Re: trias harmonica -- for Herbert Kellner and Joe Monzo

Hello, there, Herbert Kellner and everyone, and please let me that I
am honored to be quoted in your discussion of the _trias harmonica_,
and might respond by sharing some of my own historical perspective,
largely focused on medieval and Renaissance-Manneristic rather than
High Baroque practice.

Please let me also very warmly thank you Joe Monzo, our "Monz," for a most
thoughtful reply which sensitively and accurately explains many of my
views as part of your presentation, as well as offering some extensive and
invaluable documentation.

What follows was mostly written before I read your reply, Monz, and
may reflect our areas of agreement. I've added a couple of points
which your presentation suggested to me, and hope that my musical
examples may add to the dialogue.

Especially, I would like to suggest that medieval Pythagorean
practice, the meantone practice of the Renaissance, and the
well-tempered practice of Bach's era may each have their own musical
"perfection." What is fitting for one era may be "harsh," or at least
unidiomatic, for another.

First, from my perspective, if we are using the term "chord" in a
general way to suggest a combination of three or more notes or
intervals, then I would consider the basic Gothic sonority as the
_trina harmoniae perfectio_ or "threefold perfection of harmony"
described by Johannes de Grocheio in his treatise of around 1300.

The sonority manifesting this perfection, which Grocheio describes as
a kind of feminine counterpart of the Trinity, has a mother octave
(2:1), a daughter fifth (3:2), and an upper fourth (4:3) proceeding
from both of these consonances. An example of this sonority, here
heard as the resolution of a standard 13th-14th century cadence, would
be F3-C4-F4, with C4 as middle C:

MIDI example: <http://value.net/~mschulter/py3ei004.mid>

In English, I refer to Grocheio's sonority manifesting "perfect
harmony" (which he also describes as _consonantia perfectissima_,
requiring at least three voices) as a _trine_.[1]

Another theorist of the same epoch notes that the trine can be
expressed by the "natural" series of numbers 2-3-4, with a 2:3 fifth
placed below a 4:3 fourth in arranging the outer 2:1 octave. As it
happens, this "natural series" suggests the series of harmonic
partials 2:3:4, although partials seem to have been recognized in
Western Europe theory only around the early 17th century, the era of
Galileo, Mersenne, and Descartes.

In a system of music for three or four voices based on the stable
Gothic trine, unstable combinations with complex Pythagorean thirds
and sixths play a vital role in directed progressions as well as
coloristic sonority. For example, consider our cadence above:

E4 F4
B3 C4
G3 F3

<http://value.net/~mschulter/py3ei004.mid>

Here the major third G3-B3 at 81:64 (~407.82 cents) very nicely
expands to a fifth, while the major sixth G3-E3 at 27:16 (~905.87
cents) expands to the outer octave of a complete trine. All three
voices and intervals unite in this purposeful and compelling
resolution.

As Carl Dahlhaus has written, in Gothic Pythagorean tuning there is an
admirable concord between acoustical structure and musical language.

This is not to say that a standard Pythagorean tuning was the only
possibility in Continental Europe during the 14th century. In England
around 1300, for example, Walter Odington comments that the
Pythagorean ratios for major and minor thirds are close to the simple
ratios of 5:4 and 6:5, and that singers can maximize the concord of
these intervals. A "bending" of intonation toward the simplest ratios
seems to fit at least some dialects of English polyphony.

In contrast, around the same epoch, Marchettus of Padua (1318) seems
to describe a system in which cadential major thirds and sixths are
_larger_ than Pythagorean, and cadential semitones _narrower_ than the
already compact Pythagorean diatonic semitone or _limma_ at 256:243
(~90.22 cents).

While the interpretation of his system remains an uncertain and often
debated topic, here is a short composition of mine, "Salutation for
Mary Beth Ackerley, envision'd as Lysistrata," inspired by one
possible interpretation with cadential dieses at around 48 cents:

MIDI example: <http://value.net/~mschulter/mary002.mid>

Without excluding such variations, a conventional Pythagorean tuning
seems to me very nicely to fit most 14th-century music. In a thread
with much discussion of "perfection," I might add that certain
theorists describe a Pythagorean major third as "perfected," meaning
that it actively seeks to expand to the stable fifth.

The nature of musical style, however, is to change, and by around
1400-1420, as you mention, Herbert Kellner, a type of interval present
in a 12-note Pythagorean tuning of the kind common on 14th-century
keyboards has captured the imaginations of musicians.

