trias harmonica - for Margot Schulter

Dear Mrs. Margot Schulter, dear members of the group,

A recent question by Margot Schulter to quote:

"I would much agree that the _trias harmonica_ is a vital
feature of the Baroque era we are discussing.

1) Here my question might be: while the _trias harmonica_ is at the heart
both of the late modal practice around 1600 and the early tonal practice
around 1680-1720 which Werckmeister helps to define, does the concept of
the triad necessarily favor any one well-temperament, given that all are
compromises?
*********** underlining by HA Kellner

2) Similarly, while Zarlino's concept of _harmonia perfetta_ is central to
late 16th-century practice, does it necessarily favor one specific
meantone tuning?"

I unquote the large citation of Margot Schulter.

Ad 1)
Does the concept of the triad necessarily favor any one well-temperament,
given that all are compromises?
***********
The music under consideration proceeds via the harmonies of the
chords. (except, e. g. the "bicinium", etc.). These chords comprise thus
at least 3 tones. Whereas earlier times "tolerated" even the somewhat
harsh Pythagorean thirds, tastes and preferences changed and developed in
thecourse of history. One adopted the extreme of going for PERFECT thirds in
mean-tone.

(When tuning consistently and very accurately - why not - a harpsichord,
one fell automatically at one point into a major third about
384,36 cent wide, i.e . smaller by 2 cent than pure. This discovery,
pleasant to the ear, certainly gave rise to the desire to hear more of this
sort).

One took thus PERFECT thirds for mean-tone. This, not only in
contradistinction to Pythagorean, but at the same time mean-tone is set
off from the system of natural harmony that puts basically a perfect
triad 4:5:6 (in terms of frequencies), say upon C-major. Constructing this
system further, produces the annoying fact that the tones C-D and D-E are
different and drastically reduced fifths, by one comma, show up.

Once having placed instead the perfect third on C-E and filled it up by
tempering four EQUAL fifths, equalizes the sizes of C-D and D-E.
Within the straightforward principle of mean-tone (with pure thirds and
payung no attention or concern for anything else) let us consider
the TRIAD C-E-G.

Wasn't in the natural harmonic system the triad the pertinent feature?
Mean-tone concentrates on the single and unique interval of the major third.
But the music under consideration proceeds via the harmonies of chords.
Within mean-tone, the overall purity of C-E-G can be "improved", in analogy to
the step of improvement that went from the Pythagorean third to the "better"
mean-tone third.

Despite its pure third, the mesotonic triad will BEAT. To get things
"purer", shouldn't one attempt to reduce the beats? These triads
beat: not because of the third that is pure, but due to the fifth C-G.
This is one of the 4 fifths that has to be necessarily tempered smaller
in order to fit into its perfect third.
As the triad beats anyway, does it really make sense to insist on a pure
third? Obviously, NO! and the third may be relaxed, up to the order
of its beat-rate becoming comparable to the rapidity of beats of
the fifth. Enlarging the third somewhat will slow down the beats of the
fifth. The ideal "meeting/compromise point" will be the situation
in which - within a tempered triad - both its principal constituent
intervals third and fifth beat at the UNISON. What sufficient reason could
produce a BETTER COMPROMISE??
******************

Conclusion: the basis of musical progressions - in some sense - are chords
and not just intervals. This is borne out when basing the music upon
triads, the most elementary cords. And here, the major chord - trias
harmonica perfecta. As to the overall tempering of this basic musical entity,
the triad "wohltemperirt" is the best and most natural one to attain, derived
here heuristically, not via the detailed mathematics. Quality criterion for
purity and consonance was the behaviour of beats, i. e. their rapidity.

Having derived this triad, the step to extend this triad to a system
"wohltemperirt" follows, But this is very simple. Due to a surprising and
incredible mathematical co-incidence, the fifth of the triad "wohltemperirt"
explained above, is reduced by 1/5 of the Pythagorean Comma!!!

