historical precedence of 55-EDO (was: retuning midi files)

Hi Bob,

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, October 08, 2001 1:23 PM
> Subject: [tuning] Re: retuning midi files
>
>
> Hi! You're welcome, Robert! Yes, 1/4-comma was the first meantone
> from which all its variants later were derived and began appearing in
> the early renaissance. 1/6-comma appeared long before Mozart, though,
> and along with all the other variants prevailed through the baroque
> and 1/4-comma even well into the 19th century in some media
> (especially organ).
>
> As to the n-tET equivalents of the meantone variants, most of the
> them (exceptino: the 31-tET of Vicento), are modern substitutes that
> have the advantage of being closed cycle temperaments with no falling
> off the end of the earth at any point. So modulation is free, but
> will drift vis-a-vis 12-tET.

This is not entirely correct. 55-EDO takes conceptual precedence
over 1/6-comma meantone, and thus is a second exception to your
generalization (along with 31-EDO).

While 55-EDO is audible indistinguishable from 1/6-comma meantone
(and perhaps even from 1/5-comma as well), *conceptually*, 55-EDO
was the mathematical measurement used by tons of European musicians
during the Baroque and Classical periods.

They recognized that the "mean tone" was composed of two different
types of semitones, the diatonic and chromatic, and thought of the
55-EDO "comma" as the basic interval measurements, with other basic
scalar intervals as follows:

whole-tone = 9 commas
diatonic semitone = 5 commas
chromatic semitone = 4 commas

Assuming the age-old Pythagorean dictum, and its Baroque/Classical
meantone transmogrification, that the "octave" is divided as a scale
into not 6 whole-tones but rather 5 whole-tones and 2 diatonic semitones,
the total number of commas is (5 * 9) + (2 * 5) = 45 + 10 = 55.

(By "transmogrification", what I'm referring to is that the relative
sizes of the chromatic and diatonic semitones are reversed in Pythagorean
and meantone.)

This simple way of understanding the subtleties of meantone tuning
was well-established for a least a couple of centuries, far more
so than any understanding of fraction-of-a-comma deviations from
JI or Pythagorean.

I believe there's a reference to this somewhere in Boethius's
book (written c. 505 AD)... will have to search for it. It's
possible that I may be misremembering this... perhaps a later
medieval (Frankish?) author cited Boethius's explanation of the
(9/8)^5 * (256/243)^2 division in his own explanation of 55-EDO.

love / peace / harmony ...

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

_________________________________________________________
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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Hi Bob,
>
>
> > From: <BobWendell@t...>
> > To: <tuning@y...>
> > Sent: Monday, October 08, 2001 1:23 PM
> > Subject: [tuning] Re: retuning midi files
> >
> >
> > Hi! You're welcome, Robert! Yes, 1/4-comma was the first meantone
> > from which all its variants later were derived and began
appearing in
> > the early renaissance. 1/6-comma appeared long before Mozart,
though,
> > and along with all the other variants prevailed through the
baroque
> > and 1/4-comma even well into the 19th century in some media
> > (especially organ).
> >
> > As to the n-tET equivalents of the meantone variants, most of the
> > them (exceptino: the 31-tET of Vicento), are modern substitutes
that
> > have the advantage of being closed cycle temperaments with no
falling
> > off the end of the earth at any point. So modulation is free, but
> > will drift vis-a-vis 12-tET.
>
>
> This is not entirely correct. 55-EDO takes conceptual precedence
> over 1/6-comma meantone, and thus is a second exception to your
> generalization (along with 31-EDO).

