chromaticism" in the sense of direct chromatic semitones of the kind
appearing in adventurous 14th-century or 16th-century music, although
16th-century as well as modern writers often use such terms more
freely. Vicentino goes so far as to regard any writing of a direct
interval of a minor third, found in the Greek chromatic genus, as a
momentary shift from the diatonic to that genus -- a point disputed by
such theorists as Lusitano and Zarlino.

4. Blackwood (see n. 1 above), pp. 186-187.

5. Nicola Vicentino, _Ancient Music Adapted to Modern Practice_,
tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale
University Press, 1996), ISBN 0-300-06601-5, with Vicentino's
discussion and examples of "enharmonic cadences for four voices"
at pp. 207-209, and enharmonic madrigal excerpts and sections plus a
complete motet at pp. 209-222.

6. Carlo Gesualdo di Venosa, Samtliche Madrigale fur funf Stimmen
(Hamburg: Ugrino Verlag, 1958), vol. 6, p. 21.

7. Ibid., p. 94.

8. See Vicentino (n. 5 above), pp. 336-337 and 436-437 on the
"proximate minor third" at an "irrational" ratio of approximately
"5-1/2:4-1/2"; on the third "smaller than the minor third," which
Vicentino remarks is "discordant" and "resembles a second," and the
comparable "proximate of the major sixth" with an added enharmonic
diesis which he would "set aside," see pp. 339-340. At pp. 338-339, he
finds that the proximate major sixth A-Gb (~12:7) "produces less
harshness" than A-Gb* (~7:4), "since it is smaller by one diesis."
One is tempted to paraphrase: "The proximate major sixth more
resembles a major sixth, concordant in a 16th-century setting, while
the minimal seventh more resembles a usual minor seventh, in this
setting a clear discord generally treated with some caution" -- in
contrast to the freer treatment of minor sevenths in 13th-14th or
17th-19th century styles.

9. Ibid., p. 207.

10. Blackwood, n. 1 above, p. 185. The substance of this remark is in
keeping with Zarlino's advice about modal transpositions that one
should take care that an instrument has all the notes required for a
given transposition, since some chromatic or enharmonic steps are
available on only a few "artificial instruments"; but that since
singers are not limited to any fixed set of notes, such restraints
need not always be followed in vocal compositions.

11. One sampling of wind instruments from this era showed an average
pitch of A4=466, with wide variations on both sides of A=440; a
description of one "chamber pitch" by Michael Praetorius around 1619
has been read to suggest A4=424.

12. Claude V. Palisca, _Humanism in Italian Renaissance Musical
Thought_ (Yale University Press, New York and London, 1985),
pp. 257-264.

13. Ibid., p. 259.

14. Ibid., Palisca's paraphrase of Benedetti.

15. Ibid., p. 261

16. Ibid., p. 262.

17. Ibid., p. 263-264, Figures 10.9 and 10.10.

Thank you Margo for the very informative article. I hesitate to get further
involved in this issue because the question of what pitch system a composer
"had in mind" when composing for a non-fixed-pitch ensemble, besides being
somewhat speculative by its nature, also potentially assigns a particular
model to a mental process which may not have been as simple or as
consistent as we might like to think. Nevertheless, I agree that the late
humanist interest in chromatic and enharmonic genera, the regional
availability of >12 keyboards, and the implication of an "open" meantone
system are all suggestive of a certain intonational freedom in this music
that is worth study insofaras it sheds light on his works.

A more concrete question is what tuning would have worked in the couple of
his chromatic keyboard works that have survived. I haven't seen these
works, but I would be very interested in knowing more about them:

Canzon francese, a 4, kbd, London, British Library Add.30491; W x, 16

Gagliarda, a 4, Naples, Conservatorio di Musica S Pietro a Majella,
Biblioteca 4.6.3; W x, 22

Has anyone seen them?

