Yahoo Tuning Groups Ultimate Backup tuning New Music in 62 Tones (for Manuel Op de Coul)

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New Music in 62 Tones: Enharmonicism with pure concords
Essay in Honor of Manuel Op de Coul
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Hello, everyone, and it is my great pleasure to share a development
drawing on the tradition of three great xenharmonicists: Nicola
Vicentino, Christiaan Huygens, and Adriaan Fokker. This tradition is
also carried forward by a participant on this List with a very special
perspective: Manuel Op de Coul, author of the Scala scale definition
and analysis program most generously made available via the Internet
to interested lovers of music throughout the world.

Here, combining two systems described by the most sage and intrepid
Vicentino in 1555, I would like to show how the expressive power of
enharmonic or fifthtone music may be united to the beauties of purely
intoned concords.

The result at which I have arrived is a gamut of 62 notes per octave,
uniting two 31-note cycles of pure major thirds and tempered fifths,
often known as 1/4-comma meantone. If these two cycles are placed at
the distance of 1/4 syntonic comma, the amount by which the fifths
within each cycle are tempered or "foreshortened," then we may
combine notes from the two cycles so as to achieve the pure concords
which Vicentino so extols.[1] [2]

Since it may be not unfitting that the first examples of a possibly
new thing should be small and modest, so that others, in the noble
words of Johann Walther, may be encouraged to do better, I present a
small piece taking its notes from this 62-tone gamut, _Invocatio in
Quarto Tono_. As explained below, I have taken the liberty of using
tempered sonorities in a few places to avoid certain complications[3]:

MIDI: <http://value.net/~mschulter/invoc4a.mid>

Also, here is a Scala file of the complete 62-tone system with its two
31-note cycles:

<http://value.net/~mschulter/qcm62a.scl>

With due circumspection, I should caution that I do not know whether
or how many times over the last 450 years or so people may have
proposed or devised such a gamut of 62 tones; let this description be
a tribute to all, known or unknown, who may have discovered it
before.

Further, since such a system (whether new or old) should serve as an
invitation to much musical invention, I would propose an experiment
I have not yet attempted myself: to see what kind of art and ingenuity
might be expressed in a composition using all 62 notes of the gamut.

In what follows, first placing this system in a more just historical
perspective while honoring one who has fostered our craft, I shall
explain the nature of the tuning and of the example I have chosen to
illustrate it.

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1. Dedication to Manuel Op de Coul
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Just 500 years ago, in 1501, Ottavio Petrucci published the anthology
of music entitled _Odhecaton_, thus inaugurating a new and high
technology making this art more accessible to a larger public.

Only 54 years later, in 1555, Nicola Vicentino used this same
technology to publish his _Ancient Music Adapted to Modern Practice_,
exhibiting by precept and example his visionary xenharmonic art, and
describing his _archicembalo_ or superharpsichord used in making its
steps and ratios a sonorous reality.

While Vicentino took as the foundation for his basic 31-note cycle the
common practice of "good tuners," likely meaning in this connection a
temperament with pure major thirds, in the latter part of the
following century your illustrious compatriot Christiaan Huygens,
scientist and learned arbiter of beauty, proposed a "New Harmonic
Cycle" dividing the octave into 31 precisely equal parts.

Moving forward about 250 years to one of the most terrible times in
the history of Europe, we find the physicist and friend of humanity
Adriaan Fokker engaged in research which will not aid the National
Socialist occupiers of the Netherlands and many other countries: the
development of new systems of music based on purer concords.

This musical quest makes Fokker a leading theorist and advocate of two
great systems: just intonation, and the 31-tone cycle introduced by
Vicentino and presented in its ideally symmetrical form by Huygens.
The recent Netherlands traditions of 31-tone organs and ensemble music
translating this theory into practice stands as a monument to Fokker's
genius and most eloquent advocacy.

Like Petrucci, you have used a new technology to a high musical
purpose: as the author of Scala, you have provided an invaluable tool
by which we may not only search out, define, and share tunings, but
indeed may publish new compositions as well as reinterpret old ones by
specifying not only the notes but the desired intonation.

