FW: Unequal Temperaments book, 3rd revised edition
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Claudio Di Veroli
Aug 30, 2013, 12:42:07 PM8/30/13
to micro...@googlegroups.com
Dear Friends and
Colleagues:
In the four years
elapsed after the successful and favourably-reviewed 2009 edition, the
Unequal Temperaments book has undergone a final thorough revision for
completeness and consistency: the result is the recently
launched definitive 3rd edition. In the process a few topics were
identified that deserved improvement, the text has been fully reviewed and
expanded, many figures and charts improved and new ones added. The following
topics have undergone significant additions and/or enhancements:
- Werkmeister III temperament
- Vallotti/Young temperament
- Neidhardt's temperaments
- The search for J.S. Bach's temperament
- Tuning for keyboard duets or trios
- Historical review of temperaments in performance
- History of variable intonation for winds and strings
- Historical trends in the French tempérament ordinaire
- Leigh-Silver's Equal-Beating temperament
- Temperament and harpsichord/clavichord stringing
- Unequal Temperament Spreadsheet (download)
There are also quite a few minor additions or enhancements:
- Schlick's temperament
- Historical genesis of meantone temperament
- Gallimard's logarithmic tunings
- Intonation for brass wind instruments
- Mathematical coincidences in music
- Literature Cited (44 new entries)
... and many others.
- Werkmeister III temperament
- Vallotti/Young temperament
- Neidhardt's temperaments
- The search for J.S. Bach's temperament
- Tuning for keyboard duets or trios
- Historical review of temperaments in performance
- History of variable intonation for winds and strings
- Historical trends in the French tempérament ordinaire
- Leigh-Silver's Equal-Beating temperament
- Temperament and harpsichord/clavichord stringing
- Unequal Temperament Spreadsheet (download)
There are also quite a few minor additions or enhancements:
- Schlick's temperament
- Historical genesis of meantone temperament
- Gallimard's logarithmic tunings
- Intonation for brass wind instruments
- Mathematical coincidences in music
- Literature Cited (44 new entries)
... and many others.
Please find more
details in the Unequal Temperaments webpage:
I have especially
prepared a small sample of the eBook, which includes the full
Contents list, the Introduction and the final list of Literature Cited. You
are welcome to download it:
Bray Baroque books are sold exclusively by Lulu.com:
http://www.lulu.com/spotlight/BrayBaroque
I wish to express my gratitude to Fred Sturm and other temperament-savvy musicians who have suggested valuable ideas for improvement, to John O'Hagan of Bray Baroque who edited the new text, and to the readers of previous editions for their comments and encouragement. The new edition has already been endorsed not only by Fred Sturm but also by the world-leading tuning and temperaments scholar Prof. Patrizio Barbieri.
Best Regards,
Claudio Di Veroli
Bray Baroque
Ireland
Margo Schulter
Sep 2, 2013, 6:48:33 PM9/2/13
to Abridged Recipients
> "Claudio Di Veroli" <d...@braybaroque.ie> Aug 30 05:42PM +0100 �
Please let me take this occasion, Claudio, to congratulate you on
the new edition of your monumental contribution, which I look forward
to reading and reviewing.
Unfortunately, I find that Google Groups is no longer accessible
with my text-based web browsers, so I can only read the e-mail
summaries, at least unless/until I can get access on some other
machine and ask for e-mail delivery of full message texts. However,
looking at the website for Unequal Temperaments gives me a good idea
of the scope of the new edition.
Reading some of the material introducing the new edition, and also
some reviews and debates relating to the 2009 edition, has not only
prepared me for my coming encounter with the 2013 edition, but
given me a deeper appreciation of some basic dilemmas in seeking
out historical tunings, or in crafting new tuning systems which
may draw on historical possibilities, but in new rather than
established ways.
For example, I regard 17-note and 24-note unequal temperaments
drawing on medieval European and Near Eastern music as very
valuable and creative, indeed a main focus for much of my
own endeavor both in designing intonational systems and in
applying them to practical music -- but such schemes are
certainly "un-historical," in the sense that they may fit
some alternative history, but not what has been recorded
or can reasonably be inferred from what has come down to
us. I'll enlarge on this in another message, but my purpose
is simply to draw a distinction between what is known of
previous centuries and what might be designed based on a
knowledge of those centuries plus the influence of various
current trends, well-known or otherwise.
One of the most fascinating aspects of Unequal Temperaments,
and one I much look forward to reading about more in the
2013 edition, is a proposed tuning of Couperin which I
understand involves tuning nine fifths in a standard
1/4-comma meantone (F-G#), with G#/Ab-D#/Eb-A#/Bb-F all
equally wide, so that F-G# remains at 269 cents or a
near-7:6, and there are three large fifths at around
710.3 cents, or 8.3 cents wide -- a bit more than 1/3
of a Pythagorean comma.