Bill

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^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

While I'm on the topic, here's an interesting quote from Lorenzo Bianconi's
Groves article on Gesualdo:

Gesualdo shared Luzzaschi's interest in the chromatic arcicembalo made by
Vicentino and kept at the court of Ferrara. The chronicler Sardi related
that Luzzaschi played this instrument during the Este-Venosa wedding
celebrations, and it is known that Stella and Gesualdo later tried, in
vain, to construct a similar chromatic instrument in Naples. The practice
and theory of such an instrument had an undoubted influence on Gesualdo's
stylistic evolution; his writing encompassed an almost complete chromatic
scale (the only chromatic change which never appears is Fb), and frequently
used variations on the ancient chromatic tetrachord. Had the arcicembalo
been less impractical, it would have constituted the one possible link
between chromatic counterpoint and the newer forms of mixed vocal and
instrumental music; thus Gesualdo's coherent choice of the madrigal style
based on artifice rather than any kind of 'nuova musica' should be seen in
the light of the inability of contemporary keyboard instruments to cope
with extreme chromaticism. It also destroys the myth, believed by Ambros
among others, of an empirical, irrational Gesualdo, trying out his
chromaticism 'auf dem Klavier oder der Orgel'.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> "Benedetti then arrives at the law which states that
> that the product of the number representing the string
> length and the number of periods of the longer portion
> of the string will equal the product of the number
> representing the string length of the shorter portion
> and the number of periods of this portion....
>
> He proceeds to calculate the products for each of the
> consonances recognized by Fogliano, which are:
> diapason [2:1] 2, diapente [3:2] 6, diatessaron [4:3] 12,
> major sixth [5:3] 15, ditone [5:4] 20, [...]
> semiditone [6:5] 30, and minor sixth [8:5] 40. He notes
> that these numbers agree among themselves with a wonderful
> reasonableness (_mirabili analogia_).
> (Ratios in brackets added for convenience -- M.S.)

Thanks for digging this up, Margo! Benedetti certainly deserves to be
mentioned in a "consonance measures" FAQ -- he anticipated Tenney by
four centuries!
>
> [Benedetti] proceeds to give examples of what might now be termed
two "comma
> pump" progressions, each not surprisingly involving the maintenance
of
> pure fifths at both G-D and D-A. These patterns, causing a rise or
> descent by a syntonic comma (81:80, ~21.51 cents), are typical of
such
> modes as G Mixolydian or D Dorian[17]:
>
> 1 2 | 1 2 | 1 ... 1 & 2 & | 1 & 2 & |
1 ...
> G4 A4 G4 r D4 C#4 D4 E4 G4
> D4 E4 D4 r G3 G3 A3 D3 G3 r
> G3 D4 C4 G3 G3 E3 A3 r G3
>
> (a) Rise of 81:80 (b) Fall of 81:80
> Upper voice: 9:8 - 10:9 Upper voice: 27:25 - 16:15
> G4 A4 G4 D4 C#4 D4

Thanks for these early comma-pump examples! I don't think I'd heard
of Benedetti before -- clearly a sharp thinker of the 16th century.
>
> While Benedetti concludes that temperament of the vertical concords
is
> a practical necessity for performers, if one wishes to avoid such
> drifting of pitch then another solution is the adaptive JI tuning
> described by Vicentino (1555, 1561) as one alternative for his
> archicembalo and arciorgano with "perfect fifths and perfect >
thirds."

Yes, it seems somewhat likely that Benedetti would have taken back
his claim had he been familiar with Vicentino's adaptive JI system.

Margo, as I am unlikely to read all your posts in the future, let me
take this opportunity to praise the incredible efforts you've put
into educating us here on this list and the high level of both
scholarship and decorum you've maintained thoughout. Let me add that
it is my sincere hope that you contribute as much as possible to all
the FAQs that we are developing -- the virtues of honest scholarship
and patient illustration are exactly the ones we need most of for a
project like this.