In past months and years, I have celebrated in words Vicentino's
system for pure intonation, but with the reservation that it can only
partially and somewhat awkwardly be realized on a conventional
keyboard instrument which a player must negotiate in real time.

With the benefit of Scala, however, it is quite easy to implement this
system for old or new compositions, and indeed to combine it with
Vicentino's other tuning based on a 31-note meantone cycle, enjoying
the wonders of enharmonic progressions and of pure concords alike.

Further, as if at once to make Vicentino's music better known and to
show the variety of intonational choices open in performances of this
music, you have yourself admirably arranged a portion of his madrigal
_Madonna il poco dolce_ using Wuerschmidt's 31-note scale based on a
classic 5-limit system, reminding us of the contributions of Adriaan
Fokker to the art of integer-based just intonation.[4]

By such gifts, reflecting a most august musical tradition and opening
the way to so many musical possibilities in a new century, you have
enriched our xenharmonic community in a way we may celebrate through
words and notes alike.

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2. A dual harmonic cycle: the 62-note system
--------------------------------------------

As explained above, the 62-note system simply combines two 31-note
cycles in 1/4-comma meantone at the distance of 1/4 syntonic comma, a
distance of about 5.38 cents (this comma, at 81:80, having a size of
around 21.51 cents).

It is a fascinating if moot question of history whether Vicentino
himself might have favored and implemented this full 62-note system if
he had been able to build such a large instrument, and find a player
(or mechanism?) capable of realizing its musical potential.

His actual designs were limited to 38 notes per octave, with two
19-note manuals, and his practical instruments to 36 notes, since he
could fit only 17 of the desired 19 notes on his second manual,
omitting the keys corresponding to the E# and B# found on his lower or
regular manual.

In considering his two tuning systems, however, we may find it helpful
to consider his full 38-note gamut.

The first system includes 31 notes forming a complete meantone
system, likely in a 1/4-comma temperament with pure major thirds,
although 31-note equal temperament (31-tET) would result from a
precise implementation of his concept that each whole-tone is divided
into five "minor dieses" or fifthtones, if we take these steps to be
equal. This leaves seven notes on the instrument free for another
purpose.

Vicentino uses them as "comma" keys, tuned a "comma" higher than their
usual diatonic counterparts on the lower manual (F-B) so as to aid or
perfect certain concords. While the term "comma" as used by Vicentino
can mean various small intervals, here it is tempting to read it in
its meaning as the amount by which the fifth is tempered, 1/4 syntonic
comma if we assume pure major thirds at 5:4.

Here is a keyboard diagram for this interpretation of the first
tuning, showing the two manuals and six "ranks" of the instrument,
with an asterisk (*) showing a note raised by a diesis or fifthtone,
typically 128:125 (~41.06 cents) in my version of the gamut, and an
apostrophe (') showing a note raised by a syntonic comma. Vicentino's
36-note implementations do not include C' and F':

Rank 6: C' D' E' F' G' A' B'
Rank 5: Db* Eb* Gb* Ab* Bb*
Rank 4: C* D* E* F* G* A* B* C* Manual 2
----------------------------------------------------------------------
Rank 3: Db D# E# Gb Ab A# B# Manual 1
Rank 2: C# Eb F# G# Bb
Rank 1: C D E F G A B C

In Vicentino's second tuning, the 19 notes of the lower manual are
identical (Gb-B#), but instead of completing a 31-note fifthtone
cycle, all the notes of the upper manual are tuned in "just fifths"
with those of the lower manual (Db'-F##'). This produces a result
_almost_ identical to that of the lower manual raised by 1/4-comma:
instead of Gb' (which would not be a pure fifth to any note on the
lower manual), we get F##' (forming the pure fifth B#-F##').[5]

Rank 6: Db' D#' E# F##' Ab' A#' B#'
Rank 5: C#' Eb' F#' G#' Bb'
Rank 4: C' D' E' F' G' A' B' C' Manual 2
----------------------------------------------------------------------
Rank 3: Db D# E# Gb Ab A# B# Manual 1
Rank 2: C# Eb F# G# Bb
Rank 1: C D E F G A B C

Vicentino's first tuning offers a full 31-note meantone cycle for
fifthtone music, plus what is now often known as "adaptive just
intonation" for a few sonorities where the five or seven "comma keys"
are available (if this is the correct interpretation of these keys).

