Yahoo Tuning Groups Ultimate Backup tuning 55edo in 1700s (was: Mozart 31-equal??)

hi Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 6:51 PM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > in this particular Mozart case, Tosi described
> > tone = 9 commas = diatonic + chromatic semitone,
> > diatonic semitone = 5 commas, and
> > chromatic semitone = 4 commas; and Tosi was cited
> > by Leopold Mozart, in a letter to Wolfgang, as an
> > "authoritative source" on the matter of intonation.
>
> This does give three equations in three unknowns leading to
> 55-equal; did Tosi know this was a consequence?
>
> meantone equation: -4a + 4b - c = 0
> diatonic semitone equation: 4a - b - c = 5
> chromatic semitone equation: -3a - b + 2c = 4
>
> Solving this for a, b, and c leads to
> a = 55, b = 87, c = 128, which is 55-equal.

i haven't read Tosi, so i don't know how aware
he was of all the algebraic equivalences.

however, i *can* tell you that 1/6-comma meantone
was quite popular during the 1700s. it was:

- noted by Sauveur around 1700 as "the temperament
most favored by musicians in general" (as opposed to
keyboardists in particular),

- advocated by Tosi in 1723

- advocated by Telemann in his old age in 1742-3,
and thereafter by Sorge.

- advocated by Quantz in 1752 in his flute method
(Quantz's description is identical to Tosi's)

- described by L. Mozart in 1756 in his violin method
(the year his son W.A. was born), and advocated by
him in 1778 in a letter to W.A.

- described by Tï¿½rk in 1791 (the year W.A. died)

Manuel's page about Telemann and 55edo is here:
http://www.xs4all.nl/~huygensf/doc/telemann.html

it was right after Mozart's death that the "expressive
intonation" resembling Pythagorean began to supplant
meantone in orchestral playing, and not long after
that Beethoven began treating his pitch resources
as in 12edo.

except in England, where meantone persisted until
about 1850, this was pretty much the end of meantone
as a tuning paradigm.

(... except for the fact that Mahler was longing for it
a century later.)

it seems to me that the historical popularity of
the various different meantones sort of followed
a narrowing of the tempering of the "5th".

1/4-comma was of course the earliest and longest-lasting,
and i think that's for obvious reasons (if 4 "5ths"
are supposed to equal the "major-3rd" a comma below,
then you narrow each "5th" by 1/4-comma ... duh).

but other early descriptions of meantones are for
1/3- and 2/7-comma, which are on the "wide" end of
the tempering spectrum, then 1/5-comma became popular
in the later 1600s and 1/6-comma in the 1700s, until
finally 1/11-comma (i.e., 12edo) in the 1800s.

yes, i know that's an oversimplification ... but i
think there's some validity in seeing that trend.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i haven't read Tosi, so i don't know how aware
> he was of all the algebraic equivalences.

There are other ways of looking at it, which might have been easier.
We can regard the exteme ranges of meantone as being defined by 5-
equal, in which the chromatic semitone is one step and the diatonic
semitone disappears, and 7-equal, where the chromatic semitone
disappears and the diatonic semitone is a step. Any meantone may be
regarded as a combination, using two relatively prime postive
integers, of 5 and 7; which immediately tells us how many steps there
are in the diatonic and chromatic semitones. While no one would have
looked at the matter in quite this way back then, they could have
understood in something like an equivalent fashion.

12 = 5 + 7, the semitones are the same
19 = 5 + 2*7, the diatonic semitone is twice the chromatic semitone
26 = 5 + 3*7, the diatonic semitone is three times the chromatic
semitone
31 = 2*5 + 3*7, we have a 2:3 ratio
43 = 3*5 + 4*7, a 3:4 ratio
50 = 3*5 + 5*7, a 3:5 ratio
55 = 4*5 + 5*7, a 4:5 ratio

🔗monz <monz@attglobal.net>

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 16, 2003 12:35 AM
> Subject: [tuning] Re: 55edo in 1700s (was: Mozart 31-equal??)
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i haven't read Tosi, so i don't know how aware
> > he was of all the algebraic equivalences.
>
> There are other ways of looking at it, which might have been easier.
> We can regard the exteme ranges of meantone as being defined by 5-
> equal, in which the chromatic semitone is one step and the diatonic
> semitone disappears, and 7-equal, where the chromatic semitone
> disappears and the diatonic semitone is a step. Any meantone may be
> regarded as a combination, using two relatively prime postive
> integers, of 5 and 7; which immediately tells us how many steps there
> are in the diatonic and chromatic semitones. While no one would have
> looked at the matter in quite this way back then, they could have
> understood in something like an equivalent fashion.
>
> 12 = 5 + 7, the semitones are the same
> 19 = 5 + 2*7, the diatonic semitone is twice the chromatic semitone
> 26 = 5 + 3*7, the diatonic semitone is three times the chromatic
> semitone
> 31 = 2*5 + 3*7, we have a 2:3 ratio
> 43 = 3*5 + 4*7, a 3:4 ratio
> 50 = 3*5 + 5*7, a 3:5 ratio
> 55 = 4*5 + 5*7, a 4:5 ratio

just so that no-one becomes confused, i thought
it should pointed out that Gene is not using
"ratio" in its usual tuning sense here. he's
referring to a comparison of *logarithmic* interval
sizes.

-monz