--- In tuning@y..., Bill Alves <ALVES@O...> wrote:

> While I'm on the topic, here's an interesting quote from Lorenzo
Bianconi's
> Groves article on Gesualdo:
>
> Gesualdo shared Luzzaschi's interest in the chromatic arcicembalo
made by
> Vicentino and kept at the court of Ferrara. The chronicler Sardi
related
> that Luzzaschi played this instrument during the Este-Venosa wedding
> celebrations, and it is known that Stella and Gesualdo later tried,
in
> vain, to construct a similar chromatic instrument in Naples. The
practice
> and theory of such an instrument had an undoubted influence on
Gesualdo's
> stylistic evolution; his writing encompassed an almost complete
chromatic
> scale (the only chromatic change which never appears is Fb),

This implies that a "complete chromatic scale" has 21 tones per
octave -- a rather odd definition! Is this made precise in the
preceding portion of the article?

>This implies that a "complete chromatic scale" has 21 tones per
>octave -- a rather odd definition! Is this made precise in the
>preceding portion of the article?

I'm not sure I'm following the implication you see, but, no there is no
further explanation that I see in the article. Perhaps what he means is
"every possible chromatic alteration" or "every possible use of an
accidental."

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Margo Schulter wrote:

> While Blackwood, to my recollection, does not discuss the direct
> enharmonicism of Vicentino, his eloquent words about Gesualdo might
> nicely convey the kinship between these practices[4]:
>
> "The availability of more than twelve notes makes possible
> the placement in close proximity of two notes that differ
> by a diesis (41.059 cents). This unusual and expressive
> melodic device is strikingly used in _Merce grido
> piangendo_.
>
> In bars 1 and 2, the soprano makes the succession C-B-B#
> and a listener hears at once that B# is very substantially
> lower than C.... Subjectively, this device is perceived as
> an expressive inflection of a sort totally impossible
> within the confines of conventional 12-note equal tuning."

I had another half hour in the library last night, before I'd read this.
I couldn't find such a motif in the opening. The first B# I could see was
in bar 28. And it is in the soprano, at the end of a C-B-B#! So I wonder
why I missed it at the opening. If I'd had the exact bar numbers, I'd
have taken a photocopy to check later on if it really wasn't there.

The progression I did find, in jazz notation, is Gm-Am7-Em-G7-Em-G#. The
diesis is between the C in Am7 and the B# in G# major. There's no easy
way of respelling it to avoid the diesis. Although the Em and G# chords
have no notes in common, so you could re-write G# as Ab, Gesualdo seems to
prefer the indirect diesis to the direct bad thirds.

Note that, although C and B# are two notes apart in the soprano, there are
two chords in between them. Also, there's a move from a root of G to G#,
so a chromatic semitone, with one chord in between. This may not be
relevant, as root progressions weren't so important in those days(?)
Three voices move by that chromatic semitone, one indirectly (D-B-D#)

> >From this perspective, let us return to the "almost-direct" diesisism
> of Gesualdo. In addition to Blackwood's example from Gesualdo's
> madrigal _Merce, grido piagendo_ (Book V, 1611 and 1613), I have found
> an instance at the end of this phrase from _Tu piangi, o Filli mia_
> (Book VI, 1611 and 1613), p. 21, bars 24-27 of the same edition used
> by Blackwood[6], here read in 4/2 meter, with a Bb signature.
>
> 25
> 1 2 3 4 | 1 2 3 4 & | 1 & 2 3 & 4 | 1 2 3 4 |
> Eb5 D5 r r
> r Bn4 Bb4 A4 A4 A4 G4 F4. F#4
> _C5 Bn4 r G#4 G4 F4 F4 F4. Eb4 D4. D#4
> G4 r r Db4 C4 Bb3 Bb3 Bb3 Bb3 A3 A3 Bn3
> G3 r E3 F3 F3 G3 G3 G3 D3 D3 Bn2
>
> Here we have lots of conventional late 16th-century dissonances and
> idioms, and characteristic traits such as frequent motions of the bass
> by a third up or down, as well as the striking figure Eb4-D4-D#4 at
> the conclusion of the middle voice.
>
> Like C-B-B# in Blackwood's example, we have diatonic semitone
> (3/5-tone) down immediately followed by a chromatic semitone
> (2/5-tone) up, with an enharmonic diesis between the first note and
> the last.

And like my example in that respect. The chord progression that involves
the diesis is Eb-Bb-Dm-B. Again there's a root progression by a chromatic
semitone, with another chord in the middle. And there the chords are the
same kind, so it may be more relevant. Three parts move by that chromatic
semitone, two of them directly, as the notes are held.