His second tuning offers a basic 19-tone meantone subset of the full
31-note cycle, over which comma keys are available to obtain justly
intoned sonorities on every step, or almost every step (in the 36-note
version without E#' or B#').

We may take either of these tunings as the basis for a 62-note system
providing a complete 31-note fifthtone cycle with just sonorities
available on the great preponderance of steps -- 27 rather than all
31, as we shall see, owing to the slightly asymmetrical nature of a
31-note cycle based on pure major thirds.

Starting with the first tuning in its 38-note form, we have a complete
31-tone cycle plus "comma keys" for seven of these notes; adding 24
additional comma keys for the remaining notes of the cycle, we have
a gamut of 62 notes in all.

Starting with the second 38-note tuning, we have 19 notes of the basic
31-note cycle, plus comma keys for each of these notes: adding the
remaining 12 notes to complete the basic cycle, plus the 12 comma keys
for these notes, we again have 24 additional notes, or 62 in all.

We now come to a minor asymmetry of a 31-note cycle based on fifths
tempered so that a chain of four (less two octaves) form a pure major
third: 31 such fifths fall short of 18 pure octaves by about 6.07
cents. Curiously, the 31st or "odd" fifth of the cycle is almost pure,
6.07 cents larger than a regular fifth tempered by about 5.38 cents.

Major or minor thirds within each cycle having this near-pure fifth in
their chains will be respectively 6.07 cents larger or narrower than
their usual meantone sizes, so that even when notes from the two
cycles are combined, sonorities on these steps will have tempered
rather than pure concords. One might call this a kind of gentle
"well-temperament" at a few remote places in the system.

Thus four major thirds in a 31-note cycle will have a size of around
392.38 cents, wide by approximately 6.07 cents, while three minor
thirds will be narrowed by this amount plus their usual tempering of
1/4-comma, or about 11.45 cents, at a size of around 304.20 cents.

This curious asymmetry, which means that our 62-note system actually
provides sonorities with pure fifths and pure thirds at 27 locations
rather than 31 locations, might be considered not so much a flaw as an
ornament somewhat imitating nature, which as Zarlino has remarked may
tend toward variety rather than uniformity.

Thus the Earth, although often described as a sphere, in fact slightly
varies from this ideal shape; and our harmonic cycle might also be
said poetically to emulate the shifting phases of the Moon as seen
from this planet.

However, this is by no means to deprecate the cycle of 31 equal steps
as advocated by such illustrious champions as Huygens and Fokker,
which might serve as the basis of a very similar 62-tone system.

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3. An enharmonic composition with pure concords
-----------------------------------------------

To demonstrate the 62-tone system based on pure major thirds in
practice, I wlll provide a score of _Invocatio in Quarto Tono_, with
an asterisk sign (*) raising a note by a diesis or fifthtone of
128:125 (~41.06 cents), and an apostrophe or "quartercomma" sign (')
raising it by 1/4-comma (~5.38 cents):

Here "r" shows a rest; a pause sign (,) in the line showing the rhythm
(which might be taken as 2/2) indicates the conclusion of a phrase:

7
1 2 | 1 2 | 1 2 | 1 2 | 1 2 + | 1 2 | 1 2 ,|
G#4 A4 A4 A4 A4 A4 A4 G#*4
E4 E'4 E'4 F4 E'4 D4 C4 D4 E*4
B'3 C'4 C'4 C'4 C'4 A3 A3 B*'3
E3 A3 A3 F3 A3 F3 F3 E*3

1 2 + | 1 2 + | 1 2 | 1 2 | 1 2 + | 1 2 |
G#*4. G#*4 A*4 B*4 B*4 C'5 C#5 D5
E*4. E*4 E*'4 G*4 G*4 A4 A4 G4 F4 G4
B*'3. B*'3 C*'4 D*'4 D*'4 F4 E'4 D4
E*3. E*3 A*3 G*3 G*3 F3 A3 Bb3

