History again - first mention of 55-division?

In 1707, and possibly a few years before, Sauveur mentioned 55-EDO as
the system used by 'ordinary musicians'. I am trying to interpret what
this remark means and whether it can really tell us anything about the
tunings used at that period.

My suspicion is that it is a symptom of the persistence of the old
theory that whole tones were made up of 9 commas. I think it is well
established that the 53-division corresponding closely to Pythagorean
intonation, in which the tone is also 9 units, was known for a long
time. Aaron for example mentioned that the tone was divided into four
dieses and a comma, with one semitone getting two dieses and the other
the rest, the comma also being one ninth of a tone.

Now this theory is of course not appropriate for meantone tunings.
However, the division of the tone into a smaller and a larger semitone
persisted from the Middle Ages into the meantone era, thus anyone who
carried on teaching the old theory of 4+5 division could not have been
proved obviously (or audibly) wrong unless a test was made on a very
precisely marked monochord. Even then the correct division of a tone
into 9 parts would be beyond the geometrical methods of the day. The
use of logarithms in the 17th century would make it possible to find
such divisions very precisely, but I think virtually no ordinary
musicians would do so (which would also require considerable work in
marking out the intervals very precisly on a monochord).

The big difference is the diatonic semitones (E-F and B-C) which were
small in Pythagorean and large in meantone. But it may easily be that
the theoretical 9-comma division survived this shift unscathed. When
Telemann 'invented' - actually reinvented - 55-EDO he made the
assumption that these diatonic semitones were the larger size. Sauveur
based his EDO systems on having 7 larger and 5 smaller semitones in
the octave - also a correct assumption for the meantone era.

Starting from the old (and at that time quantitatively untestable)
theory of 9 commas, taking some music textbooks or theoreticians at
their word rather than trying to actually measure what musicians did,
Sauveur would easily end up with 55 comma-like divisions in the octave
- which neatly enough approximated syntonic commas.

So Sauveur's claim that 55-EDO was used by 'ordinary musicians' would
be comprehensible even if no musicians actually used that system to
tune or play in. Sauveur's obituary pointed out that he relied on the
ears and knowledge of others, since before starting his investigations
he had known nothing of music. If people were being taught that the
semitones were 4 and 5 commas Sauveur might well have noted it down
and believed it true of all ordinary musicians.

If in fact quarter-comma (for example) had been the norm then still it
would have been difficult to disprove the 9-comma theory in its 55-EDO
guise - *until* Sauveur had actually compared 55-EDO with the pure
intervals and deduced what degree of temperament thirds and fifths
actually had. This would require the use of logarithms to divide the
octave into 55 parts - which I think is very unlikely anyone did
before Sauveur.

In fact in 1707 Sauveur explicitly noted the 31-EDO of Huyghens as
being close to quarter-comma - which he called 'the tempered system
which everyone uses'!

What would be very interesting is if anyone before Sauveur referred to
a 55-division, or said explicitly that the tone was divided into 4
plus 5 parts and the diatonic semitone was 5 parts?

What Sauveur said in 1701 would be good to know too, though I'm
waiting for that article on order.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> In 1707, and possibly a few years before, Sauveur mentioned 55-EDO
as
> the system used by 'ordinary musicians'. I am trying to interpret
what
> this remark means and whether it can really tell us anything about
the
> tunings used at that period.
>
> My suspicion is that it is a symptom of the persistence of the old
> theory that whole tones were made up of 9 commas.

From "Temperament," in A supplement to Mr. Chambers's cyclopædia
(1753),
"The last _Temperature_ we have mentioned is that of 55 parts, which
Mr. Sauveur calls the _Temperature_ of practical musicians. Its
foundation lies in assuming the proporition of the semi-tones, as 5 to
4, so the tone will be 9..."

http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?
type=turn&entity=HistSciTech001000270619&isize=M&q1=temperament

Clark

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_