And again, it's a seemingly innocent progression that happens to throw up
a diesis. We can only guess at whether Gesualdo fixed it to give this
result. That a voice was held over two chords to give that -3 +2 both
times is suggestive.

> Here's an example of a "diagonal" near-direct diesis relationship from
> _Ancor che per amarti_, at p. 94, mm. 25-26, of the same modern
> edition of the Sixth Book of Madrigals[7], here a piece without
> signature, so that I show the two forms of the step B/Bb simply as B
> and Bb. This passage sets the text _Poi che vil fango anchor_:
>
> 25
> | 1 2 3 & 4 | 1 & 2 ...
> r
> r A#4 A#4 A#4 B4 D5 C#5
> r C#4 C#4 C#4 B3 G4 A4
> r F#4 F#4 F#4 E4 D4 E4
> r F#3 F#3 F#3 G3 Bb3 A3
>
> We have an almost-direct diesis contrast between A# in the highest
> sounding voice of the repeated opening sonority and Bb at the
> beginning of the second measure, with only a single sonority
> intervening.

Ah yes, so more direct in one sense, but the notes are further apart.

> The expressive tritone suspension Bb3-E4 opening the second measure
> leads to a conventional cadence of the remissive variety (with
> descending semitone Bb3-A3) to A3-E4-A4-C#5, concluding on Zarlino's
> harmonic division of the fifth (major third above bass, in contrast to
> the arithmetic division with the minor third above the bass). This
> cadence features the traditional progressions expanding from major
> third to fifth (Bb3-D4 to A3-E4) and from major sixth to octave
> (Bb3-G4 to A3-A4).

Seemingly innocent again? No root progression by a chromatic semitone
this time. Indeed, no parts move by chromatic semitones.

> Gesualdo's use of A# here makes possible the more euphonious harmonic
> division of the fifth F#3-C#4-F#4-A#4, while also setting the stage
> for the near-direct diesis contrast with Bb. The melodic diminished
> fourth outlined in the bass, F#3-G3-Bb3, is also noteworthy, with
> other composers such as Giaches de Wert and Claudio Monteverdi using
> this interval expressively.

That is an interesting interval! In the bass, as well. Do you have
examples of it being used directly (without the G3 in the middle)?

> In sum, I heartily agree with Blackwood that diesis contrasts are a
> vital part of Gesualdo's music, and with you, Paul, that someone
> encountering melodic progressions such as C-B-B# or Eb-D-D# might well
> speak of "quartertones" -- which I would amend in a friendly manner to
> the more precise "fifthtones." Such figures are distinguished by only
> one intervening note from the direct fifthtone music of Vicentino or
> Colonna based more expressly on a full 31-note cycle.

Yes, but they're also distinguished in that the dieses arrive as a
side-effect of progressions with standard intervals. Whereas Vicentino
deliberately spiced up standard progressions to get different intervals to
come out.

> On a philosophical note, I would caution against an excessive reliance
> on evolutionary concepts of history both older and newer which tend to
> propose a more or less progressive development from the simple to the
> complex. Gesualdo's art is an immensely complex and sophisticated one,
> and the challenges as well as opportunities for ensembles seeking
> felicitous intonations are at once nontrivial and exhilarating.

I do find it interesting how closely linked Gesualdo's music appears to be
to that which came after. Of course, the conventions of spelling with a
closed circle of fifths were forged in this meantone era when they did
make a difference to the sound.

The very ending of Merce grido piangendo is also interesting. It happens
to be Ex. 42 in the Watkins book (Clarendon Press, your edition may vary)
although not in the accidentals chapter, so I don't know what he has to
say about it.

The progression from bar 30 is G-Gm-A-G#m-Bb-B-Em-E. So all nice 5-limit
triads. The Eb of C minor is interesting in that it throws up a
diminished third with the C# of A major. Although they are in different
voices. There's also an indirect wolf between Eb and the G# that starts
the melody.