What immediately strikes me about this temperament is
that it preserves a pure 5:4 at E-G#, the vital closing
major third for the Fourth Tone on E, also known as the
Phrygian mode.
A glance at the compositions of Chaumont, for example,
will show that the Fourth Tone remains of vital importance,
with its characteristic cadences involving a descending
semitone -- known in Renaissance terminology as "remissive,"
which in Thomas Morley's English (1597) is expressed as a
"flat cadence."
While at least one review has challenged Claudio's
description of 1/4-comma as "Standard Meantone," this
understanding was apparently common in the early to
middle 17th century in a number of authors -- for example,
in 1666, when Lemme Rossi used the general term _sistema
participata_ (which might be translated "tempered system")
for 1/4-comma, in contrast to the minutely different equal
31 division. Likewise, Jean Denis in France (1643, 1650)
says that the fifths on well-tuned harpsichords are
narrowed by a "point" -- which Mersenne, who relied on
Dennis as a source, tells us is a term of art for 1/4 comma.
Remarks both of Vicentino (1555) and Colonna (1618) that
the first 12 or 19 notes of their 31-note meantone cycles
are tuned as in common practice also suggest the widespread
(although not universal or invariable) use of 1/4-comma.
A critical feature of Claudio's proposed Couperin tuning
is that it provides pure 5:4 major thirds above the finals
or resting notes of all the traditional modes, and their
counterparts in the 17th-century system of "Eight Tones":
F-A; C-E; G-B; D-F#; A-C#; and E-G#. These six regular
steps -- in contrast to the mutable step Bb/B of the
medieval and Renaissance diatonic, as well as the added
accidentals -- are a focus of theoretical interest as
early as Prosdocimus de Beldemandis and Ugolino of Orvieto
in the earlier 15th century, who advocate a 17-note
Pythagorean division (Gb-A#) to permit cadences with either
ascending or descending semitones, and regular Pythagorean
intervals, on all six of these steps.
Another important point is that French discussions of the
"transposed modes" near the end of the 17th century, including
remarks that that some musicians much relish these remote
transpositions where usual diatonic semitones serve as
chromatic semitones, and vice versa, very much fit a scheme
such as 1/4-comma meantone with only one or two notes altered
in order to alleviate the wolf fifth or certain wolf thirds.
Both for Claudio and for the renowned Patrizio Barbieri, we
sometimes find confirmatory evidence in intriguing places.
For example, the Lucca organ of the 1480's, early in the
meantone era, is widely interpreted as having split keys
at G#/Ab and Eb/D#. Barbieri explains that transposing
the First Tone or Dorian Mode on D to F was a compelling
motivation for Ab -- a statement that lends new import
to an amusing account by Jean Denis (1643, 1650).
According to Denis, a strong advocate of 1/4-comma who
urges that lute players who want consistent intonation
with keyboards should correct the imperfect equal
semitones on _their_ instruments, an organist should
_never_ agree to play the First Tone on F (this assumes
a usual Eb-G# instrument), even if singers ask for it!
He tells a story of an organist who, rather than play
in such a transposition, would change the pitch level
for the organ interludes when performing with a choir.
The organist's explanation: "You sing at your pleasure,
and I play at mine!"
Another caution comes from a number of 16th-17th century
theorists, with Huygens as an interesting exception: We
should not assume that because a 7:6 minor third is a
delightful interval to many modern ears, it was equally
delightful even to the very avant-garde 16th-century ears
of Nicola Vicentino, who found a neutral third at around
11:9 rather consonant and made this third part of his
enharmonic cadential technique!
Vicentino himself (1555), although quite charmed by
neutral thirds, cautions that a "minimal third" at
7/5 of a tone in his approximate 31 division tends
toward the perceived dissonance of a major second,
and should be used with caution.
Praetorius, some 50 years later, actually remarks
that the term "wolf," while it can relate to an
augmented sixth such as G#-Eb viewed as a dissonant
fifth, often applies to F-G#, an augmented second
(equivalent to a near-just 7:6 minor third).
Denis categorically urges that such a "superfluous
second" not be confused with an acceptable minor
third, one of the reasons a wise organist will
refuse to transpose the First Tone from D to F
(Barbieri's very credible explanation for a
split key at Ab/G#).
Praetorius also raises a possibility very strongly
championed by Werckmeister for those will not
move beyond the "Praetorian temperament" -- that is,
1/4-comma meantone, again seen as the old standard.
This is the idea of finessing G# so as to make it
a more or less marginally acceptable Ab also. The
basic idea, of course, goes back to Arnold Schlick
(1511), who may have had a somewhat easier task
because of his own taste for a bit less temperament
than 1/4-comma, and also for tempering the
accidentals somewhat less than the diatonic notes.
