Yahoo Tuning Groups Ultimate Backup tuning Tosi, what he did or didn't say

I now realise that my expose of the falsely precise scholarship based
on Tosi was made in the clavichord group. So anyone who tried to
search the archives here would be disappointed.

Anyway, here is a precis and link to the source. I was reacting to the
Duffin book which places a lot of emphasis on the supposed evidence
for 1/6 comma meantone.

http://www.chmtl.indiana.edu/smi/settecento/TOSOPI_TEXT.html

When Duffin (like Haynes!) quoted Tosi he omitted words in a way which
distorts the meaning and gives a false impression of precision.

Tosi wrote about singing. There was no reason for him, or his readers,
to know or care about details of keyboard tuning, such as varieties of
meantone. (Still, he did know about 'split sharps'.)
He first said that the tone is divided into nine, almost inaudible,
equal parts called 'commas', of which the large semitone is 5 and the
small one 4. [This is an ancient idea going back to medieval
Pythagorean intonation, which seems to have been passed on
uncritically through the generations.] Tosi then says that some people
think the tone is only made up of 7 parts. But he, relying on his
admittedly poor understanding, preferred to count 9, because he thinks
the ear could easily distinguish 1/7 of a tone, but would have much
more difficulty with 1/9 tone.

Duffin has only:
"A whole tone is divided into nine almost imperceptible intervals
which are called commas, five of which constitute the major semitone,
and four the minor semitone..."

Thi omits how Tosi doesn't understand the function of a comma, how
Tosi is uncertain about how many commas there are in a tone, and how
he prefers 9 to 7 on grounds which are highly subjective or nonsensical.
Then Duffin uses Tosi as one of his major sources to 'deduce' that
18th century vocal and instrumental intonation should be 1/6 comma
meantone - a tuning which could only be practiced on keyboard
instruments!

His source is a singer who says that a comma is the smallest interval
that can possibly be distinguished - and then Duffin uses it to
'prove' the use of a temperament which divides this comma into six!

Tosi relates intonation to the form of accompanying instrumentation:
he says that the difference between 'major' and 'minor' semitones is
not evident on the organ or harpsichord, if they do not have split
keys. (Split key instruments were, though, probably tuned in
quarter-comma meantone...)

After plumping for dividing the tone into 9, he says that if one is
accompanied by keyboard instruments without split keys it is
useless to know about 'major' and 'minor' semitones. But because
composers like to introduce arias with only (bowed) string instrument
accompaniment, one must know that D# is lower than Eb.

As we know, 1/6 comma meantone is a tuning for keyboards with fixed
pitches; yet Tosi only requires the singer to differentiate between D#
and Eb when accompanied by strings without keyboard. In that case
there would be no reason for any of the performers to adopt a meantone
tuning. Duffin claims to 'deduce' a keyboard-based tuning from what
Tosi said about musical performances without keyboard accompaniment.

The practical problem, as with all these alleged multiple divisions of
the tone for singers and instrumentalists, is that the sizes of
semitones are very hard to identify accurately by ear, and
particularly in harmonic progressions. 80 and 120 cents can both be
perfectly acceptable semitones in appropriate musical contexts. How
could one possibly know by listening if one had played a 'correctly'
uneven chromatic scale? How could one learn the 9-fold division at all
accurately? It is, and was, totally impractical to ask violinists or
singers to produce sizes of semitone according to a mathematical formula.

(If anyone doubts this, let them try to tune a harpsichord by
listening to tones and semitones alone!)

What is not impractical is to ask for *harmonic* intervals to be pure,
or not too far from pure. A singer can easily recognise if her Eb is
too low or her D# too high simply by listening to the chord; there is
no need to worry about the exact size of tones or semitones in themselves.

The semitone is a dissonance which is the difference between two
consonant harmonic intervals. To teach intonation by starting with the
semitone is putting the cart before the horse. Yet that is what the
scholars (Haynes, Duffin, Lehman...) seem to think Tosi and Leopold
Mozart wanted.
The practical truth is the reverse: if one can tune the harmonic
consonances well, the semitones come of their own accord. That is what
happens in keyboard tuning and in non-keyboard performance. People who
talked about dividing the tone into so and so parts, or the exact size
of difference between major and minor semitones, were, from first to
last, engaging in theoretical speculation without any useful
application to music-making.

~~~T~~~

🔗monz <monz@tonalsoft.com>

Hi Tom and Leonardo,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> I now realise that my expose of the falsely precise
> scholarship based on Tosi was made in the clavichord
> group. So anyone who tried to search the archives here
> would be disappointed.
>
> Anyway, here is a precis and link to the source.
> I was reacting to the Duffin book which places a lot
> of emphasis on the supposed evidence for 1/6 comma
> meantone.
>
> http://www.chmtl.indiana.edu/smi/settecento/TOSOPI_TEXT.html

A welcome back to you too, Leonardo ... um, how about
a quick English translation of this?

> When Duffin (like Haynes!) quoted Tosi he omitted words
> in a way which distorts the meaning and gives a false
> impression of precision.
>
> Tosi wrote about singing. There was no reason for him,
> or his readers, to know or care about details of keyboard
> tuning, such as varieties of meantone. (Still, he did
> know about 'split sharps'.)

But Tosi apparently was familiar with a tuning which
bore at least some resemblance to meantone. More below ...

> He first said that the tone is divided into nine,
> almost inaudible, equal parts called 'commas', of
> which the large semitone is 5 and the small one 4.
> [This is an ancient idea going back to medieval
> Pythagorean intonation, which seems to have been passed
> on uncritically through the generations.]

Indeed.

The earliest music-theoretical writing which could
form a basis for it is the surviving fragments
from Philolaus of c.400 BC:

http://tonalsoft.com/enc/p/philolaus.aspx

(Also see my fuller treatment, published in
_Xenharmonikon_ 18.)

Philolaus's small divisions have the whole-tone of
9:8 ratio divided into 4 "diaschismata" and 1 "comma",
and the diaschisma is approximately twice the size
of the comma, thus the 9-fold division.

It may be that this division is as old as Sumerian
music theory c.2500 BC, but i haven't yet found
numerical/mathematical corroboration in the Sumerian
or Babylonian texts.

> Tosi then says that some people think the tone is
> only made up of 7 parts.

It very well may be that Tosi is here referring to
Sauveur's recent writings on _merides_, 43-edo, and
1/5-comma meantone. These tunings divide the whole-tone
into 7 equal or nearly-equal parts, respectively.

> But he, relying on his admittedly poor understanding,
> preferred to count 9, because he thinks the ear could
> easily distinguish 1/7 of a tone, but would have much
> more difficulty with 1/9 tone.
>
> Duffin has only:
> "A whole tone is divided into nine almost imperceptible
> intervals which are called commas, five of which constitute
> the major semitone, and four the minor semitone..."
>
> Thi omits how Tosi doesn't understand the function of a
> comma, how Tosi is uncertain about how many commas there
> are in a tone, and how he prefers 9 to 7 on grounds which
> are highly subjective or nonsensical. Then Duffin uses Tosi
> as one of his major sources to 'deduce' that 18th century
> vocal and instrumental intonation should be 1/6 comma
> meantone - a tuning which could only be practiced on
> keyboard instruments!
>
> His source is a singer who says that a comma is the
> smallest interval that can possibly be distinguished
> - and then Duffin uses it to 'prove' the use of a
> temperament which divides this comma into six!

Hmm, that all does sound like shoddy scholarship to me.

> Tosi relates intonation to the form of accompanying
> instrumentation: he says that the difference between
> 'major' and 'minor' semitones is not evident on the
> organ or harpsichord, if they do not have split keys.
> (Split key instruments were, though, probably tuned in
> quarter-comma meantone...)
>
> After plumping for dividing the tone into 9, he says
> that if one is accompanied by keyboard instruments
> without split keys it is useless to know about 'major'
> and 'minor' semitones. But because composers like to
> introduce arias with only (bowed) string instrument
> accompaniment, one must know that D# is lower than Eb.
>
> As we know, 1/6 comma meantone is a tuning for keyboards
> with fixed pitches; yet Tosi only requires the singer to
> differentiate between D# and Eb when accompanied by strings
> without keyboard. In that case there would be no reason
> for any of the performers to adopt a meantone tuning.
> Duffin claims to 'deduce' a keyboard-based tuning from
> what Tosi said about musical performances without keyboard
> accompaniment.

Just playing devil's advocate here, it is true that D# is
lower than Eb in all tunings of the meantone family
(except 12-edo).

But i agree with your subsequent arguments (which i
snipped here). Tosi, in speaking of singing accompanied
only by bowed strings, would (i think) almost certainly
be referring to some form of JI or adaptive-JI.

Of course, in JI, D# may be higher or lower than Eb,
depending on which ratio was either specified by the
composer or, if not specified, which ratio fits the
harmonic context best ... and of course, that "best"
introduces a subjective opinion ...

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Tom Dent <stringph@gmail.com>

Actually, I totally agree with Tosi, to the extent that D# is probably
lower than Eb in all relevant musical contexts (i.e. strings playing
common practice harmony without enharmonic shifts). Suppose a piece in
G major. G and D are open strings tuned (practically) pure, serving as
reference. B becomes probably close to a pure third above G. D#
modulates into E minor via a third above B (eg melody G-C-B-C#-D#-E),
which is unlikely to be very much wider than pure. After a return to
the tonic, there is a brief modulation to G minor and the melody goes,
say, G-F-Eb-D over a bass G. Here the third G-Eb is also unlikely to
be much wider than pure, and Eb turns out probably quite a lot higher
than D#. The exact extent to which this happens will depend on a dozen
unquantifiable factors. (Here, D# and Eb were approached by tones:
their exact tuning depends on how large one makes the tones, not the
semitones!)

One can call the difference 'a comma' and be quite correct, if one
defines the 'comma' as a small and barely noticeable difference in
pitch. That is the level on which Tosi makes perfect sense, since he
did not give a definition that a reader could make quantitative use
of. (1/9 tone, but *which* tone, and how to divide it?)

And I totally agree that this excess of Eb over D# also happens in all
relevant historical meantone tunings, including those that string
players and singers were probably used to hearing from organs, etc.
However, I think it is extremely unlikely that such 'commas' in vocal
or non-keyboard performances were at all consistent between different
performers (I mean on different occasions) - let alone that anyone had
ever measured them accurately.

No singer is likely to remember the exact pitch D# from the first
passage and try to reproduce it as Eb in the second - let alone to try
and sharpen it by one-N'th of a tone. The advice that Eb is higher
than D# by one-N'th-tone might be true enough, but practically not
useful, since to sing in tune one has to learn pure fifths, thirds,
fourths etc. The purity of these consonances (in relation to the
accompaniment) is always a better guide than trying to control the
sizes of tones and semitones without regard to other voices.

Another possibility is that Tosi was mentioning this enharmonic
difference to show off his knowledge and subtlety. It 'proves' that he
must be a good teacher because he can detect things as small as a
nineth of a tone... whereas those ignorant modern singers can't tell
any difference. (Tosi doesn't suspect that keyboard tuners could
detect a fortieth of a tone!)

Consider this passage:

'Proccuri il Maestro, che nel solfeggiar la Scaletta le note sieno
dallo Scolaro perfettamente intonate. Chi non ha delicatezza d'
orecchio non dovrebbe impegnarsi, nè d' insegnar, nè di cantare, non
essendo assolutamente tollerabile il difetto d' una voce, che cresce,
e cala come il flusso, e il riflusso del Mare. Vi rifletta con tutta
l' attenzione l' Istruttore, perchè ogni Cantante, che stuona perde
immediatamente tutte le più belle prerogative, che avesse. Io posso
dir senza mentire, che (a riserva di pochi Professori) la moderna
intonazione è assai cattiva.'

The master should get the student to sing the scale with the notes
perfectly tuned. Whoever does not have a sensitive ear should not
either teach or sing, a voice which rises and falls like the ebb and
flow of the sea being absolutely intolerable. The instructor should
turn all his attention to it, because every singer who mistunes
immediately loses all the good repute that (he or she) had. I can say
honestly that, apart from a few professors, modern intonation is quite
bad.

Two more interesting quotations:

'Il quarto è il Trillo cresciuto, che insegnasi col far ascendere
impercettibilmente la voce trillando di Coma in Coma senza che si
conosca l' aumento.

'Il quinto è il Trillo calato, che consiste nel far discendere
insensibilmente la voce a Coma per Coma col Trillo in forma che [-27-]
non si distingua il declivo. Questi due Trilli da che s' introdusse il
vero buon gusto non sono più in voga, anzi bisogna scordarsi di
saperli fare. Chi ha l' orecchio dilicato egualmente abborre le
seccaggini antiche, e gli abusi moderni.'

i.e. trills ascending or descending imperceptibly by commas (!) which
Tosi says are no longer in fashion.

Also, in the 'appoggiatura' section:

'La Teorica insegna, che la suddetta Ottava essendo composta di
dodici Semituoni ineguali bisogna distinguere i maggiori da i minori,
e invia chi studia a consultare i Tetracordi. Gli Autori più cospicui,
che ne trattano non son tutti d' una opinione, perchè trovasi chi
sostiene, che fra il C sol fa ut, e il D la sol re, come fra 'l F fa
ut, e il G sol re ut i loro Semituoni sieno eguali, e in tanto si
languisce nel dubbio.'

Theory teaches that this octave is composed of twelve unequal
semitones, the major being distinguished from the minor, and sends
whoever is studying to consult the Tetrachords. The most famous
authors who consider it are not all of one opinion, since some are
found who hold that between C and D, as between F and G, the semitones
are equal, such that [the matter] languishes in doubt.

... This would, if taken seriously, indicate a scheme which is not
meantone at all! For example this could be a somewhat sloppy reference
to JI in which C-D and F-G are major tones divided up as 135/128 and
16/15, the difference being too small to care about, whereas D-E and
G-A were divided as 25/24 plus 16/15.

Tosi then casts aside the theory and throws himself on the naked ear...

'Ma se ha quel transito libero di mezza voce ascendendo, da che
procede, che dall' istesso Fa non può [-22-] salire al Diesis vicino,
che pur il passo è di un Semituono? Egli è minore risponde l' Udito;
Dunque suppongo di poter conchiudere, che la cagione, che toglie all'
Appoggiatura una gran parte della libertà deriva, ch' ella non può
passar di grado da un Semituono maggiore ad un minore, nè da questo a
quello; Rimettendomi sempre però al giudizio di chi intende.'

I'm not sure about the exact translation, but it seems to say, F-F# is
a minor semitone, just because the ear says so. And we forbid these
semitones in making appoggiaturas, simply because we are guided by the
judgement of whoever is listening. A reasonable, but not logical or
useful explanation, whose relevance to tuning is highly debatable.
Tosi now finds the difference between major and minor semitones
obvious to the ear. But this is probably due to their musical context
(within a given tonality) rather than a tiny difference in interval sizes.

It always remains unclear *how* the teacher should judge whether the
pupil is exactly in tune or not. Unlike, say, Ramos, Tosi never
specifies any particular scale or interval. In the end he just says
you gotta have good ears. Again, true, but useless for us, whose ears
have been trained in a quite different environment.