Therefore, given the 5 tempered fifths, one can complete the circle with the
finality of closure by the further 7 perfect fifths.Musico-theological aspects
now enter into the play: If we insist that the "best major triad" feasible in a
balanced system for all 24 keys just occurs just ONCE, and not more, then B-F#
must be the 5th tempered fifth.

As regards MINOR: provided a major triad is acceptable or supportable, the
interpolating minor third following the basic major third will also be
acceptable. If the tempered fifths themselves are acceptable, the minor
chords constructed using this material will as well be acceptable. Thus,
"wohltemperirt" yields and represents a temperament for all 24 keys.

Ad 2)
Having said the above, I can see nothing in Zarlino's concept of "harmonia
perfetta", that would favor any one specific meantone tuning. The mesotonic
system definitely gives room for improvement, technologically/mathematically
and spiritually under the musico-theological aspects, perhaps best enounced
in almost all of Werckmeister's writings.

I am grateful and have very much appreciated Mrs. Margot Schulter's
excellent expos� about the trias harmonica and even more, her critical
query I hope to have answered here more or less satisfactorily.

(When pretending music progresses by CHORDS I do not attempt to offer
a treatise on counterpoint).

Kind regards

Herbert Anton Kellner

Dear Herbert-Anton,

> From: <ha.kellner@t-online.de>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 13, 2001 1:59 AM
> Subject: [tuning] trias harmonica - for Margot Schulter
>
>
> Dear Mrs. Margot Schulter, dear members of the group,
>
> A recent question by Margot Schulter to quote:

A spelling correction: Her first name is spelled "Margo".

> Ad 1)
>
> > [Margo Schulter:]
> > Does the concept of the triad necessarily favor any one
> > well-temperament, given that all are compromises?
> ***********
> The music under consideration proceeds via the harmonies of
> the chords. (except, e. g. the "bicinium", etc.). These chords
> comprise thus at least 3 tones. Whereas earlier times
> "tolerated" even the somewhat harsh Pythagorean thirds,
> tastes and preferences changed and developed in the course
> of history. One adopted the extreme of going for PERFECT thirds
> in mean-tone.

While the description you give here and in what follows is
basically correct, there are a few points I'd like to observe.

As Margo, whose main interest is music and tuning of the "Gothic"
(i.e., medieval) period, will be well aware, it is not really
correct to express the viewpoint that 'earlier times "tolerated"
even the somewhat harsh Pythagorean thirds'. Use of the word
"tolerate" betrays a bias based on later harmonic practice,
in which the "pure" 5:4 ratio was considered the most desirable
type of "major 3rd".

Many theorists, Margo perhaps cheif among them, have argued
quite persuasively that medieval European musical sensibilites
did not merely "tolerate" the "somewhat harsh Pythagorean thirds",
but rather, *embraced* them enthusiastically as an important part
of the overall musical aesthetic. The "rules" of harmony and
counterpoint during the "Gothic" period are based in large part
on the dissonance or "imperfect consonance" of the Pythagorean
"major 3rd" with ratio 81:64 = ~407.8200035 cents.

It's very important to keep this in mind when analyzing the
*later* historical development of harmonic practice. It was only
around the 1300s (at the earliest) to 1500s in standard European
theory that the "pure" consonance of the 5:4 "major 3rd" became
a desirable part of the harmonic vocabulary.

At that point, rather than "tolerate" the Pythagorean 3rds, tuning
systems were advocated which were adjusted from the 3-limit to the
5-limit (see <http://www.ixpres.com/interval/dict/limit.htm> if
unclear about "limits"), so that the 5:4 and 6:5 "3rds" could
be employed. There was an overall paradigm shift at this time
in which these "3rds" became the basis of harmonic practice.
Note that _a capella_ vocal music was considered the stylistic
paradigm at this time.