(Schlick conversely wanted to make a usual Ab into
also a marginally acceptable G# for cadences on A.)
From this point of view, Claudio's Couperin
temperament may be seen partly as an attempt to
alter 1/4-comma meantone as little as possible
while arriving at a "semi-circulating system."
For the musician who either shares the more
favorable view of 7:6 thirds urged by Huygens,
who celebrates them as an ornament of the 31
division; or who enjoys the intriguing (although
not necessarily "consonant") color of the
"transposed modes" where augmented or diminished
intervals substitute for usual diatonic ones,
such a solution will have great allure. And it
preserves pure 5:4 thirds above all six of the
traditional and stable finals for the modes,
a consideration still relevant for a modal
worldview like Chaumont's.
In such a context, Claudio's comment which I
have seen quoted in a review of the 2009 edition
of Unequal Temperaments, predicting that pieces
venturing into remote transpositions will likely
have quite "dissonant" passages in an irregular
French temperament like the proposed Couperin
scheme, is quite sound. The term "dissonant"
is meant to be read in period context, where
a minor third near 7:6 is the "wolf" of
Praetorius, and likewise the most remote major
thirds (or diminished fourths) of French schemes
interpreted by Mark Lindley, tuning these thirds
at around 415 cents, are indeed radical departures
from the usual thirds of meantone at or around
5:4 and 6:5.
Out of this historical context, of course, judgments
might be quite different. For example, many of my
(decidedly un-historical!) temperaments have regular
major and minor thirds at around 415/289 cents, so
that the temperament ordinaire extreme becomes my
diatonic norm! However, I would be rather incautious
to read this norm back into the later 17th century!
Similarly, we sometimes see statements in some
books and articles on historical European temperaments
proposing that "a narrow minor third does not become
a wolf, but simply approximates a 7:6 third." While
Huygens (and later Euler) might well agree with this
statement, we find that the quite avant-garde Vicentino,
as well as Praetorius, Denis, and Werckmeister, take a
decidedly different view.
As it happens, in my un-historical (or perhaps historical
21st-century) temperaments, regular major thirds near 14:11
(418 cents) and minor sixths thus near 11:7 (782 cents) are
a kind of trademark. However, this does not necessarily
imply that, as urged in one article on 17th-18th century
tunings, 1/6-comma meantone would be especially acceptable
to Bach's contemporaries because the augmented fifth closely
approximates 11:7. In fact, my own temperaments favoring
11:7 (and 14:11) do not attempt to support the ratios 5:4
and 6:5 upon which Renaissance and Baroque music is, of
course, mainly premised, although a few such intervals may
sometimes occur in remote positions.
One of the challenges facing not only Claudio but any
historian seeking to reconstruct or at least suggest some
likely parameters for Renaissance or Baroque intonational
practice is that the language of theorists leaves room
for interpretation. Here I will briefly mention one
instance.
In 1511, Arnold Schlick gives a fascinating description
of an irregular organ tuning, which Mark Lindley has
analyzed in great detail. One of things that we learn
from Schlick is that, in his scheme for making a
serviceable Ab (favored by Ramos in 1482) also a
marginally useable G# in ornamented cadences to A,
the fifth C#-Ab/G# will be too large -- but this is
of little concern, such this fifth or fourth is rarely
used.
To Lindley, this means we must have a "wolf fifth" at
C#-Ab/G#. However, I would propose that while C#-Ab/G#
was evidently intended by Schlick to be notably more
impure and less attractive than the other fifths, it
need not be an absolute "wolf" as judged by some
later standards.
Here an important bit of evidence is that Schlick
evidently feels that a full 1/4-comma of temperament,
which would be needed for pure 5:4 major thirds, is
too much to ask the fifths to "suffer" (Schlick's
telling expression as translated by Lindley) for
their thirds. In other words, 5.38 cents would be
excessive for Schlick's taste.
This leaves the possibility that if the other fifths
are tempered by around 3-4 cents (narrow or wide),
then C#-Ab/G# at around 6-7 cents wide might be
impure enough to justify Schlick's caution, although
still acceptable by some more flexible meantone-era
standards like those of Costeley or Salinas (who
favor the 19-division with its tempering of the
fifths at 7.22 cents narrow, or just over 1/3 sytonic
comma at 7.17 cents).
In other words, Schlick's system might not be fully
circulating by his own criteria (as applied to the
widest diminished fourths or wolf thirds, "little
esteemed and seldom used," as well as the notably
impure C#-Ab/G#), and yet circulating by some more
relaxed criteria. While Lindley's own standard that
C#-Ab/G# should be a "wolf fifth" specifies many
legitimate historical possibilities that Schlick
and his followers may well have tuned, it does not
necessarily exhaust these possibilities.