~~~T~~~

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> (big cuts!)
>
> Of course, in JI, D# may be higher or lower than Eb,
> depending on which ratio was either specified by the
> composer or, if not specified, which ratio fits the
> harmonic context best ... and of course, that "best"
> introduces a subjective opinion ...
>
>
> -monz
>

🔗Brad Lehman <bpl@umich.edu>

> (...) (Here, D# and Eb were approached by tones:
> their exact tuning depends on how large one makes the tones, not the
> semitones!)
>
> One can call the difference 'a comma' and be quite correct, if one
> defines the 'comma' as a small and barely noticeable difference in
> pitch. That is the level on which Tosi makes perfect sense, since he
> did not give a definition that a reader could make quantitative use
> of. (1/9 tone, but *which* tone, and how to divide it?)

Whaddaya mean, *which* tone? *The* tone. The 9-comma tone. In the
system Tosi was describing in this book, there aren't two or more
different sizes of tone. There's only the 9-comma tone. The diatonic
scales are made up of five of these 9-comma tones, and two 5-comma
semitones.

And Tosi described not only that 9-comma tone, but also pointed out
(correctly) that a sharped note and the enharmonic flat (such as D# vs
Eb) are one comma apart from one another. Tosi explained carefully,
especially in the section of his book about properly improvised
ornamentation, that a singer dare not use the wrong enharmonic in
passing within an ornament. That is, the singer's supposed to sing
the correctly-spelled notes, and to be sensitive to these commatic
differences, employing notes that belong to the prevailing diatonic
scale of a passage rather than bringing in chromatic notes (off by a
comma) from outside the diatonic scale.

Tosi was writing near the end of a 50-year career of teaching and
performing. He was describing not only the current common practice at
the beginning of the 18th, but a method of understanding diatonic
scales (and intonation) stretching back into the 17th century.
Practical musicianship. Why is it so hard to believe that Tosi really
did know what he was talking/writing about? *That* is a level on
which Tosi makes perfect sense.

Brad Lehman

🔗Tom Dent <stringph@gmail.com>

Well, once more:

If there is one thing that Tosi did *not* intend in his textbook, it
is to explain a system of musical tuning for the benefit of 20th
century theorists. Actually, there is no reason to believe that he
intended to explain or refer to any 'system' of tuning at all. The
text is also not intended to teach the basics of musical theory:

'Non parlerò della cognizione delle note, del loro valore, della
battuta, dello spartire, dè tempi, delle pause, degli accidenti, né
d'altri principj triviali; perché sono generalmente noti.'

He assumes that people already know their scales, etc. etc.

What he did say was
1) it is important to sing in tune
2) most singers are bad at this
3) you have to have a sensitive ear
4) his own ear is so good that he can tell the difference between D#
and Eb, even though this is only a nineth of a tone, or maybe a
seventh, depending on which theorists you read, anyway there really is
a difference, even though it's only a little tiny one, and if you're
not convinced you can read books about it and ask the best violinists.
5) But on keyboard instruments which don't have divided sharps you
can't hear the difference so it is not necessary to bother when
accompanied by these.

And on those appoggiaturas:
6) You should sing appoggiaturas with diatonic semitones rather than
chromatic ones. Chromatic ones are smaller, at least that is what most
of the theorists say, except for some of them who say that the tones
C-D and F-G are divided into two equal semitones, but anyway when you
sing the semitone you should be able to hear immediately if it is
large or small, and the ear is the final judge of this question.

In other words his 'definition' of large and small semitones is
circular. You should make E-F large and F-F# small, because the one is
a large semitone and the other a small one; the reason why, is that
E-F sounds large when you sing it and F-F# sounds small.

What information about tuning one can glean is purely circumstantial -
and, when one looks at the text as a whole, also vague and apparently
contradictory.

He never once refers to the actual size of any musical interval, let
alone how to realize it in practice.

He never once gives any objective rule as to how fifths, thirds, etc.
etc. should be tuned.

He never says that whole tones, or fifths, or thirds (etc. etc.)
should be the same size as each other. That would be rather important
if he had been trying to communicate something about a meantone system.

He never contemplates dividing a comma into parts, as would be
necessary to produce meantone; indeed according to his definition (the
smallest possible part) this would produce something inaudible.

It is worth noting that all JI and Pythagorean schemes also have
unequal semitones! Tosi would certainly have found some of these when
reading his theorists.

One ought to remember that there were lots of theorists who did not
concern themselves at all with keyboard tuning. They would have little
or no reason to consider meantone. Even Mattheson defined all the
intervals via integer ratios...