But the really interesting chord is that Bb major. It gives a Wolf
between Bb and D# in the lowest two voices. Different voices, but a
direct melodic step. That's along with the other intervals thrown up.
Really, it should be spelt as a A# major. So why isn't it? Possibly
because Cx would lie outside the gamut.

However, it does mean a direct progression of similar chords by a
chromatic semitone is brought in. With the two lowest voices both moving
by a chromatic semitone. In fact, this cadence is full of chromatic
semitones! So it's exactly what I was looking for to support the
hypothesis that chromatic semitones should be preferred at cadences.

Of course, one example only goes so far. But maybe it isn't a coincidence
that it occurs so close to a diesis pump. Perhaps Gesualdo saved his
"quartertone" experiments for certain pieces. Because those were the ones
he wrote with access to a split key keyboard? So here's another
hypothesis to test: look at the ends of the other pieces with semi-naked
diesis pumps, and look for chromatic semitones!

The wolf rules out any 9-limit interpretation, BTW.

So far in my limited analysis, I haven't found a root progression by a
diatonic semitone. The statistically invalid sample off one fits the
hypothesis perfectly. I know it's an anachronistic concept, but I would
be interested in other examples of semitonal root progressions,
particularly the early examples of Neapolitan sixths.

Graham

--- In tuning@y..., Bill Alves <ALVES@O...> wrote:
>
> >This implies that a "complete chromatic scale" has 21 tones per
> >octave -- a rather odd definition! Is this made precise in the
> >preceding portion of the article?
>
> I'm not sure I'm following the implication you see,

The article said that the only tone of the "complete chromatic scale"
that Gesualdo didn't use was Fb. Clearly this implies (and we know
from other contributors) that he used seven nominals, seven sharps,
and six flats -- twenty tones in all. Fb would complete that at 21.
Yet the "chromatic" keyboards of Vicentino, Stella, and Colonna had
31, not 21, tones. It seems likely that the author of the article
didn't fully understand this last point.

Hello, there, Paul Erlich, and since your very moving message about your
somewhat changing status in regard to this List was part of the Gesualdo
thread, I'm responding there, although this really deserves a thread of
its own.

Please let me say that I have deeply honored to exchange ideas and
creative energy with you over the past not quite three years, and that
your latest words to me make this experience all the more precious. I
hope that we can continue this dialogue, at least from time to time,
and would like to express my indebtedness to you in many directions.

You very successfully communicated to me the contagious urge to tune
22-tET -- and find out how a literal quartertone (1/22 octave) could
serve as a very nice diatonic semitone. I'm tempted to call it a
"tempered 28:27," maybe revealing my JI/RI streak -- but whatever I
call it, it has a really neat effect, whether in my neo-Gothic
cadences for three or four voices, or just in a melodic improvisation,
maybe with the element of a drone.

One evening I sat down to improvise in 22-tET, and decided to focus on
the melodic aspect: a drone on D, and a simple melody in the Dorian
mode. That step from major sixth to minor seventh somehow epitomized
the scale for me, something that might have been documented in a paper
on ethnomusicology describing the traditional tuning practices of some
intonationally very wise culture.

Of course, your tour de force in the Aeolian mode has stuck with me
these several months, a kind of legendary exploit in the chronicles of
xenharmonicism.

The cross-cultural interplay between 22-tET and the 22 srutis of India
is also fascinating to observe; I hope that it enriches musical
perspectives and traditions on all sides.

While I might have uttered these words on various occasions, your
"change of pace" in accessing this List seems an appropriate moment.

By the way, I would invite you and everyone to make the most of the
Benedetti material: it's a fascinating chapter in the history of
consonance/dissonance theory. I'm in agreement that whoever writes
this part of the FAQ should include Benedetti.

Maybe as a fitting conclusion to this message, I would say that your
writings (and graphics!) have shown the way to fertile valleys and
elevating plateaus of intonational delight.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <
MSCHULTER@V...> wrote:

> One evening I sat down to improvise in 22-tET, and decided to focus on
> the melodic aspect: a drone on D, and a simple melody in the Dorian
> mode.