(See my long tuning list post of Mon Jun 7, 1999 10:01 pm
</tuning/topicId_3380.html#3380>
and Margo's reply of Tue Jun 8, 1999 8:03 pm
</tuning/topicId_3422.html#3422>.)

> (When tuning consistently and very accurately - why not
> - a harpsichord, one fell automatically at one point into
> a major third about 384,36 cent wide, i.e . smaller by
> 2 cent than pure. This discovery, pleasant to the ear,
> certainly gave rise to the desire to hear more of this sort).

As you describe above, tuning a series of eight "4ths" and "5ths"
as pure Pythagorean ratios 4:3 and 3:2 respectively, results
in the ratio (2^11)*(3^-8) = ~384.3599931 cents, which is
~1.953720788 cents narrower than the 5:4 ratio, but audibly
indistinguishable from it under almost all circumstances.

This extension of the Pythagorean system to the point where
skhismatic equivalents of 5-limit pitches appear is indeed a
very plausible explanation for the introduction of 5-limit
ratios themselves. [See Lindley/Turner-Smith 1993, p 136-138]

>
> One took thus PERFECT thirds for mean-tone. This, not only
> in contradistinction to Pythagorean, but at the same time
> mean-tone is set off from the system of natural harmony that
> puts basically a perfect triad 4:5:6 (in terms of frequencies),
> say upon C-major. Constructing this system further, produces
> the annoying fact that the tones C-D and D-E are different
> and drastically reduced fifths, by one comma, show up.

Of course as you describe here, the use of strict 5-limit
just-intonation will often lead to problems in dealing with
the appearance of the syntonic comma. The first solution to
this was to temper the primary offending note, which was the
supertonic (2nd degree of the diatonic scale), thus: meantone.

>
> Once having placed instead the perfect third on C-E
> and filled it up by tempering four EQUAL fifths,
> equalizes the sizes of C-D and D-E. Within the
> straightforward principle of mean-tone (with pure
> thirds and payung no attention or concern for anything
> else) let us consider the TRIAD C-E-G.
>
> Wasn't in the natural harmonic system the triad the
> pertinent feature? Mean-tone concentrates on the single
> and unique interval of the major third. But the music
> under consideration proceeds via the harmonies of chords.
> Within mean-tone, the overall purity of C-E-G can be
> "improved", in analogy to the step of improvement that
> went from the Pythagorean third to the "better"
> mean-tone third.

Again, Herbert-Anton, when discussing historical tunings,
the idea of "improving" an interval must be considered in
light of the harmonic/contrapuntal theory and practice
of the time under consideration. I suggest using caution.

>
> Despite its pure third, the mesotonic triad will BEAT.
> To get things "purer", shouldn't one attempt to reduce
> the beats? These triads beat: not because of the third
> that is pure, but due to the fifth C-G. This is one of
> the 4 fifths that has to be necessarily tempered smaller
> in order to fit into its perfect third. As the triad
> beats anyway, does it really make sense to insist on a
> pure third? Obviously, NO! and the third may be relaxed,
> up to the order of its beat-rate becoming comparable to
> the rapidity of beats of the fifth. Enlarging the third
> somewhat will slow down the beats of the fifth. The ideal
> "meeting/compromise point" will be the situation in which
> - within a tempered triad - both its principal constituent
> intervals third and fifth beat at the UNISON. What sufficient
> reason could produce a BETTER COMPROMISE??
> ******************

Well, I've already noted here (and would think that you'd
agree) that *many* other considerations (numerological,
theological, etc.) may go into a person's preference for a
particular tuning, besides simply the obvious acoustical ones.

Note also that 1/4-comma meantone, the type you describe
here, where the "5th" is calculated as ( (3/2) / (81/80)^(1/4) )
= ~696.5784285 cents (nearly identical to 31-EDO), was only
one early solution. Its "major 3rd" is of course the 5:4
ratio = ~386.3137139 cents. Its "whole-tone" is ~193.1568569
cents, which is *precisely* the mean between the 9:8 and 10:9
ratios. This is the only tuning which can be labeled *exactly*
as "mesotonic".