This is the "schisma third," as it is now often called: actually a
Pythagorean diminished fourth (e.g. C#-F or F#-Bb in a typical
14th-century tuning of Eb-G#).

By the early 15th century, this interval with its curiously smooth
qualities had apparently motivated many musicians to tune the written
sharps of notated music as Pythagorean flats in a chain from Gb to B:

(F#) (C#) (G#)
Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B

Thus major thirds involving written sharps will actually be realized
in this tuning as Pythagorean diminished fourths very close to a pure
ratio of 5:4 (~386.31 cents), e.g. D-Gb for written D-F#. Minor thirds
involving sharps likewise will be realized as augmented seconds very
close to 6:5, e.g. Gb-A for written F#-A. The precise ratios are
8192:6561 (~384.36 cents) and 19683:16384 (~317.60 cents), differing
from pure 5-based ratios by a schisma of 32805:32768 (~1.95 cents).

Thus in early 15th-century pieces such as those found in the older
repertories of the Buxheim Organ Book, we find progressions like this:

E4 Gb4 G4
Db4 D4
A3 G3

<http://value.net/~mschulter/py5ei001.mid>

By around 1450, scholars such as Mark Lindley have interpreted the
compositional style of Conrad Paumann to suggest a meantone
temperament where all regular thirds are fairly close to 5:4 or 6:5,
while by around 1482, the treatise of Bartoleme Ramos seems to imply
such a temperament as a keyboard standard.

In 1496, Franchinus Gaffurius reports that the fifths are narrowed on
organs by a "small and hidden quantity," and in the early 16th century
various writers give practical advice on how to go about it.[2]

My own experience is that just as a 14th-century Pythagorean tuning
with the regular and active thirds and sixths fits the style of this
era, or the modified Pythagorean tuning of the early 15th century fits
the epoch of the young Dufay, so a meantone with pure or near-pure
thirds fits the 16th century.

Mark Lindley suggests that maximum "resonance" for Zarlino's _harmonia
perfetta_ of a fifth divided into major and minor third may occur in
the general region of 1/4-comma (pure 5:4 major thirds) or 2/7-comma
(major and minor thirds equally impure by 1/7 comma).

Certain later theorists, including Paul Erlich here, have suggested an
optimal meantone somewhere between these two temperaments described by
Zarlino, at around 7/26-comma -- also very close to Kornerup's Golden
Meantone as advocated in the 1930's (where the ratio between the
whole-tone and diatonic semitone is equal to the Golden Mean, ~1.618).

It seems to me that this kind of meantone fits Renaissance style,
where the sweetness of the thirds tends largely to overshadow the
impurity of the fifths. The large diatonic semitones seem to fit with
the overall vertical ethos of the style, with a typically rather
smooth flow between _harmonia perfetta_ sonorities, sometimes
punctuated by the subtle dissonance form of the suspension.[2]

By 1680 and the Werckmeister era, however, we have entered into a
tonal style based on the systematic and structural use of bold
dissonances (such as tritonic seventh sonorities) to establish keys. A
kind of tuning with somewhat narrower semitones, for example, might
fit this kind of texture quite apart from issues of how much to
compromise vertical fifths and thirds.

With the new well-temperaments, as with Renaissance meantones, I would
see various shadings possible. Of course, if Bach had mathematical
reasons for favoring a specific temperament consistent with the style
of his music, then that factor could be decisive in his choice.

One helpful distinction to make in such discussions is between
statements made by or attributed to a composer such as Bach, and
inferences drawn from musical patterns or other factors.

What I would emphasize is that in my view the musical and intonational
changes from Gothic to Renaissance and Manneristic to Late Baroque
represent shifting fashions; each tuning might be considered optimal
in its own stylistic setting.

-----
Notes
-----

1. Curiously, as late as 1650, Kircher uses the 17th-century term
"triad" to refer to this traditional Gothic sonority, a term Johannes
Lippius had introduced to refer to a different kind of three-voice
sonority dividing the fifth into major and minor third.

2. For one view of tuning instructions in this era, see Mark Lindley,
"Early Sixteenth-century Keyboard Temperaments," _Musica Disciplina_
28:129-151 (1974).