To place Claudio in the same league with Lindley
and Barbieri is, I hope, rightly taken both as
a high compliment to the erudition and devoted
practical musicianship of our author, and a
statement of my pleasant anticipation of reading
the new _Unequal Temperaments_.
With many thanks,
Margo Schulter
msch...@calweb.com
>
> Dear Friends and Colleagues:
> In the four years elapsed after the successful and favourably-reviewed
> 2009 edition, the Unequal Temperaments book has undergone a final
> thorough revision for ...more
> Dear Friends and Colleagues:
> In the four years elapsed after the successful and favourably-reviewed
> 2009 edition, the Unequal Temperaments book has undergone a final
Please let me take this occasion, Claudio, to congratulate you on
the new edition of your monumental contribution, which I look forward
to reading and reviewing.
Unfortunately, I find that Google Groups is no longer accessible
with my text-based web browsers, so I can only read the e-mail
summaries, at least unless/until I can get access on some other
machine and ask for e-mail delivery of full message texts. However,
looking at the website for Unequal Temperaments gives me a good idea
of the scope of the new edition.
Reading some of the material introducing the new edition, and also
some reviews and debates relating to the 2009 edition, has not only
prepared me for my coming encounter with the 2013 edition, but
given me a deeper appreciation of some basic dilemmas in seeking
out historical tunings, or in crafting new tuning systems which
may draw on historical possibilities, but in new rather than
established ways.
For example, I regard 17-note and 24-note unequal temperaments
drawing on medieval European and Near Eastern music as very
valuable and creative, indeed a main focus for much of my
own endeavor both in designing intonational systems and in
applying them to practical music -- but such schemes are
certainly "un-historical," in the sense that they may fit
some alternative history, but not what has been recorded
or can reasonably be inferred from what has come down to
us. I'll enlarge on this in another message, but my purpose
is simply to draw a distinction between what is known of
previous centuries and what might be designed based on a
knowledge of those centuries plus the influence of various
current trends, well-known or otherwise.
One of the most fascinating aspects of Unequal Temperaments,
and one I much look forward to reading about more in the
2013 edition, is a proposed tuning of Couperin which I
understand involves tuning nine fifths in a standard
1/4-comma meantone (F-G#), with G#/Ab-D#/Eb-A#/Bb-F all
equally wide, so that F-G# remains at 269 cents or a
near-7:6, and there are three large fifths at around
710.3 cents, or 8.3 cents wide -- a bit more than 1/3
of a Pythagorean comma.
What immediately strikes me about this temperament is
that it preserves a pure 5:4 at E-G#, the vital closing
major third for the Fourth Tone on E, also known as the
Phrygian mode.
A glance at the compositions of Chaumont, for example,
will show that the Fourth Tone remains of vital importance,
with its characteristic cadences involving a descending
semitone -- known in Renaissance terminology as "remissive,"
which in Thomas Morley's English (1597) is expressed as a
"flat cadence."
While at least one review has challenged Claudio's
description of 1/4-comma as "Standard Meantone," this
understanding was apparently common in the early to
middle 17th century in a number of authors -- for example,
in 1666, when Lemme Rossi used the general term _sistema
participata_ (which might be translated "tempered system")
for 1/4-comma, in contrast to the minutely different equal
31 division. Likewise, Jean Denis in France (1643, 1650)
says that the fifths on well-tuned harpsichords are
narrowed by a "point" -- which Mersenne, who relied on
Dennis as a source, tells us is a term of art for 1/4 comma.
Remarks both of Vicentino (1555) and Colonna (1618) that
the first 12 or 19 notes of their 31-note meantone cycles
are tuned as in common practice also suggest the widespread
(although not universal or invariable) use of 1/4-comma.
A critical feature of Claudio's proposed Couperin tuning
is that it provides pure 5:4 major thirds above the finals
or resting notes of all the traditional modes, and their
counterparts in the 17th-century system of "Eight Tones":
F-A; C-E; G-B; D-F#; A-C#; and E-G#. These six regular
steps -- in contrast to the mutable step Bb/B of the
medieval and Renaissance diatonic, as well as the added
accidentals -- are a focus of theoretical interest as
early as Prosdocimus de Beldemandis and Ugolino of Orvieto
in the earlier 15th century, who advocate a 17-note
Pythagorean division (Gb-A#) to permit cadences with either
ascending or descending semitones, and regular Pythagorean
intervals, on all six of these steps.
Another important point is that French discussions of the
"transposed modes" near the end of the 17th century, including
remarks that that some musicians much relish these remote
transpositions where usual diatonic semitones serve as
chromatic semitones, and vice versa, very much fit a scheme
such as 1/4-comma meantone with only one or two notes altered
in order to alleviate the wolf fifth or certain wolf thirds.