By which you mean, 4 1 4 4 4 1 4 in 22-tET?
Randy Winchester's 22-tET piece from
"Comets over Flatland" is also in this mode
over a drone -- you might like it!

P.S. Have you had much chance to check out
my decatonic scales in 22-tET?

Hello Margo, Ibo, and anybody else with knowledge on this.

Margo wrote:

> This might easily be amended: "Gesualdo's compositions frequently
use
> more than 12 notes, thus exceeding the range of a standard meantone
> keyboard, but nicely according with a musical environment where
> harpsichords or organs offering 19 notes or even a full 31-note
cycle
> inspired their own repertory of keyboard music by composers such as
> Trabaci and, in at least one apparent instance, Gesualdo himself."

I've ordered volume 17 of the Faber Early Organ Series, which covers
the early 17th century and specifically works by Trabaci. Also Book
VI of the Gesualdo madrigals, so I don't have to get the bus into
Bristol every time I want to look at them.

Margo again:

> In my view, it is unnecessary to turn to Vicentino or Gesualdo in
> order to experience the assertive qualities of 16th-century
meantone,
> although their music exemplifies this quality _par excellence_. The
> diminished fourths or augmented fifths of Spanish vocal and keyboard
> music, including the beautiful pieces of Antonio de Cabezon for
organ
> or clavier (harpsichord or clavichord), provide one prime example.

Volume 4 of the same series contains:

Tento do 8o tom, Antonio Carreira
Tento do 3o tom, Heliadorus de Paiva
Tento do 4o tom, Heliadorus de Paiva
Veni creator spiritus, Juan Bermudo
Pange lingua, Huan Bermudo
Tiento de 5o tone, anonymous
Tiento do 7o tono super Philomena, Francisco Fernandez Palero
Del modo de taner a corcheas, Tomas de Santa Maria
Fuga a dos voces, Tomas de Santa Maria
Fuga a tres voces, Tomas de Santa Maria
Fuga a cuatro voces, Tomas de Santa Maria
Beata viscera Mariae vriginis, Antonio de Cabezon
Versos del 8o tono de Magnificat, Antonio de Cabezon
Tiento sobre Cum sancto spiritu, Antonio de Cabezon
Versos do 3o tom, Manuel Rodrigues Coelho
Tento do 2o tom, Manual Rodrigues Coelho

Cabezon's "Magnificat" and "8 toni Psalmorum" are in the "Altspanishe
Orgelmeister" volume of Schott's "Liber Organi" series.

I ask for guidance as to which of these would be worth getting to
investigate interesting uses of meantone.

> This is not to mention the audacious enharmonic sonorities of an
early
> 17th-century musician such as Fabio Colonna, who includes ratios
such
> as 17:12 in his theoretical system and favors fifthtone "slides"
> producing vertical intervals such as a fourth a diesis wider than
the
> usual interval (not too far from 11:8).

How would I find out about this theory, and does his music reflect it?

Here are the contents of Volume 18 of the Faber series:

Toccata sesta per l'organo sopra i pedali, e senza -- Frescobaldi
Recercar sesto sopra fa, fa, sol, la, fa -- Frescobaldi
Toccata prima -- Michelangelo Rossi
Toccata seconda del nono tuono naturale -- Giovanni Salvatore
Versi sopra il Kyrie (Messa della Domenica) -- Giovanni Salvatore
Canzon quarta del quarto tono naturale -- Giovanni Battista Fasolo
Hinno per la Ascensione del terzo tono: Jesu nostra ... -- Fasolo
Ricercare nono con tre soggetti -- Luigi Battiferri
Capriccio -- ?Pietro Andrea Ziani
Toccata -- Bernado Pasquini

I request advice as to how exciting this would be in the light of my
interest in meantone temperament and Neapolitan sixths.

Say, "quarto tono" doesn't mean "quartertone" does it?

There you go. Advice on CDs to accompany my research is also
welcomed. I can try getting the Stembridge articles through the
library.

Does anybody have the ISBN for Easley Blackwood's book? I can try
ordering that through the library. I don't expect it'll work, but
there's only one way to find out. They do like ISBNs.