3. However, as the Monz has noted, the use of meantone or related
tunings in the 16th century with major thirds somewhat wider than pure
is also documented. Mark Lindley has suggested that the temperament of
Arnold Schlick (1511) may have involved narrowing fifths between
diatonic notes (F-C-G-D-A-E-B) by about 4 cents, and the fifths Eb-Bb,
Bb-F, B-F#, and F#-C# by about 3 cents. He suggests that Ab-Eb may
have been about 2 cents _wide_ of pure. Schlick's declared purpose is
a 12-note tuning with Ab-Eb serviceable and E-G# marginally acceptable
in an ornamented cadence. See his "Early Sixteenth-century Keyboard
Temperaments," n. 2 above. While Lindley categorically regards C#-Ab
(or C#-G#) as a "Wolf fifth," and Schlick indeed suggests that it is
best avoided, one could argue that this fifth might be at least
"semi-playable," making the scheme a kind of "semi-well-temperament."
Schlick's system suggests that in 1511, cadences in the Third and
Fourth Modes (Phrygian/Hypophrygian) did not yet follow a preference
that any closing third above the bass should be major.

Most appreciatively,

Margo Schulter
mschulter@value.net

Dear MARGO Schulter, (sorry for my bad spelling/typing)

Not to forget: Thanks for your musical examples! The proof of the pudding
is in the eating and the proof of music is, after all, in hearing.

Thanks as well again for your excellent accout on the history of the
subject, in particular about the trias harmonica perfecta and the TRINITY.

I "do have nothing", be the way, against Pythagorean intonation. Strictly
speaking, there is nothing to "tolerate" and wilst C-major even fore Bach
boasts of "quality in terms of near-perfecet intervals", C#-major with Bach
is strictly PYTHAGOREAN.

Look at the Ab-major prelude in WTC I. This plunges straight into the
Ab-major triad. And such, I feel the style to interpret this piece is
determined at the same time.

As regards the transition period of changing fashion and preference,
let me also remark, how I like the "sumer is icomen in" - when at that
time they apparently started to appreciate the thirds! (I liked to sing
it when I was in our childrens' chorale at about 9 years).

mschulter schrieb:
> Hello, there, Herbert Kellner and everyone, and please let me that I
> am honored to be quoted in your discussion of the _trias harmonica_,
> and might respond by sharing some of my own historical perspective,
> largely focused on medieval and Renaissance-Manneristic rather than
> High Baroque practice.
>
> Please let me also very warmly thank you Joe Monzo, our "Monz," for a most
> thoughtful reply which sensitively and accurately explains many of my
> views as part of your presentation, as well as offering some extensive and
> invaluable documentation.
>
> What follows was mostly written before I read your reply, Monz, and
> may reflect our areas of agreement. I've added a couple of points
> which your presentation suggested to me, and hope that my musical
> examples may add to the dialogue.
>
> Especially, I would like to suggest that medieval Pythagorean
> practice, the meantone practice of the Renaissance, and the
> well-tempered practice of Bach's era may each have their own musical
> "perfection." What is fitting for one era may be "harsh," or at least
> unidiomatic, for another.
>
> First, from my perspective, if we are using the term "chord" in a
> general way to suggest a combination of three or more notes or
> intervals, then I would consider the basic Gothic sonority as the
> _trina harmoniae perfectio_ or "threefold perfection of harmony"
> described by Johannes de Grocheio in his treatise of around 1300.
>
> The sonority manifesting this perfection, which Grocheio describes as
> a kind of feminine counterpart of the Trinity, has a mother octave
> (2:1), a daughter fifth (3:2), and an upper fourth (4:3) proceeding
> from both of these consonances. An example of this sonority, here
> heard as the resolution of a standard 13th-14th century cadence, would
> be F3-C4-F4, with C4 as middle C:
>
> MIDI example: <http://value.net/~mschulter/py3ei004.mid>
>
> In English, I refer to Grocheio's sonority manifesting "perfect
> harmony" (which he also describes as _consonantia perfectissima_,
> requiring at least three voices) as a _trine_.[1]
>
> Another theorist of the same epoch notes that the trine can be
> expressed by the "natural" series of numbers 2-3-4, with a 2:3 fifth
> placed below a 4:3 fourth in arranging the outer 2:1 octave. As it
> happens, this "natural series" suggests the series of harmonic
> partials 2:3:4, although partials seem to have been recognized in
> Western Europe theory only around the early 17th century, the era of
> Galileo, Mersenne, and Descartes.
>
> In a system of music for three or four voices based on the stable
> Gothic trine, unstable combinations with complex Pythagorean thirds
> and sixths play a vital role in directed progressions as well as
> coloristic sonority. For example, consider our cadence above:
>
> E4 F4
> B3 C4
> G3 F3
>
> <http://value.net/~mschulter/py3ei004.mid>
>
> Here the major third G3-B3 at 81:64 (~407.82 cents) very nicely
> expands to a fifth, while the major sixth G3-E3 at 27:16 (~905.87
> cents) expands to the outer octave of a complete trine. All three
> voices and intervals unite in this purposeful and compelling
> resolution.
>
> As Carl Dahlhaus has written, in Gothic Pythagorean tuning there is an
> admirable concord between acoustical structure and musical language.
>
> This is not to say that a standard Pythagorean tuning was the only
> possibility in Continental Europe during the 14th century. In England
> around 1300, for example, Walter Odington comments that the
> Pythagorean ratios for major and minor thirds are close to the simple
> ratios of 5:4 and 6:5, and that singers can maximize the concord of
> these intervals. A "bending" of intonation toward the simplest ratios
> seems to fit at least some dialects of English polyphony.
>
> In contrast, around the same epoch, Marchettus of Padua (1318) seems
> to describe a system in which cadential major thirds and sixths are
> _larger_ than Pythagorean, and cadential semitones _narrower_ than the
> already compact Pythagorean diatonic semitone or _limma_ at 256:243
> (~90.22 cents).
>