Both for Claudio and for the renowned Patrizio Barbieri, we
sometimes find confirmatory evidence in intriguing places.
For example, the Lucca organ of the 1480's, early in the
meantone era, is widely interpreted as having split keys
at G#/Ab and Eb/D#. Barbieri explains that transposing
the First Tone or Dorian Mode on D to F was a compelling
motivation for Ab -- a statement that lends new import
to an amusing account by Jean Denis (1643, 1650).
According to Denis, a strong advocate of 1/4-comma who
urges that lute players who want consistent intonation
with keyboards should correct the imperfect equal
semitones on _their_ instruments, an organist should
_never_ agree to play the First Tone on F (this assumes
a usual Eb-G# instrument), even if singers ask for it!
He tells a story of an organist who, rather than play
in such a transposition, would change the pitch level
for the organ interludes when performing with a choir.
The organist's explanation: "You sing at your pleasure,
and I play at mine!"
Another caution comes from a number of 16th-17th century
theorists, with Huygens as an interesting exception: We
should not assume that because a 7:6 minor third is a
delightful interval to many modern ears, it was equally
delightful even to the very avant-garde 16th-century ears
of Nicola Vicentino, who found a neutral third at around
11:9 rather consonant and made this third part of his
enharmonic cadential technique!
Vicentino himself (1555), although quite charmed by
neutral thirds, cautions that a "minimal third" at
7/5 of a tone in his approximate 31 division tends
toward the perceived dissonance of a major second,
and should be used with caution.
Praetorius, some 50 years later, actually remarks
that the term "wolf," while it can relate to an
augmented sixth such as G#-Eb viewed as a dissonant
fifth, often applies to F-G#, an augmented second
(equivalent to a near-just 7:6 minor third).
Denis categorically urges that such a "superfluous
second" not be confused with an acceptable minor
third, one of the reasons a wise organist will
refuse to transpose the First Tone from D to F
(Barbieri's very credible explanation for a
split key at Ab/G#).
Praetorius also raises a possibility very strongly
championed by Werckmeister for those will not
move beyond the "Praetorian temperament" -- that is,
1/4-comma meantone, again seen as the old standard.
This is the idea of finessing G# so as to make it
a more or less marginally acceptable Ab also. The
basic idea, of course, goes back to Arnold Schlick
(1511), who may have had a somewhat easier task
because of his own taste for a bit less temperament
than 1/4-comma, and also for tempering the
accidentals somewhat less than the diatonic notes.
(Schlick conversely wanted to make a usual Ab into
also a marginally acceptable G# for cadences on A.)
From this point of view, Claudio's Couperin
temperament may be seen partly as an attempt to
alter 1/4-comma meantone as little as possible
while arriving at a "semi-circulating system."
For the musician who either shares the more
favorable view of 7:6 thirds urged by Huygens,
who celebrates them as an ornament of the 31
division; or who enjoys the intriguing (although
not necessarily "consonant") color of the
"transposed modes" where augmented or diminished
intervals substitute for usual diatonic ones,
such a solution will have great allure. And it
preserves pure 5:4 thirds above all six of the
traditional and stable finals for the modes,
a consideration still relevant for a modal
worldview like Chaumont's.
In such a context, Claudio's comment which I
have seen quoted in a review of the 2009 edition
of Unequal Temperaments, predicting that pieces
venturing into remote transpositions will likely
have quite "dissonant" passages in an irregular
French temperament like the proposed Couperin
scheme, is quite sound. The term "dissonant"
is meant to be read in period context, where
a minor third near 7:6 is the "wolf" of
Praetorius, and likewise the most remote major
thirds (or diminished fourths) of French schemes
interpreted by Mark Lindley, tuning these thirds
at around 415 cents, are indeed radical departures
from the usual thirds of meantone at or around
5:4 and 6:5.
Out of this historical context, of course, judgments
might be quite different. For example, many of my
(decidedly un-historical!) temperaments have regular
major and minor thirds at around 415/289 cents, so
that the temperament ordinaire extreme becomes my
diatonic norm! However, I would be rather incautious
to read this norm back into the later 17th century!
Similarly, we sometimes see statements in some
books and articles on historical European temperaments
proposing that "a narrow minor third does not become
a wolf, but simply approximates a 7:6 third." While
Huygens (and later Euler) might well agree with this
statement, we find that the quite avant-garde Vicentino,
as well as Praetorius, Denis, and Werckmeister, take a
decidedly different view.