Then we should ask why 20th century theorists (primarily Barbour) were
so obsessed with interpreting everything in the framework of regular
temperaments, even when the source and its intended readers had little
or nothing to do with keyboard tunings.

~~~T~~~

🔗Klaus Schmirler <KSchmir@online.de>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> > (...) (Here, D# and Eb were approached by tones:
> > their exact tuning depends on how large one makes the tones, not the
> > semitones!)
> >
> > One can call the difference 'a comma' and be quite correct, if one
> > defines the 'comma' as a small and barely noticeable difference in
> > pitch. That is the level on which Tosi makes perfect sense, since he
> > did not give a definition that a reader could make quantitative use
> > of. (1/9 tone, but *which* tone, and how to divide it?)
>
> Whaddaya mean, *which* tone? *The* tone. The 9-comma tone. In the
> system Tosi was describing in this book, there aren't two or more
> different sizes of tone. There's only the 9-comma tone. The diatonic
> scales are made up of five of these 9-comma tones, and two 5-comma
> semitones.

Oi Brad! Haven't we been all through this once before, somewhere,
where? The harpsichord list? Here? I don't remember.

How do you know what system Tosi was describing? It could just as well
be that he was using the 4+1+4 comma whole tone construct to
illustrate the general difference between sharps and flats, perhpas
within the context of different sizes whole tones. Point is, he
doesn't say, so how can you affirm with such conviction?

>
> Tosi was writing near the end of a 50-year career of teaching and
> performing. He was describing not only the current common practice at
> the beginning of the 18th, but a method of understanding diatonic
> scales (and intonation) stretching back into the 17th century.

The point you persist in glosing over with statements like this is the
conflict with what other sources recommend, like Tatinni, Leopold
Mozart, Rameau, and Bremner, who all adivse the musician to practice
or employ PURE thirds, sixths, and fifts in performance, even to the
point of DISagreeing with the same note on the haprischord (Rameau).
So where does that leave us with your touted common practice?

> Practical musicianship. Why is it so hard to believe that Tosi really
> did know what he was talking/writing about?

Because it doesn't jive, that's why. Let's assume for the sake of
argument that Tosi did mean what you would have us believe he meant.
As everyone here knows, the closest white-bread equivalent to 55-EDO
is 1/6 comma meantone, which by definiton does NOT produce pure
intervals, not thirds, not sixths, not fifths. THAT'S why it is so
hard to take Tosi literally, because taking him so would conflict with
the advice of other - dare I say - far more prominent musicians.

I have no trouble believing that the 9-comma tone model was used
extensively to help people get a handle on the idea of the difference
between sharps and flats, but as Thomas pointed out, it doesn't have
to all add up to some neat system that describes the whole octave. I
think that if you are honest, you will have to admit that the evidence
that we can automatically link a general application of a theoretical
basis of 55-EDO to every mention of a 9 comma tone, or a 1 comma diff
between sharps and flats, is scant if not nil. Frankly I think it
would better serve us to acknowledge this conflict and seek for
possible resolutions rather than just pretend it doesn't exist.

Ciao,

🔗Brad Lehman <bpl@umich.edu>

"To be honest", I say the things I do because they really do work. I
have no trouble believing that Tosi really knew what he was talking
about, and was indeed reporting the way he sang and taught with
common-practice accompaniment, whether that was keyboard-based or
ensemble-based or both.

No, Tosi's report isn't any incontrovertible evidence that people
*were* doing things resembling regular 1/6 comma, on keyboards or
otherwise, in this common practice. There are additional
possibilities. But it's better than outright contrary assumptions
(i.e. that they were doing something radically different from 1/6 C,
and that Tosi was teaching the 9-comma stuff *despite* the things good
ensembles and good keyboard players were doing in everyday practice).

As for Ross Duffin's published recommendation from all this, from Tosi
and Telemann and the Mozarts et al, why not just try it out? That is:
assume regular 1/6 comma (or the 55EDO) as at least a conceptual basis
for everybody in the ensemble, and maybe also have it on the
keyboard...and then play, with the freedom to tighten up the
occasional major 3rd to a pure 5:4, or open up the occasional 5th to a
pure 3:2, or whatnot. The point as Duffin explains it is to have
something down-the-middle like that, and then *allowing* that
intonational flexibility for good musicians in both directions, not
having to hit all the 55EDO points exactly (or match the tyranny of
any keyboard), but merely as a basis for good overall ensemble
intonation.

If that means tuning the strings of the violin and/or viola-da-gamba
families to 1/6 comma tightness or environs (which Duffin *measured*
Joachim doing on recordings, 698-cent open 5ths in 1903, but we
digress...), to help establish that basis, give it a try. Duffin has
provided evidence for that from a variety of 18th century sources, and
not only from Tosi.

I did try it out myself in ensemble, and it happened to work
marvelously. This was in a concert earlier this year with two
professors who spend most of their time doing mainstream
ensemble/orchestral/teaching, not Baroque specialty. I played
harpsichord, and the other two musicians played bassoon and cello. It
was for a bassoon recital, and she played Baroque bassoon on this, but
modern bassoon for everything else. We did a sonata by Telemann in A
minor, melody and continuo, and it happened to fit exactly into
Bb-F-...-G#-D#. So, I tuned that way in regular 1/6 comma and leaving
a wolf at D#-Bb. I didn't do any explaining to my colleagues, other
than remarking offhand that I had set up "Telemann's own tuning" on
the harpsichord (which is true: the subset of 12 55EDO notes for this
composition, from Telemann's published system), and then I gave each
of the open strings to the cellist, one by one. He tuned to these
without fuss, and then we all just played.

In each rehearsal and the performance, that's all that was said and
done about intonation; we used musical instincts from there on. I
noticed the bassoonist occasionally lowered her notes to make decently
near 5:4 major 3rds wherever it seemed important, while the 5ths all
sounded fine no matter what she did, and we simply PLAYED MUSIC and
listened with our ears. The harpsichord's resonance did what it's
supposed to do, as to establishing nicely solid and present major and
minor chords in all the keys that came up. The pure tritones were an
added attraction, coming up as they do in 7th-chords and diminished
triads. This overall intonation scheme made the music easy and
unproblematic to play, which is of course part of the point of
selecting an appropriate temperament for the music. It all sounded
natural enough to me, too, as to what that Baroque bassoon was already
inclined to do with its intonation. It certainly wasn't set up in or
bowdlerized to equal.

So, really, what's the point of trying to discredit Tosi's expert
witness in the first decades of the 18th C? Or to discredit the
writings of any other suitably expert musician/teacher, of the time?

Brad Lehman

>
> > Tosi was writing near the end of a 50-year career of teaching and
> > performing. He was describing not only the current common practice at
> > the beginning of the 18th, but a method of understanding diatonic
> > scales (and intonation) stretching back into the 17th century.
>
> The point you persist in glosing over with statements like this is the
> conflict with what other sources recommend, like Tatinni, Leopold
> Mozart, Rameau, and Bremner, who all adivse the musician to practice
> or employ PURE thirds, sixths, and fifts in performance, even to the
> point of DISagreeing with the same note on the haprischord (Rameau).
> So where does that leave us with your touted common practice?
>
> > Practical musicianship. Why is it so hard to believe that Tosi really
> > did know what he was talking/writing about?
>
> Because it doesn't jive, that's why. Let's assume for the sake of
> argument that Tosi did mean what you would have us believe he meant.
> As everyone here knows, the closest white-bread equivalent to 55-EDO
> is 1/6 comma meantone, which by definiton does NOT produce pure
> intervals, not thirds, not sixths, not fifths. THAT'S why it is so
> hard to take Tosi literally, because taking him so would conflict with
> the advice of other - dare I say - far more prominent musicians.
>
> I have no trouble believing that the 9-comma tone model was used
> extensively to help people get a handle on the idea of the difference
> between sharps and flats, but as Thomas pointed out, it doesn't have
> to all add up to some neat system that describes the whole octave. I
> think that if you are honest, you will have to admit that the evidence
> that we can automatically link a general application of a theoretical
> basis of 55-EDO to every mention of a 9 comma tone, or a 1 comma diff
> between sharps and flats, is scant if not nil. Frankly I think it
> would better serve us to acknowledge this conflict and seek for
> possible resolutions rather than just pretend it doesn't exist.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

It is most intriguing, that Antoine Murat, a westernized Turk, has explained
Maqam Music tuning of the 18th century in terms of 9 commas per whole tone,
55 to the octave. Don't you think there is a parallel here?

Oz.

----- Original Message -----
From: "Brad Lehman" <bpl@umich.edu>
To: <tuning@yahoogroups.com>
Sent: 18 Ekim 2007 Perï¿½embe 21:04
Subject: [tuning] Re: Tosi, what he did or didn't say

> "To be honest", I say the things I do because they really do work. I
> have no trouble believing that Tosi really knew what he was talking
> about, and was indeed reporting the way he sang and taught with
> common-practice accompaniment, whether that was keyboard-based or
> ensemble-based or both.
>
> No, Tosi's report isn't any incontrovertible evidence that people
> *were* doing things resembling regular 1/6 comma, on keyboards or
> otherwise, in this common practice. There are additional
> possibilities. But it's better than outright contrary assumptions
> (i.e. that they were doing something radically different from 1/6 C,
> and that Tosi was teaching the 9-comma stuff *despite* the things good
> ensembles and good keyboard players were doing in everyday practice).
>
> As for Ross Duffin's published recommendation from all this, from Tosi
> and Telemann and the Mozarts et al, why not just try it out? That is:
> assume regular 1/6 comma (or the 55EDO) as at least a conceptual basis
> for everybody in the ensemble, and maybe also have it on the
> keyboard...and then play, with the freedom to tighten up the
> occasional major 3rd to a pure 5:4, or open up the occasional 5th to a
> pure 3:2, or whatnot. The point as Duffin explains it is to have
> something down-the-middle like that, and then *allowing* that
> intonational flexibility for good musicians in both directions, not
> having to hit all the 55EDO points exactly (or match the tyranny of
> any keyboard), but merely as a basis for good overall ensemble
> intonation.
>
> If that means tuning the strings of the violin and/or viola-da-gamba
> families to 1/6 comma tightness or environs (which Duffin *measured*
> Joachim doing on recordings, 698-cent open 5ths in 1903, but we
> digress...), to help establish that basis, give it a try. Duffin has
> provided evidence for that from a variety of 18th century sources, and
> not only from Tosi.
>
> I did try it out myself in ensemble, and it happened to work
> marvelously. This was in a concert earlier this year with two
> professors who spend most of their time doing mainstream
> ensemble/orchestral/teaching, not Baroque specialty. I played
> harpsichord, and the other two musicians played bassoon and cello. It
> was for a bassoon recital, and she played Baroque bassoon on this, but
> modern bassoon for everything else. We did a sonata by Telemann in A
> minor, melody and continuo, and it happened to fit exactly into
> Bb-F-...-G#-D#. So, I tuned that way in regular 1/6 comma and leaving
> a wolf at D#-Bb. I didn't do any explaining to my colleagues, other
> than remarking offhand that I had set up "Telemann's own tuning" on
> the harpsichord (which is true: the subset of 12 55EDO notes for this
> composition, from Telemann's published system), and then I gave each
> of the open strings to the cellist, one by one. He tuned to these
> without fuss, and then we all just played.
>
> In each rehearsal and the performance, that's all that was said and
> done about intonation; we used musical instincts from there on. I
> noticed the bassoonist occasionally lowered her notes to make decently