Thanks all,

Graham

>Does anybody have the ISBN for Easley Blackwood's book? I can try
>ordering that through the library. I don't expect it'll work, but
>there's only one way to find out. They do like ISBNs.
>
>
>Thanks all,
>
> Graham
>

The ISBN for Blackwood's book is: 0-691-09129-3

Regards,

Ralph Lorenz

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:

/tuning/topicId_19306.html#19552

> > [Margo]
> > This is not to mention the audacious enharmonic sonorities of
> > an early 17th-century musician such as Fabio Colonna, who
> > includes ratios such as 17:12 in his theoretical system and
> > favors fifthtone "slides" producing vertical intervals such
> > as a fourth a diesis wider than the usual interval (not too
> > far from 11:8).
>
> How would I find out about this theory, and does his music
> reflect it?

Hi Graham,

I do not know of any music by Colonna... I think some exists,
but I believe there are only a few small pieces, if any.

The most accessible source I know of for info on Colonna
is "Fabio Colonna's Sambuca" by Lynn Wood Martin, in
_Xenharmonikon_ 7 + 8 (Spring 1979).

See the _Xenharmonikon_ pages:
http://www.tiac.net/users/xen/xh/

I found a few interesting mathematical errors in this article
and began a webpage about Colonna's _Sambuca_ (a microtonal
keyboard instrument), but never got very far with it.

"Stay tuned" to my website... eventually it will show up.

If you'd like me to email what I have to you, I can do that...
after I get the time to find my work on it. It was a couple
of years ago.

-monz
http://www.monz.org
"All roads lead to n^0"

Joe Monzo wrote:

> I do not know of any music by Colonna... I think some exists,
> but I believe there are only a few small pieces, if any.
>
> The most accessible source I know of for info on Colonna
> is "Fabio Colonna's Sambuca" by Lynn Wood Martin, in
> _Xenharmonikon_ 7 + 8 (Spring 1979).

I don't have that, but maybe it's time for another bulk order anyway.

Incidentally, it's in Xenharmonikon 4 where Ivor Darreg advocates
mis-spelling meantone to get smaller semitones. Page 6 of Xenharmonic
Bulletin no. 5.

> I found a few interesting mathematical errors in this article
> and began a webpage about Colonna's _Sambuca_ (a microtonal
> keyboard instrument), but never got very far with it.
>
> "Stay tuned" to my website... eventually it will show up.
>
> If you'd like me to email what I have to you, I can do that...
> after I get the time to find my work on it. It was a couple
> of years ago.

Could be interesting, getting the corrections before I have the article.

Graham

--- In tuning@y..., MONZ@J... wrote:

/tuning/topicId_19306.html#19620

> I found a few interesting mathematical errors in this article
> and began a webpage about Colonna's _Sambuca_ (a microtonal
> keyboard instrument), but never got very far with it.
>
> "Stay tuned" to my website... eventually it will show up.

OK - here it is, in somewhat aborted format. The webpage has
been reworked from an email I sent John Chalmers, and includes
his reply at the end (thanks in advance for permission, John):

http://www.ixpres.com/interval/monzo/colonna/sambuca.htm

-monz
http://www.monz.org
"All roads lead to n^0"

First of all, Graham would not have a problem if you realized his progressions in 72-tET --
Graham is just a little more flexible about the possibilities that we are.

Secondly, Graham's decatonic notation is based on the scale

Decimal notation Monz 72-tET notation
--------------------- ----------------------------
0 C
1 C#+
2 D>
3 Ev
4 F<
5 F#-
6 G
7 G#+
8 A>
9 Bv
0v C<
1v C#-
2v D
3v Eb+
4v E>
5v F^

etc. The generator is 7/72 octave.

Thirdly, I think we need more sequences to aid our discussion of comma shift/drift. For example,
Benedetti's two examples, as presented by Margo Schulter.