As it happens, in my un-historical (or perhaps historical
21st-century) temperaments, regular major thirds near 14:11
(418 cents) and minor sixths thus near 11:7 (782 cents) are
a kind of trademark. However, this does not necessarily
imply that, as urged in one article on 17th-18th century
tunings, 1/6-comma meantone would be especially acceptable
to Bach's contemporaries because the augmented fifth closely
approximates 11:7. In fact, my own temperaments favoring
11:7 (and 14:11) do not attempt to support the ratios 5:4
and 6:5 upon which Renaissance and Baroque music is, of
course, mainly premised, although a few such intervals may
sometimes occur in remote positions.
One of the challenges facing not only Claudio but any
historian seeking to reconstruct or at least suggest some
likely parameters for Renaissance or Baroque intonational
practice is that the language of theorists leaves room
for interpretation. Here I will briefly mention one
instance.
In 1511, Arnold Schlick gives a fascinating description
of an irregular organ tuning, which Mark Lindley has
analyzed in great detail. One of things that we learn
from Schlick is that, in his scheme for making a
serviceable Ab (favored by Ramos in 1482) also a
marginally useable G# in ornamented cadences to A,
the fifth C#-Ab/G# will be too large -- but this is
of little concern, such this fifth or fourth is rarely
used.
To Lindley, this means we must have a "wolf fifth" at
C#-Ab/G#. However, I would propose that while C#-Ab/G#
was evidently intended by Schlick to be notably more
impure and less attractive than the other fifths, it
need not be an absolute "wolf" as judged by some
later standards.
Here an important bit of evidence is that Schlick
evidently feels that a full 1/4-comma of temperament,
which would be needed for pure 5:4 major thirds, is
too much to ask the fifths to "suffer" (Schlick's
telling expression as translated by Lindley) for
their thirds. In other words, 5.38 cents would be
excessive for Schlick's taste.
This leaves the possibility that if the other fifths
are tempered by around 3-4 cents (narrow or wide),
then C#-Ab/G# at around 6-7 cents wide might be
impure enough to justify Schlick's caution, although
still acceptable by some more flexible meantone-era
standards like those of Costeley or Salinas (who
favor the 19-division with its tempering of the
fifths at 7.22 cents narrow, or just over 1/3 sytonic
comma at 7.17 cents).
In other words, Schlick's system might not be fully
circulating by his own criteria (as applied to the
widest diminished fourths or wolf thirds, "little
esteemed and seldom used," as well as the notably
impure C#-Ab/G#), and yet circulating by some more
relaxed criteria. While Lindley's own standard that
C#-Ab/G# should be a "wolf fifth" specifies many
legitimate historical possibilities that Schlick
and his followers may well have tuned, it does not
necessarily exhaust these possibilities.
To place Claudio in the same league with Lindley
and Barbieri is, I hope, rightly taken both as
a high compliment to the erudition and devoted
practical musicianship of our author, and a
statement of my pleasant anticipation of reading
the new _Unequal Temperaments_.
With many thanks,
Margo Schulter
msch...@calweb.com
Ozan Yarman
Sep 3, 2013, 5:59:16 PM9/3/13
to micro...@googlegroups.com
Dear Margo,
I quite enjoyed your comments on Claudio Di Veroli's book. I share with you similar tastes regarding the riches and utility of Modified Meantone Temperaments both harmony-wise and melody-wise, as exemplified by my preference for such circulating solutions also contained in my three proposals for Maqam music: Yarman-24, Yarman-36, and 79 MOS 159-tET. In each instance, one finds a "Temperament Ordinaire" nested among the other microtonal pitches minimally needed for maqam intonation.
I would like to iterate upon this oppurtunity my conclusive Yarman-24c tuning solution with a 12-tone Modified Meantone Temperament core in the style of Rameau (here featuring MIDI pitch bend units):
=====================
C -240
B‡ 773 white key
=====================
C# -852
Dƀ 1547 black key
=====================
D -80
Dd -577 white key
=====================
D# -551
Eƀ 1740 black key
=====================
E -914
Ed -1776 white key
=====================
=====================
F -320
E‡ -387 white key
=====================
F# -1003
Gƀ 1160 black key
=====================
G -350
Gd 1754 white key
=====================
G# -702
Aƀ 1933 black key
=====================
A 0
Ad 1547 white key
=====================
A# -400
Bƀ 1740 black key
=====================
B -834
Bd -1389 white key
=====================
Following is the FINAL Yarman24-c and its natural extension Yarman-31c...
12-tone Modified Meantone Temperament core cycle between notes C and c (from perde rast):
695.88538026004839 cents
695.88538026004839 cents
695.88538026004839 cents
695.88538026004839 cents
701.955000865387092 cents (3/2)
695.88538026004839 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
Cycle of 17 fifths onward from C (again perde rast):
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
whereby, in order to complete the tuning to 24-tones per octave, the extra-cyclic perdes (dik kurdi & dik ajem) can verily be connected from a 853.063064827076914 cent perde hisar via acceptably tempered fifths with a size of 695.2801937769087317.