But while 1/4-comma meantone was indeed very popular during the
1500s and 1600s, it was far from being the exclusive "meantone"
tuning employed.

By the mid-1600s 1/4-comma had fallen out of general favor and was
usually replaced by 1/6-comma meantone or its equivalent 55-EDO:

>> "Sauveur ... said the 55 division was "the one which ordinary
>> musicians use" [Sauveur 1711, p 315].
>> ...
>> Sauveur's remarks when taken together with other pertinent
>> evidence [from p 54, regarding 1/5-comma meantone; see
>> quote below] suggest that the 1/4-comma temperament is not
>> the most valid model for a menatone system of his day.
>> It had been the theorists' favourite model, however, during
>> the period between Zarlino's _Dimostrationi harmoniche_
>> (1570) and Mersenne' _Harmonie Universelle_ (1637), and
>> presumably it was used a good deal during that period, even
>> though one writer said in 1613 [Cerone 1613, p 1049] that
>> master organ builders used a meantone temperament with the
>> major 3rds larger than pure." [Lindly/Turner-Smith 1993, p 149]

In 1/6-comma meantone the "5th" is calculated as
( (3/2) / (81/80)^(1/6) ) = ~698.3706193 cents
(nearly identical to 55-EDO... see
<http://www.ixpres.com/interval/monzo/55edo/55edo.htm>).
The "major 3rd" of 1/6-comma meantone = ~393.4824771 cents,
or ~7.168763199 cents wider than that of 1/4-comma meantone
(which is the "pure" 5:4). Under certain conditions this
could be an audible difference.

The 55-EDO "5th" is 2^(32/55) = ~698.1818182 cents. Its
"major 3rd" is 2^(18/55) = ~392.7272727 cents, and its
"whole-tone" is 2^(9/55) = ~196.3636364 cents.

I have not done any beat-calculations or listening tests to
determine how 1/6-comma meantone or 55-EDO beats; perhaps
you or someone else could calculate and post results.

> Conclusion: the basis of musical progressions - in
> some sense - are chords and not just intervals. This is
> borne out when basing the music upon triads, the most
> elementary cords. And here, the major chord - trias
> harmonica perfecta. As to the overall tempering of
> this basic musical entity, the triad "wohltemperirt"
> is the best and most natural one to attain, derived
> here heuristically, not via the detailed mathematics.
> Quality criterion for purity and consonance was the
> behaviour of beats, i. e. their rapidity.
>
> Having derived this triad, the step to extend this triad
> to a system "wohltemperirt" follows, But this is very simple.
> Due to a surprising and incredible mathematical co-incidence,
> the fifth of the triad "wohltemperirt" explained above,
> is reduced by 1/5 of the Pythagorean Comma!!!

Resulting in a tempered "5th" of ratio
(3/2)/(((2^-19)*(3^12))^(1/5))= ~697.2629988 cents,
four of which ("octave"-reduced) give a "major 3rd" of
~389.0519952 cents. Both of these values fall nearly
midway between those of the 1/4-comma and 1/6-comma
meantones, with "wohltermperirt" a bit close to the former
than to the latter.

> Ad 2)
>
> > [Margo Schulter:]
> > 2) Similarly, while Zarlino's concept of _harmonia perfetta_
> > is central to late 16th-century practice, does it necessarily
> > favor one specific meantone tuning?"
>
> Having said the above, I can see nothing in Zarlino's
> concept of "harmonia perfetta", that would favor any one
> specific meantone tuning. The mesotonic system definitely
> gives room for improvement, technologically/mathematically
> and spiritually under the musico-theological aspects, perhaps
> best enounced in almost all of Werckmeister's writings.