> near 5:4 major 3rds wherever it seemed important, while the 5ths all
> sounded fine no matter what she did, and we simply PLAYED MUSIC and
> listened with our ears. The harpsichord's resonance did what it's
> supposed to do, as to establishing nicely solid and present major and
> minor chords in all the keys that came up. The pure tritones were an
> added attraction, coming up as they do in 7th-chords and diminished
> triads. This overall intonation scheme made the music easy and
> unproblematic to play, which is of course part of the point of
> selecting an appropriate temperament for the music. It all sounded
> natural enough to me, too, as to what that Baroque bassoon was already
> inclined to do with its intonation. It certainly wasn't set up in or
> bowdlerized to equal.
>
> So, really, what's the point of trying to discredit Tosi's expert
> witness in the first decades of the 18th C? Or to discredit the
> writings of any other suitably expert musician/teacher, of the time?
>
>
> Brad Lehman
>
>
> >
> > > Tosi was writing near the end of a 50-year career of teaching and
> > > performing. He was describing not only the current common practice at
> > > the beginning of the 18th, but a method of understanding diatonic
> > > scales (and intonation) stretching back into the 17th century.
> >
> > The point you persist in glosing over with statements like this is the
> > conflict with what other sources recommend, like Tatinni, Leopold
> > Mozart, Rameau, and Bremner, who all adivse the musician to practice
> > or employ PURE thirds, sixths, and fifts in performance, even to the
> > point of DISagreeing with the same note on the haprischord (Rameau).
> > So where does that leave us with your touted common practice?
> >
> > > Practical musicianship. Why is it so hard to believe that Tosi really
> > > did know what he was talking/writing about?
> >
> > Because it doesn't jive, that's why. Let's assume for the sake of
> > argument that Tosi did mean what you would have us believe he meant.
> > As everyone here knows, the closest white-bread equivalent to 55-EDO
> > is 1/6 comma meantone, which by definiton does NOT produce pure
> > intervals, not thirds, not sixths, not fifths. THAT'S why it is so
> > hard to take Tosi literally, because taking him so would conflict with
> > the advice of other - dare I say - far more prominent musicians.
> >
> > I have no trouble believing that the 9-comma tone model was used
> > extensively to help people get a handle on the idea of the difference
> > between sharps and flats, but as Thomas pointed out, it doesn't have
> > to all add up to some neat system that describes the whole octave. I
> > think that if you are honest, you will have to admit that the evidence
> > that we can automatically link a general application of a theoretical
> > basis of 55-EDO to every mention of a 9 comma tone, or a 1 comma diff
> > between sharps and flats, is scant if not nil. Frankly I think it
> > would better serve us to acknowledge this conflict and seek for
> > possible resolutions rather than just pretend it doesn't exist.
>
>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> "To be honest", I say the things I do because they really do work.

Whether or not "it works" (whatever sort of subjective process of
evaluation that may involve) is NOT the point, Brad. THE point is you
stating that Tosi WAS talking about THE system of 9 commas per tone
with 4 tones of 9 each and 2 semitones of 5 each, which adds up to a
55 comma octave. When you state that "this is the system Tosi was
talking about" without any evidence whatsoever, it's just slipshod
musicology. Lord knows we've had more than enough of that already.

> I
> have no trouble believing that Tosi really knew what he was talking
> about, and was indeed reporting the way he sang and taught with
> common-practice accompaniment, whether that was keyboard-based or
> ensemble-based or both.

I also have no trouble believing that Tosi knew what he was talking
about. I just have a lot of trouble believing your interpretation of
Tosi.
>
> No, Tosi's report isn't any incontrovertible evidence that people
> *were* doing things resembling regular 1/6 comma, on keyboards or
> otherwise, in this common practice. There are additional
> possibilities. But it's better than outright contrary assumptions
> (i.e. that they were doing something radically different from 1/6 C,
> and that Tosi was teaching the 9-comma stuff *despite* the things good
> ensembles and good keyboard players were doing in everyday practice).

It's only contrary when one insists on the rigid interpretation that a
conceptual division of the tone into 9 parts means a 55-comma octave,
like you did. Personally, I have no trouble believing Tosi's 9-comma
tone in combination with pure thirds and fifths as recommended by the
other guys. I don't see why there has to be a conflict. It's just
proportional, Brad. It's just a rule of thumb. Use it to chop up any
tone of any size into two semitones.

>
> As for Ross Duffin's published recommendation from all this, from Tosi
> and Telemann and the Mozarts et al, why not just try it out?

Because I'm not interested in "just trying it out". I'm interested in
what can be gleaned from the sources without jumping to conclusions.
Sure, eventually we have to do something when we play music, but thats
another ball of wax. It's about musicology, Brad, which is supposed to
be based on objectivity, not what you like or what you THINK "works".
It's got nothing to do with what you or I like. I don't like 1/6th
comma meantone, but the only reason I question the current fad of
using it as a default temperament for this era is precisely all those
guys who said otherwise. Punkt schluss.
>
> I did try it out myself in ensemble, and it happened to work
> marvelously. This was in a concert earlier this year with two
> professors who spend most of their time doing mainstream
> ensemble/orchestral/teaching, not Baroque specialty. I played
> harpsichord, and the other two musicians played bassoon and cello. It
> was for a bassoon recital, and she played Baroque bassoon on this, but
> modern bassoon for everything else. [snip]This overall intonation
scheme made the music easy and
> unproblematic to play, which is of course part of the point of
> selecting an appropriate temperament for the music. It all sounded
> natural enough to me, too, as to what that Baroque bassoon was already
> inclined to do with its intonation. It certainly wasn't set up in or
> bowdlerized to equal.

So, two modern musicians who normally play ET found it easy to use
your intonation scheme to everyone's mutual satisfaction. Why, that
must indeed prove Tosi was right!! How very scientific! Yes, now I'm
utterly convinced.
>
> So, really, what's the point of trying to discredit Tosi's expert
> witness in the first decades of the 18th C?

Nobody is trying to discredit Tosi, I'm just pointing out problems
with your affirmations about what Tosi meant.

> Or to discredit the
> writings of any other suitably expert musician/teacher, of the time?

Uh, I don't understand your motivation for bringing that one up? Did
anybody do that?

Really, Brad, based on the way you argue points and drag out red
herrings left, right, and center, sometimes I think you listen to too
much Rush Limbaugh. Next post you'll accuse me of having said Tosi,
Tartinni, Mozart, and Bremner were all "phoney musicians".

;-)

Ciao,

🔗Tom Dent <stringph@gmail.com>

Apologies for taking all replies together ...

Klaus, if you have an author for whom anything smaller than a comma is
effectively inaudible, you can't use him as a source of information on
the use of one or other kind of meantone tuning.

> > Whaddaya mean, *which* tone? *The* tone. The 9-comma tone.

So, define a tone by the fact that it contains 9 commas, and then
define the comma as being a 9th part of the tone. (Not to censor the
reality of Tosi's description, 9 might also be 7.) That is empty, it
doesn't tell the reader what size a tone or comma should be in the
first place. Pythagorean tones, the easiest to tune, are made up of
about 9 Pythagorean commas too...

> > The diatonic
> > scales are made up of five of these 9-comma tones, and two 5-comma
> > semitones.

One can make a nice arithmetic sum and get the number 55 - *if* you
assume that every comma is an identical mathematical quantity, and
that every tone is alike, and the two remaining semitones are the same
as the 'larger' semitones that divide the five tones. But there is no
reason to believe Tosi ever did this arithmetic. (Why would he need
to, if he trusted his ears more than his theory books?)

Even if he had, it would not have helped him or his readers, since
none of them could tell what x/55 of an octave should sound like. Let
alone that, by a stretch of mathematical implication, fifths should
then be sounded barely flat, major thirds sharper, and minor thirds
yet flatter. (None of which is necessary anyway if you don't have a
keyboard accompaniment.)

I am certainly not attacking Tosi as musician. A lack of knowledge of
meantone temperament would NOT have been a handicap for a singer or
teacher. It is simply necessary to listen to the purity of
consonances, and to keyboard instruments which have been carefully
tuned. The singer's ear does the job itself much better than any formula.

So, anyone who doesn't have enough insight into the mechanics of
regular tempered scales (to the extent of being able to 'deduce' a
9-comma tone into a sixth-comma meantone) can't be a good practical
musician? If we believe Brad, every competent musician in history was
a temperament fanatic like him.

Ignorance of 'correct' tuning theory, and the inability to define a
comma, have not been a problem for thousands of excellent singers
today. They know how to sing in tune, they just don't know how to
describe it mathematically or verbally. And why should they?

Tuning theorists are at a loss to describe how choirs *really* sing
and how string quartets *really* play. We have some mathematical
models which are probably a reasonable approximation in some cases. So
what would an experienced singer do with one or more semi-accurate
mathematical models? - Throw his hands up and rely on his sense of
hearing. Tosi starts and ends the discussions of tuning and semitones
by saying: the sensitive ear is the judge of everything. But he
couldn't, and didn't attempt to, describe exactly what he heard. That
is no failing on his part - simply the reality of complex, real,
un-mathematically-tidy musical performance.

> > a method of understanding diatonic
> > scales (and intonation) stretching back into the 17th century.

Musicians back at least to the 15th century understood the diatonic
scale with chromatic inflections, and the difference between diatonic
and chromatic semitones. But I don't see where anyone showed that this
distinction meant anything definite for the tuning of consonances in
vocal or orchestral performance. The unequal division of C-C#-D into
small and large semitone is equally true for Pythagorean tuning (with
C# tuned as Db) as for meantone. In the Pythagorean era, E-F and B-C
were *small* semitones - making, if one cares to do the arithmetic, 53
divisions in an octave.

When exactly did theory catch up with meantone practice? Perhaps with
Huygens' 31-division and Rossi's compendium of meantones (didn't
include 1/6 comma and 55-equal!). But most music theorists ignored
Rossi and Huygens anyway.

What *does* tell us about tuning in practice is texts which say
whether the consonances (third, fifth, etc) should be sharp, flat or
pure. Something immediately audible. Lo and behold, most such texts in
the 17th century ask for pure major thirds, and the keyboard
instruments with their extra notes obligingly allow us to hear as many
of them as possible.

*No* known 17th century text explains or encourages the use of
1/6-comma...

> > Practical musicianship. Why is it so hard to believe that Tosi really
> > did know what he was talking/writing about?

Non Sequitur. (And what is musicianship if not practical?) Musicality
and correct statements about tuning theory are, practically,
uncorrelated. Dom Bedos was the best French 18th century writer on
organbuilding, his account of tuning theory is an absurd muddle, but
he then tells you perfectly clearly how to tune (quarter comma)
meantone. Most excellent musicians today couldn't begin to tell you
what a comma or a meantone temperament was, unless you gave them a
copy of Grove. Even then they couldn't tell you what fraction of a
comma flat or sharp they make such and such an interval in a melody.
They just l i s t e n.

Let's boil this down. If Tosi understood what 1/6 comma meantone was -
a system in which fifths are slightly flat and major thirds are
slightly more sharp - and thought it to be important - if! - why
didn't he explain it?

If he (by assumption) was relying on a previous tradition of 1/6 comma
meantone, why have all the historical explanations of how singers and
instrumentalists could use this particular system in practice simply
vanished?