I simply do not like thirds larger than Pythagorean. I am, unfortunately
completely biased towards systems for all 24 keys and it is obvious that
therein no thirds enlarlarged by more than one comme should occur.

> While the interpretation of his system remains an uncertain and often
> debated topic, here is a short composition of mine, "Salutation for
> Mary Beth Ackerley, envision'd as Lysistrata," inspired by one
> possible interpretation with cadential dieses at around 48 cents:
>
> MIDI example: <http://value.net/~mschulter/mary002.mid>
>
> Without excluding such variations, a conventional Pythagorean tuning
> seems to me very nicely to fit most 14th-century music. In a thread
> with much discussion of "perfection," I might add that certain
> theorists describe a Pythagorean major third as "perfected," meaning
> that it actively seeks to expand to the stable fifth.
>
> The nature of musical style, however, is to change, and by around
> 1400-1420, as you mention, Herbert Kellner, a type of interval present
> in a 12-note Pythagorean tuning of the kind common on 14th-century
> keyboards has captured the imaginations of musicians.
>
> This is the "schisma third," as it is now often called: actually a
> Pythagorean diminished fourth (e.g. C#-F or F#-Bb in a typical
> 14th-century tuning of Eb-G#).
>
> By the early 15th century, this interval with its curiously smooth
> qualities had apparently motivated many musicians to tune the written
> sharps of notated music as Pythagorean flats in a chain from Gb to B:
>
> (F#) (C#) (G#)
> Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B
>
> Thus major thirds involving written sharps will actually be realized
> in this tuning as Pythagorean diminished fourths very close to a pure
> ratio of 5:4 (~386.31 cents), e.g. D-Gb for written D-F#. Minor thirds
> involving sharps likewise will be realized as augmented seconds very
> close to 6:5, e.g. Gb-A for written F#-A. The precise ratios are
> 8192:6561 (~384.36 cents)

(It is difficult enough - fromthe engineering point of view - to put into
a system for all 24 keys a decently perfect major third - Yes, I'm
nagging! Therefore, as I'm "born�", please forgive me if I do not like
thirds smaller than the perfect third. My viewpoint is a provocation,
intentionally not historic, but based upon "systems engineering" for
all 24 keys!

Kellner, H.A.: Temperaments for all 24 Keys - A Systems Analysis. Acustica, Vol.
52/2, 1982/83. S. Hirzel Verlag Stuttgart. Seite 106-113. Publication of the
lecture delivered July 1980 at the Bruges 6th International Harpsichord Week).