But Zarlino himself advocated 2/7-comma meantone earlier than,
and as superior to, 1/4-comma! I quote:

>> "... the oldest quantitatively coherent account of a meantone
>> temperament is Zarlino's description (1558) of a tuning in
>> which the major (and minor) 3rds are smaller than pure
>> by 1/7 of the syntonic comma, and so the 5ths are tempered
>> by 2/7 comma ...
>> In 1571 Zarlino published a scheme which several later
>> theorists took as their only model of meantone temperament
>> (although by the end of the 17th century, it may have been
>> used less than tunings with the major 3rds tempered larger
>> than pure) ...
>> But Zarlino through the 1570s and '80s still liked the sound
>> of his original 2/7-comma meantone temperament as well...
>> Near the other limit which we have determined for a meantone
>> temperament lies this possibility: [1/6-comma meantone].
>> And there is an obvious intermediate possibility [1/5-comma
>> meantone]... Sauveur preferred it to any other kind of system,
>> and Louli� in 1698 [Louli� 1698, p 28] said it was more in
>> use than any other." [Lindley/Turner-Smith 1993, p 52-54]

The 2/7-comma "5th" is ~695.8103467 cents, and the "major 3rd" is
~383.2413868 cents. Both of these values are narrower than any
of the three tunings described above, and the "3rd" is indeed even
narrower than the Pythagorean skhismic equivalent (2^11)*(3^-8).
The "whole tone" is ~191.6206934 cents.

The 1/5-comma (that's 1/5 of the *syntonic* comma) "5th" is
~697.6537429 cents, the "major 3rd" is ~390.6149718, and the
"whole-tone" is ~195.3074859 cents.

Therefore, none of these systems are *exactly* "mesotonic"!

REFERENCES
----------

Lindley, Mark and Turner-Smith, Roland. 1993.
_Mathematical Models of Musical Scales: A New Approach_.
Band 66 der Orpheus-Schriftenreihe zu Grundfragen der Musik,
herausgegeben von Marin Vogel.
Verlag f�r systematische Musikwissenschaft GmbH, Bonn.

Zarlino, Gioseffo. 1558.
_Istitutioni harmoniche_.
Pietro da Fino, Venezia, 1558, 1562, 1573, 1589.
Reprint of 1573 edition, Ridgewood, 1966.
German translation by Michael Fend (ed.):
_Gioseffo Zarlino: Theorie des Tonsystems.
Das 1. und 2. Buch der Istitutioni harmoniche (1573)_.
PhD diss., TU Berlin, 1983.
Series: Europ�ische Hochschulschriften no.36:43.
Lang, Frankfurt a.M., 1989, 488 pages.
English translation of part 4: _On the modes_
by Vered Cohen, Claude V. Palisca (ed.),
Yale University Press, New Haven CT, 1983, 120 pages.
The Thesaurus Musicarum Italicarum gives the editions of 1558 and 1589 (WWW
and CD-ROM).

Zarlino, Gioseffo. 1571.
_Dimostrationi harmoniche, Ragionamento Quinto, Definitione prima_.
Francesco Senese, Venezia, 1571, 1573, 312 pages.
Second edition, Venice, 1588.
Reprint: Gregg Press, Ridgewood NJ, 1966.

Cerone, Pietro. 1613.
_El Melopeo y maestro_: Tractado de musica theorica y
pratica: en que se pone por extenso, lo que uno para
hazerse perfecto musico ha menester saber: y por mayor
facilidad, comodidad, y claridad del lector, esta
repartido en XXII libros. Va tan exemplificado y claro,
que qualquiera de mediana habilidad, con poco trabajo
alcan�ar� esta profession...
Juan Bautista Gargano y Lucrecio Nucci, Naples.
Reprint: Bologna, Forni, 1969,
Bibliotheca musica Bononiensis. Sezione II. n. 25.