~~~T~~~

🔗Brad Lehman <bpl@umich.edu>

"Slipshod musicology"? I think you're missing one of my other
favorite points, and it's this foray into epistemology: Those old dead
guys of the 17th century who didn't bother to write down what they
did, as to tuning.... well, the positivistic type of musicology can't
prove OR DISPROVE that they did OR DIDN'T ever use 1/6 comma as
practical musicians. It can't answer that question on written
evidence, one way OR the other.

Some houses have mice, whether or not there is a cat on duty to reveal
their presence. Even if a house does have one or several well-fit
cats on duty, they still might not know about all the present mice.
It's not a perfect system. Some things may have existed, even if
nobody troubled to write them down. We can't know for sure, by the
method of examining ONLY such writings, the existence OR the
non-existence. There is no written evidence (at least that I know of)
that anybody in the 17th century in Europe ever cut themselves a
couple thinnish slices of softish bread, stuck something tasty in
between, and had themselves a "sandwich". The name "sandwich" didn't
exist, as such, describing such a mini-meal. Does this constitute
absolute proof that nobody ever ate one before the idea or the name
got canonized? Positivism can't answer such a question, either yea or
nay, for sandwiches.

And meanwhile, the *practical* type of musicology, i.e. actually
examining and listening to extant music, and trying it out with
various plausible tuning schemes that people *could* have known about
(even if they didn't write it down in modern scientific language)...it
may be able to offer us clues into questions such as this, which
positivism cannot answer.

To me, and maybe I'm just an oddball, musicology CAN NEVER wholly
divorce itself from hands-on practice as to know what's plausible, or
even what's possible. The positivistic angle of musicology HAS in the
20th century gone in that direction, on and off, as we all know. What
about those of us who fancy some questions that that style of inquiry
cannot answer, either one way or the other? I'm a performing
musician. I have to make firm decisions every time I prepare any
performance, even if positivistic musicology (for all its other
strengths) simply cannot tell me everything I would need to know to
give a convincing interpretation of music. Given 5 to 240 minutes of
music to be played on some given occasion, I have to find a way to set
up the instrument to make plausible results, both historically and
musically, for the people who show up to hear it. Even if it's a
program of all 17th century music, it's pretty unlikely that anybody's
rigid scheme of 1/4 comma meantone ONLY (nothing else allowed, because
the evidence for anything else is insufficient!!) is going to do.
Positivistic musicology, at least when applied in that way, condemns
to mediocrity some of the music it's trying to reveal solutions for.

I have a virginal at home (among other stringed keyboard instruments),
and I keep it in regular 1/5 comma most of the time BECAUSE IT SOUNDS
BEST THAT WAY to me, the owner, playing the repertoire to which it's
best suited: 16th-17th C along with the tonal improvisations I enjoy
creating. For its tone quality, if the 5ths are any narrower than
that (to suit the allegedly "better" major 3rds of 1/4 comma), they
just get too nasty; admittedly an aesthetic decision, a musical
opinion. I know we're in a completely different culture now from the
guys 300-400 years ago, but I have to believe (at least on faith) that
some of them with musical ability *also* probably made some similar
choices toward sharpened major 3rds of some degree. Another of my
harpsichords sounds best (at least to me) left in regular 1/6 most of
the time, and its 5ths/4ths stand out as jarringly ugly any tighter
than that. These instruments themselves, the musical sound, tell me
things I can't find in books of positivistic musicology. If I cut off
such sources of understanding, from myself, I'd be less than half a
musician.

As for Rush Limbaugh, I have never once listened to even one of his
broadcasts. Ever. Not that that should matter.

And now I'm going to go make myself a sandwich and sight-read some
"English Pastime Music, 1630-1660" from the fine A-R Edition by Martha
Maas. It gives me some musical and intellectual fulfillment to go do
so, playing it (as it happens) in 1/5 comma for the joy of playing it.

Brad Lehman

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@> wrote:
> >
> > "To be honest", I say the things I do because they really do work.
>
> Whether or not "it works" (whatever sort of subjective process of
> evaluation that may involve) is NOT the point, Brad. THE point is you
> stating that Tosi WAS talking about THE system of 9 commas per tone
> with 4 tones of 9 each and 2 semitones of 5 each, which adds up to a
> 55 comma octave. When you state that "this is the system Tosi was
> talking about" without any evidence whatsoever, it's just slipshod
> musicology. Lord knows we've had more than enough of that already.
>
> > I
> > have no trouble believing that Tosi really knew what he was talking
> > about, and was indeed reporting the way he sang and taught with
> > common-practice accompaniment, whether that was keyboard-based or
> > ensemble-based or both.
>
> I also have no trouble believing that Tosi knew what he was talking
> about. I just have a lot of trouble believing your interpretation of
> Tosi.
> >
> > No, Tosi's report isn't any incontrovertible evidence that people
> > *were* doing things resembling regular 1/6 comma, on keyboards or
> > otherwise, in this common practice. There are additional
> > possibilities. But it's better than outright contrary assumptions
> > (i.e. that they were doing something radically different from 1/6 C,
> > and that Tosi was teaching the 9-comma stuff *despite* the things good
> > ensembles and good keyboard players were doing in everyday practice).
>
> It's only contrary when one insists on the rigid interpretation that a
> conceptual division of the tone into 9 parts means a 55-comma octave,
> like you did. Personally, I have no trouble believing Tosi's 9-comma
> tone in combination with pure thirds and fifths as recommended by the
> other guys. I don't see why there has to be a conflict. It's just
> proportional, Brad. It's just a rule of thumb. Use it to chop up any
> tone of any size into two semitones.
>
> >
> > As for Ross Duffin's published recommendation from all this, from Tosi
> > and Telemann and the Mozarts et al, why not just try it out?
>
> Because I'm not interested in "just trying it out". I'm interested in
> what can be gleaned from the sources without jumping to conclusions.
> Sure, eventually we have to do something when we play music, but thats
> another ball of wax. It's about musicology, Brad, which is supposed to
> be based on objectivity, not what you like or what you THINK "works".
> It's got nothing to do with what you or I like. I don't like 1/6th
> comma meantone, but the only reason I question the current fad of
> using it as a default temperament for this era is precisely all those
> guys who said otherwise. Punkt schluss.
> >
> > I did try it out myself in ensemble, and it happened to work
> > marvelously. This was in a concert earlier this year with two
> > professors who spend most of their time doing mainstream
> > ensemble/orchestral/teaching, not Baroque specialty. I played
> > harpsichord, and the other two musicians played bassoon and cello. It
> > was for a bassoon recital, and she played Baroque bassoon on this, but
> > modern bassoon for everything else. [snip]This overall intonation
> scheme made the music easy and
> > unproblematic to play, which is of course part of the point of
> > selecting an appropriate temperament for the music. It all sounded
> > natural enough to me, too, as to what that Baroque bassoon was already
> > inclined to do with its intonation. It certainly wasn't set up in or
> > bowdlerized to equal.
>
> So, two modern musicians who normally play ET found it easy to use
> your intonation scheme to everyone's mutual satisfaction. Why, that
> must indeed prove Tosi was right!! How very scientific! Yes, now I'm
> utterly convinced.
> >
> > So, really, what's the point of trying to discredit Tosi's expert
> > witness in the first decades of the 18th C?
>
> Nobody is trying to discredit Tosi, I'm just pointing out problems
> with your affirmations about what Tosi meant.
>
> > Or to discredit the
> > writings of any other suitably expert musician/teacher, of the time?
>
> Uh, I don't understand your motivation for bringing that one up? Did
> anybody do that?
>
> Really, Brad, based on the way you argue points and drag out red
> herrings left, right, and center, sometimes I think you listen to too
> much Rush Limbaugh. Next post you'll accuse me of having said Tosi,
> Tartinni, Mozart, and Bremner were all "phoney musicians".
>
> ;-)
>
> Ciao,
>
> P
>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> "Slipshod musicology"? I think you're missing one of my other
> favorite points, and it's this foray into epistemology: Those old dead
> guys of the 17th century who didn't bother to write down what they
> did, as to tuning.... well, the positivistic type of musicology can't
> prove OR DISPROVE that they did OR DIDN'T ever use 1/6 comma as
> practical musicians. It can't answer that question on written
> evidence, one way OR the other.