14
1 2 | 1 , 2 | 1 2 | 1 2 | 1 2 | 1 + 2 |
C#*5. C#*5 D*'5 D*'5 D*'5 B*4 C*5 B*4 A*'4 B*4
A*4. A*4 B*4 B*4 B*4 G*4 G'*4 G*4
E*'4. E*'4 G*4 G*4 G*4 D*'4 E*4 D*'4
A*3. A*3 G*3 G*3 G*3 G*3 C*4 G*3

1 2 ,| 1 2 | 1 2 | 1 2 | 1 2 | 1 2 | 1 2 |
C*5 C'5. A4 A'4 A'4 G#*4
G*'4 A4. E'4 D4 D4 E*4
E*4 E'4. C'4 A'3 A'3 B*'3
C*4 A3. A3 F'3 F'3 E*3

1 2 | 1 2 | 1 2 | 1 2 ||
A4 G#4
E'4 E4
C'4 B'3
A2 E3

A few sonorities associated with suspensions (measures 6-7, 12-13, 19)
are left in their regular meantone forms, with usual tempered
concords, as I shall explain below; apart from this, concords have
been adjusted to their pure ratios.

A general rule is to raise the upper note of a fifth or minor third by
a quartercomma (E3-B'3-E4-G#4, A*3-C*'4-E*'4-A*4), and likewise the
lower note of a fourth or major sixth (e.g. F'3-A'4-D4-A'4).

As one type of possible exception to this rule of pure concords, we
may consider the first cadence with its suspension:

7
| 1 2 | 1 2 + | 1 2 | 1 2 ,|
A4 A4 A4 G#*4
E'4 D4 C4 D4 E*4
C'4 A3 A3 B*'3
A3 F3 F3 E*3

Here my intuition was that the unity of the suspension with its
preparation and resolution might call for an unchanging note such as
A4 in the highest voice, resulting in some tempered concords. Also,
one might deem the slight impurity of the sonority F3-A3-D4-A4 at
measure 6, following the suspension, possibly to add a certain subtle
impetus to the progression of major sixth to octave and major third to
fifth, arriving at the pure concord of E*3-B*'3-E*4-G#*4.

However, as Thomas Morley has well said, there are more ways to the
woods than one, and those favoring pure concords wherever they do not
cause major inconvenience might prefer a version like this:

7
| 1 2 | 1 2 + | 1 2 | 1 2 ,|
A4 A4 A'4 G#*4
E'4 D4 C4 D4 E*4
C'4 A3 A'3 B*'3
A3 F3 F'3 E*3

In this version there are still some tempered concords around the
suspension itself, with a yet purer version requiring quartercomma
shifts in the very process of the resolution:

7
| 1 2 | 1 2 + | 1 2 | 1 2 ,|
A4 A4 A'4 A4 A'4 G#*4
E'4 D4 C'4 D4 E*4
C'4 A3 A'3 A3 A'3 B*'3
A3 F3 F'3 F3 F'3 E*3

Often, as at measure 19, we encounter sonorities with intervals all of
which cannot be pure within our 62-note system:

20
| 1 2 | 1 + 2 | 1 2 , |
B*4 C*5 B*4 A*'4 B*4 C*5
G*4 G'*4 G*4 G*'4
D*'4 E*4 D*'4 E*4
G*3 C*4 G*3 C*4

Neither the suspended fourth or eleventh C*5 nor the ornamental major
second or ninth A*'4 can participate in a sonority with all ratios
pure (6:9:12:16 for the first sonority, and 4:6:8:9 for the second).
In this kind of style, however, such sonorities are regarded as
discords, so that the customary tempering of some of these fifths or
fourths by 1/4 comma might be accounted of little musical moment.

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4. A conclusion, or envoi
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As an amicable and prudent counsel, I might guess that many ardent
lovers of pure concords may be less acquainted with the subtleties of
enharmonic or fifthtone music after the noble manner of Vicentino.
For such listeners, it may well be the enharmonic shifts and
progressions which especially catch the ear -- however pleasantly or
otherwise.