and 19683:16384 (~317.60 cents), differing
> from pure 5-based ratios by a schisma of 32805:32768 (~1.95 cents).
>
> Thus in early 15th-century pieces such as those found in the older
> repertories of the Buxheim Organ Book, we find progressions like this:
>
> E4 Gb4 G4
> Db4 D4
> A3 G3
>
> <http://value.net/~mschulter/py5ei001.mid>
>
> By around 1450, scholars such as Mark Lindley have interpreted the
> compositional style of Conrad Paumann to suggest a meantone
> temperament where all regular thirds are fairly close to 5:4 or 6:5,
> while by around 1482, the treatise of Bartoleme Ramos seems to imply
> such a temperament as a keyboard standard.
>
> In 1496, Franchinus Gaffurius reports that the fifths are narrowed on
> organs by a "small and hidden quantity," and in the early 16th century
> various writers give practical advice on how to go about it.[2]
>
> My own experience is that just as a 14th-century Pythagorean tuning
> with the regular and active thirds and sixths fits the style of this
> era, or the modified Pythagorean tuning of the early 15th century fits
> the epoch of the young Dufay, so a meantone with pure or near-pure
> thirds fits the 16th century.
>

(Dufay: "In to dominum speravi !!!)

> Mark Lindley suggests that maximum "resonance" for Zarlino's _harmonia
> perfetta_ of a fifth divided into major and minor third may occur in
> the general region of 1/4-comma (pure 5:4 major thirds) or 2/7-comma
> (major and minor thirds equally impure by 1/7 comma).

Werckmeister states that the more perfect an interval is, the less it may
support tempering. Thus, the octave nothing, the fifth, let me say
anachronistically, 1/4 of S (the syntonic comma) and the major third
can obviously tolerate easily tempering by one entire comma.

Coming now to my point: In case BOTH the minor and major third are
tempered by 1/7 comma, there seems to be a problem. Let's t think
of the major third, where one comma tempering is plenty. On the contrary,
just one comma is no problem whatsoever for the minor third, though it gets
pretty soft!!

>
> Certain later theorists, including Paul Erlich here, have suggested an
> optimal meantone somewhere between these two temperaments described by
> Zarlino, at around 7/26-comma -- also very close to Kornerup's Golden
> Meantone as advocated in the 1930's (where the ratio between the
> whole-tone and diatonic semitone is equal to the Golden Mean, ~1.618).
>
> It seems to me that this kind of meantone fits Renaissance style,
> where the sweetness of the thirds tends largely to overshadow the
> impurity of the fifths. The large diatonic semitones seem to fit with
> the overall vertical ethos of the style, with a typically rather
> smooth flow between _harmonia perfetta_ sonorities, sometimes
> punctuated by the subtle dissonance form of the suspension.[2]
>
> By 1680 and the Werckmeister era, however, we have entered into a
> tonal style based on the systematic and structural use of bold
> dissonances (such as tritonic seventh sonorities) to establish keys. A
> kind of tuning with somewhat narrower semitones, for example, might
> fit this kind of texture quite apart from issues of how much to
> compromise vertical fifths and thirds.
>
> With the new well-temperaments, as with Renaissance meantones, I would
> see various shadings possible. Of course, if Bach had mathematical
> reasons for favoring a specific temperament consistent with the style
> of his music, then that factor could be decisive in his choice.

UNITAS between temperament and structure of compositions can be assured
provided the structure utilizes numbers that pertain to the tuning.

The 4 Duets' 369 bars have the ratio superparticularis of 370/369 that
belongs to the tempered fifth in 1/5 P of Werckmeister/Bach "wohltemperirt".
369/370 is the first continued fraction approximation, as Monz remarked
recently, a(370/371 the next superparticular approximation).

The midpoint of the Four Duets, upon inspection, pereferrably of the original
prnt, or the HENLE edition, is clearly dsistinguished. Bach knew there were 369
bars in total.

>
> One helpful distinction to make in such discussions is between
> statements made by or attributed to a composer such as Bach, and
> inferences drawn from musical patterns or other factors.
>
> What I would emphasize is that in my view the musical and intonational
> changes from Gothic to Renaissance and Manneristic to Late Baroque
> represent shifting fashions; each tuning might be considered optimal
> in its own stylistic setting.
>
>
> -----
> Notes
> -----
>
> 1. Curiously, as late as 1650, Kircher uses the 17th-century term
> "triad" to refer to this traditional Gothic sonority, a term Johannes
> Lippius had introduced to refer to a different kind of three-voice
> sonority dividing the fifth into major and minor third.
>
> 2. For one view of tuning instructions in this era, see Mark Lindley,
> "Early Sixteenth-century Keyboard Temperaments," _Musica Disciplina_
> 28:129-151 (1974).
>
> 3. However, as the Monz has noted, the use of meantone or related
> tunings in the 16th century with major thirds somewhat wider than pure
> is also documented. Mark Lindley has suggested that the temperament of
> Arnold Schlick (1511)