Yes, we are quite familiar with this line of arguing, used by many in
the field of general history to explain away everything and justify
whatever anyone feels like proposing, including (and I am not making
this up) that assertion that the Eskimo's own mythological explanation
for how they came into existence (something akin to polar bear pulling
the first Eskimo out of a hole in the ice) is just as "valid"
(whatever that word might mean in their intellectual universe) as the
"positivistic theory" that they walked across ice bridges from
northern Asia.

Personally, I find such arguments to be intellectually vapid and
ultimately nihilistic, as they can be used to dismantle absolutely
everything, as they often are. I have little patience for them, and I
certainly place no stock in any viewpoint which relies upon them for
credibility.
>
> And meanwhile, the *practical* type of musicology, i.e. actually
> examining and listening to extant music, and trying it out with
> various plausible tuning schemes that people *could* have known about
> (even if they didn't write it down in modern scientific language)...it
> may be able to offer us clues into questions such as this, which
> positivism cannot answer.

I quite agree. But let's keep this discussion on point, because you
are drifting seriously to port and starboard, Brad. THE point is that
you stated that Tosi WAS talking about a certain system, a statement
for which you nor anyone has any proof. Now, this statement may well
mislead many here into thinking that there is a historical clarity
here which simply does not exist. I quite agree with Margo that
teaching is a responsibility, and one should be careful to state what
we know and what we only can postulate, although your comments above
would seem to indicate that you are in that camp which thinks there is
no difference between the two. If you had said, "Tosi may well have
been referring to the system of ..." there would have been nothing to
argue here. It was the cock-sure force of your assertion that is
slipshod musicology, no matter how you mince your words.

>
> As for Rush Limbaugh, I have never once listened to even one of his
> broadcasts. Ever. Not that that should matter.

You should sometime, I really mean it. I think everybody should, not
only as a exercise in clear thinking and logical discourse (NOT!), but
also as a means of understanding the mentality of a significant
portion of the American electorate. Plus it is a lot of fun, the guy
is a real scream. I'll never forget the time many years back when he
was ranting about how the social costs of employing people in Europe
drove the price of a pizza in Germany up to $15!!! Of course, we he
failed to mention was that (a) the dollar was really weak against the
Mark at that moment and (b) the average German had a salary that
comparatively, in terms of real purchasing power, was quite superior
to the average American, so that a $15 pizza was no big deal. The guy
is REALLY good and fast at cooking up such apparently true but
ultimately false constructs, and it's smoking out that sort of
rhetorical rabbit-out-a-hat that makes listening to Rush such fun!

Ciao,

🔗monz <monz@tonalsoft.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

>
> *No* known 17th century text explains or encourages the use of
> 1/6-comma...
>
Good to hear this from another source. Personally, I didn't know of
any myself, which has always been at the core of my doubts about the
historical validity of using regular 1/6 meantone, but I make no
claims to have made any sort of detailed examination of the
literature. Actually, is there ANY indication whatsoever for the
actual use of regular 1/6 mean in any historical period other than
Sorge's description of Silbermann's tuning of a couple organs?

Just for the record, we should not forget the temperament Werckmeister
recommended in 1698 for continuo realization, which is a modified
meantone. While at first reading it may appear to leave a considerable
amount of wiggle room, when you sit down and analyze what he says, the
limit of possibilities which satisfy all of his stated conditions is
actually rather narrow. For the 7 or 8 identically tempered fifths
(what I call the "main sequence", borrowing a term from astronomy),
the narrowest possibility is about 1/5 comma and the widest either 1/6
(with F-C pure) or 1/7 (with F-C narrow). Any closer to pure and
you've got no overcompensation ("harmonic waste" as Jorgy calls it -
love that term!) to cancel with the wide fifths. So one could argue
that this temperament strongly suggests the use of a basic tempering
of approximately 1/6 comma as a point of departure for a
semicirculating modified meantone. However, since Werckmeister himself
specifically states that he doesn't want to define any particular
comma (by which I assume he means any specific fraction of any comma
for the individual fifths), no matter how much the numbers seem to
suggest 1/6, we cannot pronounce anything like "the system
Werckmeister was starting from was indeed a 1/6 comma meantone."

Seriously, though, getting back to the matter of 17th century (or ANY
century, for that matter) indications for the use of 1/6 mean, of
course this text is very late to qualify as a 17th century document,
squeaking by by the hair of its chinny chin chin as it does.
Furthermore, other than "discrediting" those who persist in using 1/4
comma meantone (which he does in no uncertain terms), Werckmeister
offers no clue as to whether or not his recommendation for how the
practicing keyboard player can best temper his instrument was based
standard practice or represented a new approach.

I suppose I really ought to read Duffin's book one of these days,
although we don't have it here in the ESMUC library, and I'm loath to
spend my own hard-earned Euros just to know upon which sources he
bases his claims for 1/6 comma usage. Maybe somebody who has the book
could do us all the favor of citing the specific sources, including
the original texts, if it's not too much to ask? It would make for
interesting debate, I suspect.

I'd also really like to know Duffin's technical procedure for
analyzing recordings of violinists tuning. Having done a fair bit of
temperament and intonation deduction based on recordings of live
string and wind playing (mostly Indian and Arabic), I can say that any
real precision is quite difficult to achieve, especially since bowed
strings all suffer from subtle pitch change depending upon bow
pressure and speed. My methods have always been two fold: make a loop
of the bit to analyze and set it playing repeatedly, then compare it
to a generated tone of similar color which is tuned by ear to achieve
the minimal beating, the frequency of which is then taken as the
frequency of the recorded note; secondly, if the waveform of the tone
can be clearly viewed, without interference of other tones sounding at
the same time, simply counting the number of milliseconds for 20 or 30
periods, a process which must be done at several different places in
the sample and the results averaged. Let me postulate - Duffin doesn't
describe his techniques? Wouldn't surprise me.

Ciao,

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Brad Lehman <bpl@umich.edu>

> > *No* known 17th century text explains or encourages the use of
> > 1/6-comma...

I guess the definition of "known" is a rather loose one, then. What
about A. Kircher's _Musurgia universalis_ of 1650?

Barbour (p. 126) and Dominique Devie (p. 74) both cite a regular 1/6
comma system by "Ramarin", as explained in this Kircher tome. Devie's
table there gives it as a "temperament usuel" with that 1650 date...as
opposed to the three other options "temperament
excentrique", "temperament academique", or "temperament d'erudit" in
the same table.

Devie also (on pp 70-71) explains Sauveur's reference to the 55-
division, from 1701 and 1711. Sauveur didn't fancy it, but he said it
existed and was in ordinary usage: "Le systeme de 55 qui est celui des
musiciens ordinaires doit etre exclu, parce qu'il est exprime par un
grand nombre et que les tierces ont des differences fort grandes."
That is, apparently, Sauveur wasn't fond of the [major] thirds or
chopping up the octave into *that* many chunks (55), but he
nevertheless reported its common usage by real musicians.

Last time I had a printed copy of Kircher's book open on a table, some
years ago, I wasn't looking for that Ramarin bit in particular. I
suppose I should go take another look when opportunity presents
itself. Or, one or several of us could start pawing through all 1283
pages of the thing (two volumes) in the free online facsimile:
http://echo.mpiwg-berlin.mpg.de/ECHOdocuView/ECHOzogiLib?
mode=imagepath&url=/mpiwg/online/permanent/library/B398U3SN/pageimg
http://echo.mpiwg-berlin.mpg.de/ECHOdocuView/ECHOzogiLib?
mode=imagepath&url=/mpiwg/online/permanent/library/WFCRQUZK/pageimg

Anyway: the idea that *nobody* 17th century thought of the obvious and
practical idea of going down the middle between 1/4 comma and 1/12
comma, to strike a balance of smoothness among the major 3rds and the
5ths?! Pshaw.

Or to use 1/6 comma not as any sort of compromise, but for its own
attractively musical sake (such as the pure tritones, and the usability
of misspelled triads B major, F# major, and C# major in first
inversion. That works because the outer notes hit very close to an
11:7 sixth (Eb-B, Bb-F#, and F-C#), and therefore sound attractively
resonant...even though those same three misspelled triads in root
position are less attractive. Nobody 17th century ever bothered to try
this (whether they wrote it down or not)? Pshaw. Anybody screwing
around with "a little bit sharp" major 3rds, and gentler-than-1/4-comma
5ths, on a real harpsichord (past or present) is bound to hit on such
happy results, whether they can explain them technically with any maths
or not. It's *that* obvious, at least to me, having done so myself.
Set up the dang thing and play in it, and these niceties pop out of it.