Mersenne, Marin. 1637.
_Trait� de l'harmonie universelle, contenant la th�orie et la
pratique de la musique. O� il est trait� des Consonances, des
Dissonances, des Genres, des Modes, de la Composition, de la
Voix, des Chants, & de toutes sortes d'Instruments Harmoniques_.
Guillaume Baudry, Paris (1627).
S�bastien Cramoisy, Paris (1636/1637).
Facsimile reprint: Fran�ois Lesure (ed.), �ditions du Centre
National de la Recherche Scientifique, Paris, 1963.
English translation: _Harmonie Universelle: The Books on Instruments_
by Roger Edington Chapman, Martinus Nijhoff, The Hague, 1957.

Sauveur, Joseph. 1711.
_Table g�n�rale des syst�mes temp�r�s_. Paris.

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:
/tuning/topicId_26171.html#26173

Quoting from Lindley/Turner-Smith 1993:

>> In 1571 Zarlino published a scheme which several later
>> theorists took as their only model of meantone temperament
>> (although by the end of the 17th century, it may have been
>> used less than tunings with the major 3rds tempered larger
>> than pure) ...

I meant to make it clear that that paragraph refers to
1/4-comma meantone. Sorry.

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

--- In tuning@y..., ha.kellner@t... wrote:
......
>
> Despite its pure third, the mesotonic triad will BEAT. To get
things
> "purer", shouldn't one attempt to reduce the beats? These triads
> beat: not because of the third that is pure, but due to the fifth C-
G.
> This is one of the 4 fifths that has to be necessarily tempered
smaller
> in order to fit into its perfect third.
> As the triad beats anyway, does it really make sense to insist on a
pure
> third? Obviously, NO! and the third may be relaxed, up to the order
> of its beat-rate becoming comparable to the rapidity of beats of
> the fifth. Enlarging the third somewhat will slow down the beats of
the
> fifth. The ideal "meeting/compromise point" will be the situation
> in which - within a tempered triad - both its principal constituent
> intervals third and fifth beat at the UNISON. What sufficient
reason could
> produce a BETTER COMPROMISE??
> ******************
>
........

The concept of the ideal "trias harmonica perfecta" being realized by
a regular temperament where the _beat rate_ of the 5th of the major
triad is identical to that of the major 3rd has compeling
philosophical appeal. It chould be appreciated that 1/5-(syntonic)
comma meantone, which results in a 1/5-comma error in both of these
intervals (5th narrowed, 3rd widened) does _not_ produce equal beat
rates. The primary beat of the 5th is produces by the 3rd harmonic
of the lower tone against the 2nd of the upper, whereas that of the
major 3rd is the 5th harmonic of the lower against the 4th of the
upper.

Herein follows the derivation of the degree of temperament for this
case:

Let f(1), f(2) and f(3) be the frequencies of the triad tones in
increasing order. Then:

B(5) = 3*f(1) - 2*f(3)

where B(5) is the beat rate of the 5th. Likewise:

B(3) = 4*f(2) - 5*F(1)

where B(3) is the beat rate of the 3rd. Setting B(5) = B(3) yields:

3*f(1) - 2*f(3) = 4*f(2) - 5*f(1).

Collecting terms and transposing:

8*f(1) = 4*f(2) + 2*f(3).

Dividing each term by f(1) yields:

8 = 4*R(3) + 2*R(5) [1]

where R(3) equals the frequency ratio of the 3rd, f(2)/f(1); and R(5)
that of the 5th, f(3)/f(1).

Using the regular temperament definition for the major 3rd as four
ascending 5ths reduced by two octaves, we have:

R(3) = R(5)^4/4.

Substituting into [1]:

8 = R(5)^4 + 2*R(5)

which can be rewritten as the quartic polynomial equation:

x^4 + 2*x - 8 = 0.

An iterative numerical solution quickly converges to yield:

x = 1.4959535....

Converting to an equivalent portion of a syntonic comma:

T = ln(1.5/1.49595535)/ln(81/80) = 0.2175428.... ,

a value between 1/4 comma (0.25) and 1/5 comma (0.20). A very close
fractional approximation to this value is 5/23.

Respectfully,

Fred Reinagel