A question I have - to be critisized - not yet seriously considered: Is
"Bach's system wohltemperirt" consistent with Schlick. Or else, is
Schlick's system consistent with "Bach's system wohltemperirt" ????

may have involved narrowing fifths between
> diatonic notes (F-C-G-D-A-E-B) by about 4 cents, and the fifths Eb-Bb,
> Bb-F, B-F#, and F#-C# by about 3 cents. He suggests that Ab-Eb may
> have been about 2 cents _wide_ of pure. Schlick's declared purpose is
> a 12-note tuning with Ab-Eb serviceable and E-G# marginally acceptable
> in an ornamented cadence. See his "Early Sixteenth-century Keyboard
> Temperaments," n. 2 above. While Lindley categorically regards C#-Ab
> (or C#-G#) as a "Wolf fifth," and Schlick indeed suggests that it is
> best avoided, one could argue that this fifth might be at least
> "semi-playable," making the scheme a kind of "semi-well-temperament."
> Schlick's system suggests that in 1511, cadences in the Third and
> Fourth Modes (Phrygian/Hypophrygian) did not yet follow a preference
> that any closing third above the bass should be major.
>
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net
>
>
>
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold for
> the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest
> mode.
> tuning-normal@yahoogroups.com - change your subscription to individual
> emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

----- Original Message -----
> From: <ha.kellner@t-online.de>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, July 15, 2001 7:57 AM
> Subject: Re: [tuning] Re: trias harmonica -- for Herbert Kellner and Joe
Monzo
>
> > [Margo Schulter:]
> > Mark Lindley suggests that maximum "resonance" for Zarlino's _harmonia
> > perfetta_ of a fifth divided into major and minor third may occur in
> > the general region of 1/4-comma (pure 5:4 major thirds) or 2/7-comma
> > (major and minor thirds equally impure by 1/7 comma).
>
> Werckmeister states that the more perfect an interval is, the less it may
> support tempering. Thus, the octave nothing, the fifth, let me say
> anachronistically, 1/4 of S (the syntonic comma) and the major third
> can obviously tolerate easily tempering by one entire comma.

I know that this is a long-held belief, but my own experience,
and the experimental results published in many journal articles,
show evidence exactly to the contrary of this.

Many people prefer the sound of an "8ve" which is stretched
about 15 cents wide. And I've pointed out here many times
how the "5th" can tolerate quite a bit of "mistuning", as
for example in jazz, where a "perfect 5th" of ~700 cents
can easily be substituted with a "sharp 5th" (~800 cents)
or even more commonly with a "flat 5th" (~600 cents),
admitting a range of variation of ~2 full Semitones!,
without appreciably changing the affect of the chord in
question.

In contrast, commatic changes in the "major 3rd" generally
exhibit a change of affect. The commatically-shifted
"3rd" clearly has a different "mood" than the original
one.

> Coming now to my point: In case BOTH the minor and major third are
> tempered by 1/7 comma, there seems to be a problem. Let's t think
> of the major third, where one comma tempering is plenty. On the contrary,
> just one comma is no problem whatsoever for the minor third, though it
gets
> pretty soft!!

I think it's interesting that you wrote this, because as I was
reading the paragraphs before it, I was thinking about how
much I like using very narrow "minor 3rds" such as 75:64 and
7:6 in my own music. These are respectively ~1.909155841 and
~2.267726433 (~ 2 and 2&1/4) syntonic commas narrower than the
5-limit JI "minor 3rd" of 6:5.

> > [Margo:]
> > With the new well-temperaments, as with Renaissance meantones, I would
> > see various shadings possible. Of course, if Bach had mathematical
> > reasons for favoring a specific temperament consistent with the style
> > of his music, then that factor could be decisive in his choice.
>
> UNITAS between temperament and structure of compositions can be assured
> provided the structure utilizes numbers that pertain to the tuning.
>
> The 4 Duets' 369 bars have the ratio superparticularis of 370/369 that
> belongs to the tempered fifth in 1/5 P of Werckmeister/Bach
"wohltemperirt".
> 369/370 is the first continued fraction approximation, as Monz remarked
> recently, a(370/371 the next superparticular approximation).
>
> The midpoint of the Four Duets, upon inspection, pereferrably of the
original
> prnt, or the HENLE edition, is clearly dsistinguished. Bach knew there
were 369
> bars in total.