I know that my belligerent pshawing doesn't constitute a positive
argument, "proving" that 17th century existence in any way. At the
same time, my musical experience tells me: how could the decent
musicians (or at least one or several of them) *not* have hit upon
this, with or without the help of "Ramarin" or any others? This stuff
is just not that hard to do or to perceive, working directly at a
decent instrument with a tuning lever in hand, whether one understands
the maths or not. (And yes, I had to go cogitate for a while *why*
those sixths such as Eb-B were sounding so nice, without knowing
immediately that they were near 11:7; but the sound itself was obvious,
and immediate perception, first.)

Brad Lehman

🔗monz <monz@tonalsoft.com>

🔗Leonardo Perretti <dombedos@alice.it>

🔗Brad Lehman <bpl@umich.edu>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > Philolaus's small divisions have the whole-tone of
> > 9:8 ratio divided into 4 "diaschismata" and 1 "comma",
> > and the diaschisma is approximately twice the size
> > of the comma, thus the 9-fold division.
>
> Is any of that made precise?

Not really.

First of all, it's *all* hearsay, because nothing
actually *by* Philolaus exists. What we have are one
fragment quoted by Nicomachus and two other fragments
quoted by Boethius.

According to the last fragment in Boethius (book 3,
chapter 8), we get the following:

(click the "Option | Use Fixed Width Font" links for
viewing correctly on the stupid Yahoo web interface)

-- 2,3-monzo --
2 3 ~cents
schisma [ -19/2 6 > 11.73000519
comma [ -19 12 > 23.46001038
diaschisma [ 4 -5/2 > 45.11249784
diesis [ 8 -5 > 90.22499567
integral 1/2-tone [ -3/2 1 > 101.9550009
apotome [ -11 7 > 113.6850061
tone [ -3 2 > 203.9100017
4th [ 2 -1 > 498.0449991

Note that this definition of "diesis" is idiosyncratic
(however, it is also the earliest): it's exactly the
same as the pythagorean "limma". The "comma" is the
pythagorean-comma. The Philolaus "schisma" is defined
as 1/2 comma, and his "diaschisma" is 1/2 diesis.

But note that a logarithmic 1/2 of his diaschisma
is ~22.55 cents, quite close to the comma.

My webpage on Philolaus has all of this:
http://tonalsoft.com/enc/p/philolaus.aspx

But if you have or can get a copy of _Xenharmonikon_ 18,
i expanded this into a much fuller treatment, with a
very extensive bibliography. (Someday i plan to include
all of that in the webpage.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Paul Poletti" <paul@> wrote:
>
> > It's only contrary when one insists on the rigid interpretation that a
> > conceptual division of the tone into 9 parts means a 55-comma octave,
> > like you did.
>
> The conclusion follows from the right premises.

'Right' for what, and for whom?

Who can tell, historically, what were the 'right' premises?
We can only find out Tosi's premises by occult mind-reading - or by
trying to fit them into a preconceived idea of what he should have
been thinking, which is about equally reliable.

Actually, Gene, you need one more premise:

(0) The same interval is always to be tuned the same way, within some
margin of error much smaller than the small interval 'c'.

The argument falls down already here, since there is no reason for
this premise to be true in the absence of keyboard accompaniment.

> (1) There is only one size of tone
> (2) There are two sizes of semitone (call them diatonic and chromatic)
> which combine to give a tone
> (3) There is some small interval, call it c, such that the diatonic
> semitone is 5c and the chromatic is 4c.
>
> A tone is 5c+4c = 9c. Since a major third is two tones, it is 18c.
> Three major thirds plus a chromatic semitone is an octave, which is
> therefore 55c.

3 * 18 + 4 = 58. Some mistake, shurely.

I think Gene meant to say that three major thirds plus an enharmonic
diesis is an octave, and since (by assumption) that diesis is 1c, you
get 55.