> (Yarman24-31d is simply an algebraic equalization of the two different generator meantone fifths at 695.2801937769087317 and 695.88538026004839 cents respectively to a single fifth size of 695.424285796704 cents)
Hence, Yarman-24c is ultimately given as:
24-tone maqam music tuning with 12-tones tempered in the style of Rameau's modified meantone and 17 tones produced by cycle of super-pyth fifths
|
0: 1/1 C Dbb unison, perfect prime RAST ♥
1: 83.059 cents C# Db nim zengule
2: 143.623 cents zengule
3: 191.771 cents C## Dd dik zengule
4: 9/8 D Ebb major whole tone DÜGAH ♥
5: 292.413 cents D# Eb kürdi
6: 348.343 cents D#| Eb- dik kürdi
7: 362.503 cents nerm segah
8: 156/125 cents E SEGAH ♥
9: 415.305 cents E| Buselik
10: 4/3 F Gbb perfect fourth ÇARGAH ♥
11: 581.382 cents F# Gb nim hicaz
12: 634.184 cents hicaz
13: 695.885 cents F## Gd dik hicaz
14: 3/2 G Abb perfect fifth NEVA ♥
15: 788.736 cents G# Ab nim hisar
16: 853.063 cents hisar
17: 887.656 cents G## Ad dik hisar
18: 27/16 A Bbb Pyth. major sixth HÜSEYNİ ♥
19: 16/9 A# Bb Pyth. minor seventh acem
20: 1043.623 cents A#| Bb- dik acem
21: 1071.942 cents nerm eviç
22: 234/125 cents B EVİÇ ♥
23: 1124.744 cents B| mahur
24: 2/1 C Dbb octave GERDANİYE ♥
Exhausting the notational venues of Arel-Ezgi-Uzdilek under the Yarman-24 cast by prolonging the 695.2801937769087317 cents sized fifths up perde hisar through an auxiliary array of F‡, C‡, G‡, D‡, A‡, brings us to E‡ (perde buselik) and produces yet another complete cycle. Adding to this mixture a link to Fd and Cd starting from Gd down, we attain Yarman-31c:
Yarman24c extended to 31 notes using missing "comma" flats and sharps
31
!
34.18384
85.05893
143.62345
191.77076
9/8
224.74423
292.41297
348.34326
362.50268
156/125
415.30462
477.00616
4/3
538.90365
581.38190
634.18384
695.88538
3/2
729.46403
788.73595
853.06306
887.65614
27/16
920.02442
16/9
1043.62345
1071.94229
234/125
1124.74423
1186.44577
2/1
Rationalized and made fretting-friendly:
51/50
21/20
88/81
19/17
9/8
33/29
58/49
11/9
69/56
156/125
75/59
54/41
4/3
86/63
7/5
75/52
16/11 (16/11 for Gd instead of 148/99 at 648.682 cents for practical fretting)
187/125 (187/125 for G instead of 3/2 at 697.3322 cents for a single route to Dd & D)
32/21
41/26
18/11
117/70
27/16
17/10
16/9
42/23
13/7
234/125
67/35
125/63
2/1
As stated previously and directly above in between parantheses, the small 6 cent discrepency between G and Gd is modified to yield a single practicable/frettable perde neva whereby the "fazla flat" (d) could be re-used as a "dik saba/bestenigar" (22/15 being also acceptable). The keyboard mapping of this fretting-friendly variant is correctly reflected to the Chromatic Clavier window in SCALA when set to YA31.
I am currently collaborating with Andrew McPherson, creator of TouchKeys, so that savory maqam demonstrations will be possible on electronic keyboards using this Yarman-24 mapping.
Cordially,
Dr. Oz.
✩ ✩ ✩
www.ozanyarman.com
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I quite enjoyed your comments on Claudio Di Veroli's book. I share with you similar tastes regarding the riches and utility of Modified Meantone Temperaments both harmony-wise and melody-wise, as exemplified by my preference for such circulating solutions also contained in my three proposals for Maqam music: Yarman-24, Yarman-36, and 79 MOS 159-tET. In each instance, one finds a "Temperament Ordinaire" nested among the other microtonal pitches minimally needed for maqam intonation.
I would like to iterate upon this oppurtunity my conclusive Yarman-24c tuning solution with a 12-tone Modified Meantone Temperament core in the style of Rameau (here featuring MIDI pitch bend units):
=====================
C -240
B‡ 773 white key
=====================
C# -852
Dƀ 1547 black key
=====================
D -80
Dd -577 white key
=====================
D# -551
Eƀ 1740 black key
=====================
E -914
Ed -1776 white key
=====================
=====================
F -320
E‡ -387 white key
=====================
F# -1003
Gƀ 1160 black key
=====================
G -350
Gd 1754 white key
=====================
G# -702
Aƀ 1933 black key
=====================
A 0
Ad 1547 white key
=====================
A# -400
Bƀ 1740 black key
=====================
B -834
Bd -1389 white key
=====================
Following is the FINAL Yarman24-c and its natural extension Yarman-31c...