Herbert Anton, I have found the "wohltemperirt" tuning and your
writings about it so intriguing that I am very interested to
know what documentary evidence exists, if any, that Bach was
thinking this way. Are there any letters, church archives,
student's notes, etc., which contain information about the
relationships between number of measures and the tuning?

I wish I had the time to hunt down some of your papers,
but I don't right now. Is there any possibility that more
of them can be made accessible on the internet? I think
there would be a strong interest from the cyber-community
if your work was more accessible there.

> > [Margo:]
> > ... as the Monz has noted,

An aside...

Margo, you really like adding the "the", huh? :)

No problem with it from me... just curious how you're
the only person on the lists who uses it...

As for Arnold Schlick's description of an irregular
temperament in 1511, Margo, you certainly piqued my curiosity.
I'm exploring the mathematics of that tuning now, and will
report my findings in a bit.

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

monz schrieb:
>
> ----- Original Message -----
> > From: <ha.kellner@t-online.de>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, July 15, 2001 7:57 AM
> > Subject: Re: [tuning] Re: trias harmonica -- for Herbert Kellner and Joe
> Monzo
> >
> > > [Margo Schulter:]
> > > Mark Lindley suggests that maximum "resonance" for Zarlino's _harmonia
> > > perfetta_ of a fifth divided into major and minor third may occur in
> > > the general region of 1/4-comma (pure 5:4 major thirds) or 2/7-comma
> > > (major and minor thirds equally impure by 1/7 comma).
> >
> > Werckmeister states that the more perfect an interval is, the less it may
> > support tempering. Thus, the octave nothing, the fifth, let me say
> > anachronistically, 1/4 of S (the syntonic comma) and the major third
> > can obviously tolerate easily tempering by one entire comma.
>
>
> I know that this is a long-held belief, but my own experience,
> and the experimental results published in many journal articles,
> show evidence exactly to the contrary of this.
>
> Many people prefer the sound of an "8ve" which is stretched
> about 15 cents wide.
On the harpsichord I personally definitely prefer absolutely
accurate octaves.

And I've pointed out here many times
> how the "5th" can tolerate quite a bit of "mistuning", as
> for example in jazz,

The "dirty" play in jazz is g�nial! What an enrichement of sonority.

where a "perfect 5th" of ~700 cents
> can easily be substituted with a "sharp 5th" (~800 cents)
> or even more commonly with a "flat 5th" (~600 cents),
> admitting a range of variation of ~2 full Semitones!,
> without appreciably changing the affect of the chord in
> question.
>
> In contrast, commatic changes in the "major 3rd" generally
> exhibit a change of affect. The commatically-shifted
> "3rd" clearly has a different "mood" than the original
> one.
>
>
> > Coming now to my point: In case BOTH the minor and major third are
> > tempered by 1/7 comma, there seems to be a problem. Let's t think
> > of the major third, where one comma tempering is plenty. On the contrary,
> > just one comma is no problem whatsoever for the minor third, though it
> gets pretty soft!!
>
> I think it's interesting that you wrote this, because as I was
> reading the paragraphs before it, I was thinking about how
> much I like using very narrow "minor 3rds" such as 75:64 and
> 7:6 in my own music. These are respectively ~1.909155841 and
> ~2.267726433 (~ 2 and 2&1/4) syntonic commas narrower than the
> 5-limit JI "minor 3rd" of 6:5.

Some baroque writer - I don't know at the moment for sure, but
presumably Marpurg (I dislike strongly) criticized very small
minor thirds. On the contrary, I do like them very much, like,
incidentally, the large minor thirds.

Marpurg said, temperaments with 7 pure and 5 tempered fifths are the
worst ones of the better tunings and the best one of the worst tunings.
He didn't researched into that matter more responsibly, however.