>
> 1 2 | 1 2 | 1 ... 1 & 2 & | 1 & 2 & | 1 ...
> G4 A4 G4 r D4 C#4 D4 E4 G4
> D4 E4 D4 r G3 G3 A3 D3 G3 r
> G3 D4 C4 G3 G3 E3 A3 r G3
>
Maintaining common tones, the result would be:

> (a) Rise of 81:80 (b) Fall of 81:80
> Upper voice: 9:8 - 10:9 Upper voice: 27:25 - 16:15
> G4 A4 G4 D4 C#4 D4

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_19306.html#22593

> First of all, Graham would not have a problem if you realized
> his progressions in 72-tET -- Graham is just a little more
> flexible about the possibilities that we are.

But I thought that his whole point in illustrating that
2401:2400 pump on both lattices was to show that there
was drift in 72-EDO but not in his tuning, or whatever
it is that he's doing that's different other than the
decimal notation.

>
> Secondly, Graham's decatonic notation is based on the scale
> <snip>

Right, that one I got. Thanks anyway, for those who don't.

>
> Thirdly, I think we need more sequences to aid our discussion
> of comma shift/drift. For example, Benedetti's two examples,
> as presented by Margo Schulter.
>
> >
> > 1 2 | 1 2 | 1 ... 1 & 2 & | 1 & 2 & | 1...
> > G4 A4 G4 r D4 C#4 D4 E4 G4
> > D4 E4 D4 r G3 G3 A3 D3 G3 r
> > G3 D4 C4 G3 G3 E3 A3 r G3
> >
> Maintaining common tones, the result would be:
>
> > (a) Rise of 81:80 (b) Fall of 81:80
> > Upper voice: 9:8 - 10:9 Upper voice: 27:25 - 16:15
> > G4 A4 G4 D4 C#4 D4

Huh? I've begun making the MIDI-files, but some things
here must be wrong.

Example (a)
/tuning/files/monz/drift/benedeta.mid
is straighforward, and you describe the drift correctly as
upward by 81:80, but the upper voice ratios are reversed; they
should be 10/9 - 9/8, with exactly the same drift reproduced
in the bottom voice and again, but at a different pitch level,
in the middle voice too:

10/9 5/4 9/8
5/3 15/8 27/16
10/9 5/3 3/2 9/8

Example (b) is another story. It seems to me to be open
to a variety of different JI interpretations. And is it
really in C#?, which D-as-16/15 would imply. Can you
supply some more ratios?

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning@y..., paul@s... wrote:
>
> /tuning/topicId_19306.html#22593
>
> > First of all, Graham would not have a problem if you realized
> > his progressions in 72-tET -- Graham is just a little more
> > flexible about the possibilities that we are.
>
>
> But I thought that his whole point in illustrating that
> 2401:2400 pump on both lattices was to show that there
> was drift in 72-EDO but not in his tuning,

Not correct. There is no drift in 72-tET because the 2401:2400
(Breedsma?) vanishes in 72-tET.

> or whatever
> it is that he's doing that's different other than the
> decimal notation.

It's just lattice stuff.
> > > >
> Huh? I've begun making the MIDI-files, but some things
> here must be wrong.
>
You'll have to check with Margo, or find the Palisca book, or find
the Benedetti original.

Monz wrote:

> But I thought that his whole point in illustrating that
> 2401:2400 pump on both lattices was to show that there
> was drift in 72-EDO but not in his tuning, or whatever
> it is that he's doing that's different other than the
> decimal notation.

The idea was to show a progression that would drift in JI, but not in
any miracle/Erlich-Keenan temperament. And hence that such progressions
can be shown on my lattice.

Whether you use 31, 41, 72 or anything in between doesn't really matter.
I'm finding a third of the way from 31 to 41 (2/3 of the way to 72) works
well for Blackjack. The quommas are fat enough to work melodically, and
chords like neutral-third triads and 7:9:11 approximations sound quite
good (with the timbres I'm using).

There's also a sweet spot around as far the other side of 72 for 11:8, and
this tuning could be useful for the 31-note MOS (Miracle/Canasta) where
you want to distinguish the quomma (q) from the other melodic step (s-2q,
3 steps from 72).

11-limit optimisations always give something so close to 72-equal that the
difference isn't important. In reality, you'd probably tune to some JI
based well temperament.

Graham