12-tone Modified Meantone Temperament core cycle between notes C and c (from perde rast):
695.88538026004839 cents
695.88538026004839 cents
695.88538026004839 cents
695.88538026004839 cents
701.955000865387092 cents (3/2)
695.88538026004839 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
Cycle of 17 fifths onward from C (again perde rast):
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
709.439612446183125 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
703.67702402589925 cents
701.955000865387092 cents (3/2)
701.955000865387092 cents (3/2)
whereby, in order to complete the tuning to 24-tones per octave, the extra-cyclic perdes (dik kurdi & dik ajem) can verily be connected from a 853.063064827076914 cent perde hisar via acceptably tempered fifths with a size of 695.2801937769087317.
> (Yarman24-31d is simply an algebraic equalization of the two different generator meantone fifths at 695.2801937769087317 and 695.88538026004839 cents respectively to a single fifth size of 695.424285796704 cents)
Hence, Yarman-24c is ultimately given as:
24-tone maqam music tuning with 12-tones tempered in the style of Rameau's modified meantone and 17 tones produced by cycle of super-pyth fifths
|
0: 1/1 C Dbb unison, perfect prime RAST ♥
1: 83.059 cents C# Db nim zengule
2: 143.623 cents zengule
3: 191.771 cents C## Dd dik zengule
4: 9/8 D Ebb major whole tone DÜGAH ♥
5: 292.413 cents D# Eb kürdi
6: 348.343 cents D#| Eb- dik kürdi
7: 362.503 cents nerm segah
8: 156/125 cents E SEGAH ♥
9: 415.305 cents E| Buselik
10: 4/3 F Gbb perfect fourth ÇARGAH ♥
11: 581.382 cents F# Gb nim hicaz
12: 634.184 cents hicaz
13: 695.885 cents F## Gd dik hicaz
14: 3/2 G Abb perfect fifth NEVA ♥
15: 788.736 cents G# Ab nim hisar
16: 853.063 cents hisar
17: 887.656 cents G## Ad dik hisar
18: 27/16 A Bbb Pyth. major sixth HÜSEYNİ ♥
19: 16/9 A# Bb Pyth. minor seventh acem
20: 1043.623 cents A#| Bb- dik acem
21: 1071.942 cents nerm eviç
22: 234/125 cents B EVİÇ ♥
23: 1124.744 cents B| mahur
24: 2/1 C Dbb octave GERDANİYE ♥
Exhausting the notational venues of Arel-Ezgi-Uzdilek under the Yarman-24 cast by prolonging the 695.2801937769087317 cents sized fifths up perde hisar through an auxiliary array of F‡, C‡, G‡, D‡, A‡, brings us to E‡ (perde buselik) and produces yet another complete cycle. Adding to this mixture a link to Fd and Cd starting from Gd down, we attain Yarman-31c:
Yarman24c extended to 31 notes using missing "comma" flats and sharps
31
!
34.18384
85.05893
143.62345
191.77076
9/8
224.74423
292.41297
348.34326
362.50268
156/125
415.30462
477.00616
4/3
538.90365
581.38190
634.18384
695.88538
3/2
729.46403
788.73595
853.06306
887.65614
27/16
920.02442
16/9
1043.62345
1071.94229
234/125
1124.74423
1186.44577
2/1
Rationalized and made fretting-friendly:
51/50
21/20
88/81
19/17
9/8
33/29
58/49
11/9
69/56
156/125
75/59
54/41
4/3
86/63
7/5
75/52
16/11 (16/11 for Gd instead of 148/99 at 648.682 cents for practical fretting)
187/125 (187/125 for G instead of 3/2 at 697.3322 cents for a single route to Dd & D)
32/21
41/26
18/11
117/70
27/16
17/10
16/9
42/23
13/7
234/125
67/35
125/63
2/1
As stated previously and directly above in between parantheses, the small 6 cent discrepency between G and Gd is modified to yield a single practicable/frettable perde neva whereby the "fazla flat" (d) could be re-used as a "dik saba/bestenigar" (22/15 being also acceptable). The keyboard mapping of this fretting-friendly variant is correctly reflected to the Chromatic Clavier window in SCALA when set to YA31.
I am currently collaborating with Andrew McPherson, creator of TouchKeys, so that savory maqam demonstrations will be possible on electronic keyboards using this Yarman-24 mapping.
Cordially,
Dr. Oz.
✩ ✩ ✩
www.ozanyarman.com
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