>
>
> > > [Margo:]
> > > With the new well-temperaments, as with Renaissance meantones, I would
> > > see various shadings possible. Of course, if Bach had mathematical
> > > reasons for favoring a specific temperament consistent with the style
> > > of his music, then that factor could be decisive in his choice.
> >
> > UNITAS between temperament and structure of compositions can be assured
> > provided the structure utilizes numbers that pertain to the tuning.
> >
> > The 4 Duets' 369 bars have the ratio superparticularis of 370/369
that > belongs to the tempered fifth in 1/5 P of Werckmeister/Bach
> "wohltemperirt".
> > 369/370 is the first continued fraction approximation, as Monz remarked
> > recently, (370/371 the next superparticular approximation).
> >
> > The midpoint of the Four Duets, upon inspection, pereferrably of the
> original print, or the HENLE edition, is clearly dsistinguished. Bach
thus must have known that there > were 369 > > bars in total.
>
>
> Herbert Anton, I have found the "wohltemperirt" tuning and your
> writings about it so intriguing that I am very interested to
> know what documentary evidence exists, if any, that Bach was
> thinking this way. Are there any letters, church archives,
> student's notes, etc., which contain information about the
> relationships between number of measures and the tuning���������?
******

+ number of keystrokes, etc. ...
******

Dear Monz, let's take our time until I am in a position to
disseminate (abstracts) of my (old) and widely spread papers
here and there conveniently, into INTERNET.

���������� Nothing like this exists anywhere. But this ����������
cannot disprove, that Bach's method is not a reality a fact.
I have been thinking about musical composition since my age of
about 4 or 5 years. To a mathematician Bach's procedures are
evident - and, as I hope, from my papers to any open minded reader.
This result was possible, because I myself did never write
anything; I only study what another mathematician has done.

Dr. D�rr liked to point out to me, should Bach really had done so
many number games, how could he have composed, besides, and in
addition, so much!? But JSB was a prodiguous mental calculator,
miraculously intelligent and excellent in school.

Why should Bach have written treatises about his personal, genuine
and original method, to waste his time this way, rather than to
put it into practice and compose music according to his principles.
His music, though, is rationally designed and transparent to analyses,
as I observe. Also, one might refer in analogy to "Bauh�ttengeheimnisse",
secrets of the cathedral builders.

I consider that to Bach's esoteric numerological procedures, for himself,
applied the dictum: "arcana publicata vilescunt"! Neither his composing
method, nor even "wohltemperirt" did JSB reveal, nor his sons who were
familiar as well and tuned instriuments in Leipzig churches. Along
the same lines, with quite similar methods, Werckmeister concealed
his musical temperament invented:

Just a striking example, his treatise's title:

11 letters again 11 letters : partition, UNITAS=mediator
MUSICALISCH ETEMPERATUR
112 135
CHRISTUS 1 unitas
3 third in thoroughbass
5 fifth in thoroughbass
Specification of "wohltemperirt", digne de JSB
CHRISTUS = mediator
___________________________________________________

>
> I wish I had the time to hunt down some of your papers,
> but I don't right now. Is there any possibility that more
> of them can be made accessible on the internet? I think
> there would be a strong interest from the cyber-community
> if your work was more accessible there.
>
>
> > > [Margo:]
> > > ... as the Monz has noted,
>
> An aside...
>
> Margo, you really like adding the "the", huh? :)
>
> No problem with it from me... just curious how you're
> the only person on the lists who uses it...
>
>
>
> As for Arnold Schlick's description of an irregular
> temperament in 1511, Margo, you certainly piqued my curiosity.
> I'm exploring the mathematics of that tuning now, and will
> report my findings in a bit.
>
Schlick's description is at the same time exciting and to a
quantifying analysis exasperating. With some good will, you
can regard it in the extreme like "Werckmeister/Bach/wohltemperirt"

>
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>***********************************************************
>Dear Margo, dear Monz, dear group members,
I now go on leave, so long until later.
I wish you to spend good summer holidays,
************************************************************
Cordially, Herbert Anton Kellner

>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold for
> the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest
> mode.
> tuning-normal@yahoogroups.com - change your subscription to individual
> emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

--- In tuning@y..., ha.kellner@t... wrote:
>
> Werckmeister states that the more perfect an interval is, the less
it may
> support tempering. Thus, the octave nothing, the fifth, let me say
> anachronistically, 1/4 of S (the syntonic comma) and the major
third
> can obviously tolerate easily tempering by one entire comma.
>
> Coming now to my point: In case BOTH the minor and major third are
> tempered by 1/7 comma, there seems to be a problem.

Why? What's the problem?

Let's t think
> of the major third, where one comma tempering is plenty. On the
contrary,
> just one comma is no problem whatsoever for the minor third, though
it gets
> pretty soft!!