Anyway, if even Gene doesn't get it right first time, what hope would
the ordinary singing student or (say) flute player have? Even if they
do get to the 'right' number 55, how could this be of any conceivable
practical help?

~~~T~~~

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Brad Lehman <bpl@umich.edu>

> > I think Gene meant to say that three major thirds plus an
> > enharmonic diesis is an octave, and since (by assumption) that
> > diesis is 1c, you get 55.
>
> I meant to say three major thirds plus the difference between a
> major and a minor semitone. I reached into my brain for the name,
> and a brain fart had me write down the difference between a major
> and minor third. Anyway, the next derivation is less direct but uses
> diatonic arguments, so it's better.

Another way to think of the same thing: an octave is built of three
major 3rds plus one error that renders an enharmonic name-change to
one note. For example, Bb-D-F#-A# plus the flip from A# to Bb to get
our octave. Since each tone in this system is built of 9 chunks, and
each major 3rd is two of those tones (i.e. 18 chunks), and the
enharmonic error is one chunk: 18 + 18 + 18 + 1 = 55.

Or if you prefer: that's six tones (9 each) plus one name change.
Bb-C-D-E-F#-G#-A# plus the A# to Bb. 9 + 9 + 9 + 9 + 9 + 9 + 1 = 55.

Or, using the hexachord system that Tosi said was the normal way of
thinking about scales: we've got Ut-Re-Mi-Fa-Sol; it mutates and that
Sol becomes a new Ut; Ut-Re-Mi-Fa and we've completed the octave
(C-D-E-F-G; G-A-B-C...which could also be thought of as two
Ut-Re-Mi-Fa tetrachords next to one another). Each of those two Mi-Fa
steps is the "large" semitone, as Tosi says, i.e. 5 chunks. And
tallying all this up we have two Ut-Re steps, two Re-Mi steps, and one
Fa-Sol step.

Tosi didn't say anything about two or three different sizes of tones
here (even if some theorists before and after him *did*, or would
throw some just-intonation premises into the mix here, about major and
minor tones). Tosi's only assertion about tones was that they get
split up in 5:4 proportion to make the major and minor semitones. So,
implicitly here (and I know someone here will scream to differ!),
those Ut-Re steps, Re-Mi steps, and the Fa-Sol step are all the same
size as one another *to Tosi*, namely 9 chunks. And 9 + 9 + 5 + 9 + 9
+ 9 + 5 = 55.

> > Anyway, if even Gene doesn't get it right first time, what hope
> > would the ordinary singing student or (say) flute player have?
> > Even if they do get to the 'right' number 55, how could this be of
> > any conceivable practical help?
>
> This question misses the point, which is that if you make some
> staightforward assumptions about what *Tosi*, not a flute player,
> thought, based on what he says, the conclusion of 55 follows.

The question not only misses the point (as you've correctly pointed
out), it's also merely argumentative. If even a thoroughly smart
string-theory physicist can't (or chooses not to) grasp a
straightforward New Grove article, asserting the 17th century
existence and widespread importance IN PRACTICE of meantone systems
gentler than 1/4 comma, and the 55-division illustrated by Antonio
Fernandez in 1626 (SIXTEEN-twenty-six!), or prefers to believe it's
only the province of keyboard-based wankers instead of other
musicians, ...what hope would the ordinary singing student or flute
player have?

Tiny Tim the Voice Student, God bless us every one, doesn't or needn't
give a rip about the octave-division chunks or counting them up in any
numerical way. He only needs to know what the dang things *sound*
like properly when he's practicing his Ut-Re-Mi-Fa-Sol-La as scales
and leaps, performing "in tune" according to whatever system(s)
happen(s) to be prevalent. If he hits the right pitches he's in tune.
If he judges them incorrectly and hits the cracks between where
they're supposed to be, he's out of tune. He also needs to keep
performing correctly whenever the music modulates or mutates to put Ut
and all its brethren onto some new pitch level. All those regular
systems of whatever size exist to help make this an easy process:
regularity of relative pitch placement, wherever we've located Ut for
a given passage. And those pitches all have their properly designated
spots whether there's a keyboard or other inflexible instrument
playing, or not.

The 55-chunk division, "How could this be of any conceivable practical
help?"!! It could help the student of voice or any melody instrument
to practice and perform evenly and correctly-proportioned hexachords
and their mutations, that's how!

(...)
E F# G# A B C#
A B C# D E F#
D E F# G A B
G A B C D E
C D E F G A
F G A Bb C D
Bb C D Eb F G
Eb F G Ab Bb C
(...)

The semitone in the middle of all those is always a Mi-Fa. It's
always going to be sized 5 chunks, melodically. And all those other
whole tones in there, whether at the moment they're serving as Ut-Re,
Re-Mi, Fa-Sol, or Sol-La, are implicitly interchangeable with one
another (we can start a scale anywhere...), namely a consistent size
of 9 chunks. They're not made up of 8, 10, or fractional almost-9
sizes; they're 9.

The student can start on a G and take a step to A, Ut-Re. Or, he can
start on F and take two steps Ut-Re-Mi to G and A. The G to A step is
going to be the same size whether he stuck an F in front of it or not.
The equality of *the* step, whether it's Ut-Re or Re-Mi, makes the
system *easy* and practical to learn. A student who sings all the way
from C-D-E-F-G-A-B-C-D-E-F-G doesn't have to worry about two
potentially different choices for A's pitch in there (egads, is it La
or Re, having made a shift to supply the forthcoming B?)...he simply
sings *the* pitch for A, which happens to be 41 chunks above C, the
same as he would do if leaping directly from C to A. Continuing, he
hits 50, 55, 64, 73, 78, 87, 96....

If he's singing "pure" and correct intonation according to the
standard of this system, he's hitting those marks: the proper mark,
and not fudging it to the spaces between the marks either. Whether he
gets there by steps or leaps, a correctly-spelled and
correctly-intoned major 6th is always going to be that same distance
of 41/55 of the octave, whether it comes up in music as Ut-La, Fa-Re,
or Sol-Mi. Learn what it sounds like, as a consistent distance, and
do it. A correctly-intoned 4th, whether it's Ut-Fa or Re-Sol or
Mi-La, is always going to be 23/55 of the octave. Learn by ear and
experience *the* size of that interval, use it consistently, and it
can be applied to all musical situations whether there are ornamental
passing notes within that leap or not.

(See also Ross Duffin's footnote on page 72 of his book, about
Mattheson's 1725 citation of the "pure" (rein) layout of Silbermann's
temperament...regular 1/6 comma....)

Now if anyone, whether Galliard or a modern writer, inserts the
premise of unequally-sized tones into Tosi's model, making the Ut-Re
be bigger than the Re-Mi...now we have to deal with not only two sizes
of semitone, but four or maybe six or eight. Chop up each of those
steps Ut-Re, Re-Mi, Fa-Sol, Sol-La from the hexachord system into 5:4
proportioned semitones. Is that was Tosi was really into, a
microtonality that fine, basing it on two to four *different* sizes of
tones? Despite the fact that he didn't say so?

Brad Lehman

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> First of all, it's *all* hearsay, because nothing
> actually *by* Philolaus exists.
Fully agreed.
>
> According to the last fragment in Boethius (book 3,
> chapter 8), we get the following:

online available under:
http://www.chmtl.indiana.edu/tml/6th-8th/BOEMUS3_TEXT.html

>
> (click the "Option | Use Fixed Width Font" links for
> viewing correctly on the stupid Yahoo web interface)
>
> -- 2,3-monzo --
> 2 3 ~cents
> schisma [ -19/2 6 > 11.73000519
It's difficult to find that modern
interpretation in the original:
There's no reason to derive that present-day view
from the old source-texts.
To impute fractional 'monzo's on Philolaos is alike:
"to break a fly on the wheel"

Doubtless:
Philoaos never intended to divide the PC into 2
logarithmically equal parts.
Nobody at that time was able to perform such advanced maths.

> diaschisma [ 4 -5/2 > 45.11249784
Here the term sqrt(3) appears even
more ahistorically than the above imputed sqrt(2).
Pythagoreans alike Philolaos avoided
http://en.wikipedia.org/wiki/Irrational_number
s as far as possible, also did Euclid in his
http://ota.ahds.ac.uk/textinfo/0241.html
'sectio canonis'

> integral 1/2-tone [ -3/2 1 > 101.9550009

> The Philolaus "schisma" is defined
> as 1/2 comma, and his "diaschisma" is 1/2 diesis.
Arithmetically or harmonically bisection split generally
any none-quadratic ratio in two different parts,
amounting different in seize,
but never into 2 equal halves.
cf:
Werckmeister, 'Mus. Temp.' FFM 1691, p.37
"..und lässet sich nicht in zwey gleiche PROPORTIONES theilen"
>
> But note that a logarithmic 1/2 of his diaschisma
> is ~22.55 cents, quite close to the comma.

Martin Vogel wrote in his
"Tonbeziehungen" p.204
about allegations on supposed
logarithmically divisions
antedated back wrongly into ancient times:
"...then lacked the basis to calculate logarithms."
in a proper way.

http://www.mathsisgoodforyou.com/topicsPages/concepts/logarithmsinvention.htm
http://www.thocp.net/reference/sciences/mathematics/logarithm_hist.htm
http://www.micheloud.com/FXM/LOG/

Vogel quotes
Ingemar Düring, editor of Ptolemaios:

"Jede Rede von temperierter Stimmung in der antiken Musik
ist verfehlt, streitet sogar gegen ihr Wesen und Geist"

'Any idle rigmarole about tempering in ancient
musics misses the point, argues even against its
character and spirit"

Lit. reference:
I. Düring
Ptolemaios and Porphyros über die Musik
in: Göteborgs Högskolas Arsskrift 40, 1934, 258
footnote 1: p.204 (Vogel)
Similar concluded also:
Barbour, Marnold, Schlesinger, Schönfelder,
Torr & Winnigton-Ingram too.

hence the
>... webpage on Philolaus has all of this:
Inept modern claims on logarithmically equal divisions:
> http://tonalsoft.com/enc/p/philolaus.aspx
and needs therefore some update,
at least in discerning more clearly
P's ancient original ratios
from todays interpretations
obtained by later developed methods,
that were still unknown then at P's life-time.

sorry
for objecting such untenable attribution errors

A.S.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
> if you make some
> staightforward assumptions about what *Tosi*, not a flute player,
> thought, based on what he says, the conclusion of 55 follows.
>

Exactly. Given the kind of tuning and the kind of theory that Gene is
interested in, certain assumptions *appear* to be straightforward. If
one accepts the premise that Tosi's preoccupations were similar to
Gene's, no further objection can be raised.

But if you make other, equally straightforward assumptions about how
Tosi thought and how he proceeded in writing his book, the conclusions
you can come to are quite different.

Now since Tosi doesn't tell us what assumptions he himself had (how
could he, since the assumptions were either unconscious or
unarticulated...) how can we proceed?

The main skill to being a reasonably good historian is to realise that
the assumptions that we make today, and the points which we find more
or less important, may not be anything like the same as for people
made 250 years ago. For example the unspoken definition of 'interval'
that Gene confessed to applying. For him, the purpose of speaking
about 'intervals' is to give them a definite size and/or place within
a regular mathematical structure.

For Tosi, the idea of 'interval' could be quite different: for
example, it is what occurs between singing two different notes. Then,
for example, the correct tuning of an interval (meaning one particular
notated interval in its musical context) could mean that the resulting
melodic line sounds acceptable in sung performance. That means that
the same notated interval, in different contexts, could be correctly
tuned in several, possibly widely divergent, ways. We could experiment
ourselves with this definition, and see how wide the deviations are
that we can happily accept for a soprano with string accompaniment -
and ask ourselves where this leaves Gene's definition of an interval.

How can we tell whether or not Tosi thought it was desirable that
every tone should be the same size? Did he give any conscious
consideration to the question? Or did he only really consider the
question 'does this soprano sound in tune when singing his or her
scales and arias' - which can be answered by someone who has no
theoretical opinion whatsoever about the size of tones?

~~~T~~~
(By the way, it is *not* necessary or helpful to posit that people had
different ears as well as different assumptions in the past. A pure
third is and always will be a pure third...)

🔗Mark Rankin <markrankin95511@yahoo.com>

Monz,

When I try to send you email at <monz@tonalsoft.com>
I'm getting a Failure (to deliver) Notice from Yahoo.

Mark

--- monz <monz@tonalsoft.com> wrote:

> Hi Gene,
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@>
> wrote:
> >
> > > Philolaus's small divisions have the whole-tone
> of
> > > 9:8 ratio divided into 4 "diaschismata" and 1
> "comma",
> > > and the diaschisma is approximately twice the
> size
> > > of the comma, thus the 9-fold division.
> >
> > Is any of that made precise?
>
>
> Not really.
>
> First of all, it's *all* hearsay, because nothing
> actually *by* Philolaus exists. What we have are one
> fragment quoted by Nicomachus and two other
> fragments
> quoted by Boethius.
>
> According to the last fragment in Boethius (book 3,
> chapter 8), we get the following:
>
> (click the "Option | Use Fixed Width Font" links for
> viewing correctly on the stupid Yahoo web interface)
>
> -- 2,3-monzo --
> 2 3 ~cents
> schisma [ -19/2 6 > 11.73000519
> comma [ -19 12 > 23.46001038
> diaschisma [ 4 -5/2 > 45.11249784
> diesis [ 8 -5 > 90.22499567
> integral 1/2-tone [ -3/2 1 > 101.9550009
> apotome [ -11 7 > 113.6850061
> tone [ -3 2 > 203.9100017
> 4th [ 2 -1 > 498.0449991
>
> Note that this definition of "diesis" is
> idiosyncratic
> (however, it is also the earliest): it's exactly the
> same as the pythagorean "limma". The "comma" is the
> pythagorean-comma. The Philolaus "schisma" is
> defined
> as 1/2 comma, and his "diaschisma" is 1/2 diesis.
>
> But note that a logarithmic 1/2 of his diaschisma
> is ~22.55 cents, quite close to the comma.
>
> My webpage on Philolaus has all of this:
> http://tonalsoft.com/enc/p/philolaus.aspx
>
> But if you have or can get a copy of _Xenharmonikon_
> 18,
> i expanded this into a much fuller treatment, with a
> very extensive bibliography. (Someday i plan to
> include
> all of that in the webpage.)
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>
>

__________________________________________________
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🔗monz <monz@tonalsoft.com>

Hi Andreas,

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > First of all, it's *all* hearsay, because nothing
> > actually *by* Philolaus exists.
> Fully agreed.
> >
> > According to the last fragment in Boethius (book 3,
> > chapter 8), we get the following:
>
> online available under:
> http://www.chmtl.indiana.edu/tml/6th-8th/BOEMUS3_TEXT.html

I have already included the Latin text of Boethius's
quotations of Philolaus on my webpage, without all
the other commentary by Boethius. Bower's English
translation appears alongside.

> hence the
> >... webpage on Philolaus has all of this:
> Inept modern claims on logarithmically equal divisions:
> > http://tonalsoft.com/enc/p/philolaus.aspx
> and needs therefore some update,
> at least in discerning more clearly
> P's ancient original ratios
> from todays interpretations
> obtained by later developed methods,
> that were still unknown then at P's life-time.
>
> sorry
> for objecting such untenable attribution errors

As i said, a much fuller treatment of my work on
Philolaus was published in _Xenharmonikon_ 18,
and i fully intend to eventually incorporate all
of that additional material into my webpage.

One of the things in the published paper is an
examination of an *arithmetic* division of the
whole-tone, as would be done on a monochord.
This results in some other values for Philolaus's
small intervals.

-monz
http://tonalsoft.com
Tonescape microtonal music software
joemonz(AT)yahoo.com