In the age of Petrucci, Desiderius Erasmus of Rotterdam discoursed
most eloquently upon the adage _Dulce bellum inexpertis_, "War is
sweet to the inexperienced," offering much wisdom and humanity which
might yet serve the cause of peace and concord.

Might fifthtones, at least for some listeners, have the contrary
property of growing sweeter with experience? Given the diversity of
tastes, I would seek only to call attention to the special quality of
these small steps, which some of us along with Vicentino find
"gentle," and, I might add, at the same time often delightfully
astonishing.

It may have happened more than once that an unsuspecting theorist,
upon first encoutering this enharmonic music, has taken as a comma
shift for the sake of pure intonation what is actually a deliberate
fifthtone inflection, and has ventured to propose a solution in which
just concords may be achieved more "smoothly."

Such are the vagaries of music by which even the most learned may be
taken unawares, and here a friendly explanation may do much to nourish
both understanding and mutual charity, however tastes may agree or
diverge.

May the genius and beauty of Vicentino's music and tunings, whether
realized through a 62-tone system or some other, lend a richer flavor
to the musics of a new century and millennium.

-----
Notes
-----

1. In his circular of 1561 advertising the arciorgano, this "perfect
diatonic music" is described along with its marvellous effects, see,
Henry W. Kaufman, "Vicentino's Arciorgano: An Annotated Translation,"
_Journal of Music Theory_ 5:32-53 (1961) at 34-35.: "First there are
obtained perfect fifths above the white keys of the common organ,
which make a wonderful sound; then two kinds of thirds, one major, one
minor, and similarly, two kinds of sixths, in which case it happens
that whenever perfect fifths are struck together with perfect thirds,
they fill the ears with such harmony that no better can be heard on
earth."

2. Actually, there are two slightly different alternatives for
arranging the 62-note system based on a 31-note cycle of 1/4-comma
meantone. The first alternative would be literally to transpose the
first cycle up by a quartercomma; the second is to transpose up by a
pure 3:2 fifth, and order all notes within a single octave. Using a
31-note cycle of Fbb-G##, these alternatives would respectively yield
an additional 31-note cycle at Fbb'-G##' -- an apostrophe (') here
showing a note raised by a quartercomma -- or Cbb'-D##'. Apart from
the question of Fbb' or D##', the two methods yield identical 62-note
tunings. In 1/4-comma meantone, the notes Fbb' and D##' differ by the
"31-note diesis" of about 6.07 cents by which 31 tempered fifths fall
short of 18 pure octaves, with Fbb' higher than D##'.

3. While, with a few exceptions of a kind discussed in Section 3, the
sonorities in this piece should be pure as defined in the MIDI file
(to within the nearest tuning unit), noticeable beating within these
sonorities might nevertheless occur depending on various factors.

4. This arrangement is included with Scala as madonna.seq, a MIDI
input file which may be used with the EXAMPLE command to generate a
MIDI file. When I happened to learn of the EXAMPLE feature only about
two months ago, madonna.seq dramatized for me the power of this
feature and served as a convenient model or template for my own first
efforts.

5. Following Vicentino's statement that in this tuning every key on
the second manual is tuned in a "just fifth" to a key on the first
manual (e.g. C-G'), we have available a pure sonority without a
tempered counterpart on that lower manual: B-D#'-F##'. With the second
manual tuned in the slightly different arrangement of the first
transposed up a quartercomma (Gb'-Db'), we would instead have
available the pure sonority Eb-Gb'-Bb' in addition to the usual
tempered form Eb-Gb-Bb already available on the lower manual.

Most appreciatively,

Margo Schulter
mschulter@value.net

New Music in 62 Tones (for Manuel Op de Coul)

🔗mschulter <MSCHULTER@VALUE.NET>

🔗manuel.op.de.coul@eon-benelux.com

🔗Clark <CACCOLA@NET1PLUS.COM>

🔗jpehrson@rcn.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗manuel.op.de.coul@eon-benelux.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗manuel.op.de.coul@eon-benelux.com