Wouldn't 43-tET be an exception as well? And 19-tET, certainly in
Costeley's work in the Renaissance?
>
> While 55-EDO is audible indistinguishable from 1/6-comma meantone
> (and perhaps even from 1/5-comma as well), *conceptually*, 55-EDO
> was the mathematical measurement used by tons of European musicians
> during the Baroque and Classical periods.
>
> They recognized that the "mean tone" was composed of two different
> types of semitones, the diatonic and chromatic, and thought of the
> 55-EDO "comma" as the basic interval measurements, with other basic
> scalar intervals as follows:
>
> whole-tone = 9 commas
> diatonic semitone = 5 commas
> chromatic semitone = 4 commas
>
> Assuming the age-old Pythagorean dictum, and its Baroque/Classical
> meantone transmogrification, that the "octave" is divided as a scale
> into not 6 whole-tones but rather 5 whole-tones and 2 diatonic
semitones,
> the total number of commas is (5 * 9) + (2 * 5) = 45 + 10 = 55.
>
> (By "transmogrification", what I'm referring to is that the relative
> sizes of the chromatic and diatonic semitones are reversed in
Pythagorean
> and meantone.)

I believe you're exactly correct, Monz, and I brought up
this "transmogrification" a few months ago in a discussion with Margo
Schulter.
>
> This simple way of understanding the subtleties of meantone tuning
> was well-established for a least a couple of centuries, far more
> so than any understanding of fraction-of-a-comma deviations from
> JI or Pythagorean.

A couple of centuries? Wasn't it simply in vogue in the 18th century?

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I believe there's a reference to this somewhere in Boethius's
> book (written c. 505 AD)... will have to search for it. It's
> possible that I may be misremembering this... perhaps a later
> medieval (Frankish?) author cited Boethius's explanation of the
> (9/8)^5 * (256/243)^2 division in his own explanation of 55-EDO.

You must mean 53-tET here? 55-tET would not have fit this
description, nor would it (given its meantone nature) be anything a
medieval author would be likely to understand or care about. If
you're right, Monz, this would be a total revolution in meantone
historiography . . . which is why I think you're mistaken.

I wrote:

> Wouldn't 43-tET be an exception as well? And 19-tET, certainly in
> Costeley's work in the Renaissance?

To answer my own question:

According to your meantone page, Monz,

http://www.ixpres.com/interval/dict/meantone.htm

43-tET and 1/5-comma meantone were first conceived simultaneously by
Sauveur in 1701.

But 19-tET, put forth in 1558 by Costeley, appears to have precedence
over 1/3-comma meantone, put forth in 1577 by Salinas (on a 19-tone
keyboard, incidentally . . .)

Margo or someone else more versed in the history may be able to give
us more information than a mere table . . .

Hi, Monz! Good to have you back! Been missing you. I knew I was
taking a slight risk with this statement, but I suspect the 55-
EDO was conceptual or arithmetical and not really a practically
implemented scale in the current sense. I was referring to actual use
of the scale.

I know Vicento physically implemented his 31-EDO scale and the comma
displacement of it, but did anyone actually do that with 55-EDO? I
was referring to its actual implementation as a scale and not to its
conceptual existence. I know the Greeks knew about 53-EDO, but did
they really implemement that? Doubtful, I think, but maybe you can
tell us.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Hi Bob,
>
>
> > From: <BobWendell@t...>
> > To: <tuning@y...>
> > Sent: Monday, October 08, 2001 1:23 PM
> > Subject: [tuning] Re: retuning midi files
> >
> >
> > Hi! You're welcome, Robert! Yes, 1/4-comma was the first meantone
> > from which all its variants later were derived and began
appearing in
> > the early renaissance. 1/6-comma appeared long before Mozart,
though,
> > and along with all the other variants prevailed through the
baroque
> > and 1/4-comma even well into the 19th century in some media
> > (especially organ).
> >
> > As to the n-tET equivalents of the meantone variants, most of the
> > them (exceptino: the 31-tET of Vicento), are modern substitutes
that
> > have the advantage of being closed cycle temperaments with no
falling
> > off the end of the earth at any point. So modulation is free, but
> > will drift vis-a-vis 12-tET.
>
>
> This is not entirely correct. 55-EDO takes conceptual precedence
> over 1/6-comma meantone, and thus is a second exception to your
> generalization (along with 31-EDO).
>
> While 55-EDO is audible indistinguishable from 1/6-comma meantone
> (and perhaps even from 1/5-comma as well), *conceptually*, 55-EDO
> was the mathematical measurement used by tons of European musicians
> during the Baroque and Classical periods.
>
> They recognized that the "mean tone" was composed of two different
> types of semitones, the diatonic and chromatic, and thought of the
> 55-EDO "comma" as the basic interval measurements, with other basic
> scalar intervals as follows:
>
> whole-tone = 9 commas
> diatonic semitone = 5 commas
> chromatic semitone = 4 commas
>
> Assuming the age-old Pythagorean dictum, and its Baroque/Classical
> meantone transmogrification, that the "octave" is divided as a scale
> into not 6 whole-tones but rather 5 whole-tones and 2 diatonic
semitones,
> the total number of commas is (5 * 9) + (2 * 5) = 45 + 10 = 55.
>
> (By "transmogrification", what I'm referring to is that the relative
> sizes of the chromatic and diatonic semitones are reversed in
Pythagorean
> and meantone.)
>
>
> This simple way of understanding the subtleties of meantone tuning
> was well-established for a least a couple of centuries, far more
> so than any understanding of fraction-of-a-comma deviations from
> JI or Pythagorean.
>
> I believe there's a reference to this somewhere in Boethius's
> book (written c. 505 AD)... will have to search for it. It's
> possible that I may be misremembering this... perhaps a later
> medieval (Frankish?) author cited Boethius's explanation of the
> (9/8)^5 * (256/243)^2 division in his own explanation of 55-EDO.
>
>
>
> love / peace / harmony ...
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., BobWendell@t... wrote:

> I know Vicento physically implemented his 31-EDO scale and the
comma
> displacement of it, but did anyone actually do that with 55-EDO? I
> was referring to its actual implementation as a scale and not to
its
> conceptual existence.

As for actual implementation of the _entire_ ET, not just a subset of
it, this:

http://www.ixpres.com/interval/dict/eqtemp.htm

seems to indicate that historically, not only Vicentino, but later,
in the early 17th century, also Gonzaga, Stella, and Scipione did it
for 31-tET . . . but no one did it for 55-tET, and if Salinas did it
for 19-tET, it may have been only by accident!

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, October 08, 2001 5:29 PM
> Subject: [tuning] Re: historical precedence of 55-EDO (was: retuning midi
files)
>
>
> >
> > This simple way of understanding the subtleties of meantone tuning
> > [the 55-EDO conceptualization] was well-established for a least
> > a couple of centuries, far more so than any understanding of
> > fraction-of-a-comma deviations from JI or Pythagorean.
>
> A couple of centuries? Wasn't it simply in vogue in the 18th century?

Hmmm... I was thinking about Telemann's advocacy of 55-EDO,
and he was born in the 17th century, so I suppose that clouded my
reasoning. Of course his composing career

I think you're right.

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Oops... my bad... this one got away from me.

I'm looking into Boethius and writing a detailed response
to several posts in this thread. Ignore this incomplete post,
except for my agreement with Paul.

-monz

----- Original Message -----
From: monz <joemonz@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Tuesday, October 09, 2001 10:26 AM
Subject: Re: [tuning] Re: historical precedence of 55-EDO (was: retuning
midi files)

>
> > From: Paul Erlich <paul@stretch-music.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Monday, October 08, 2001 5:29 PM
> > Subject: [tuning] Re: historical precedence of 55-EDO (was: retuning
midi
> files)
> >
> >
> > >
> > > This simple way of understanding the subtleties of meantone tuning
> > > [the 55-EDO conceptualization] was well-established for a least
> > > a couple of centuries, far more so than any understanding of
> > > fraction-of-a-comma deviations from JI or Pythagorean.
> >
> > A couple of centuries? Wasn't it simply in vogue in the 18th century?
>
>
> Hmmm... I was thinking about Telemann's advocacy of 55-EDO,
> and he was born in the 17th century, so I suppose that clouded my
> reasoning. Of course his composing career
>
>
> I think you're right.

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For the Mteenth time, there is no evidence that Telemann supported or even
mentioned 55-tET. Check the archives for more detail on this. Paul, how
could you?

Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:

> For the Mteenth time, there is no evidence that Telemann supported
or even
> mentioned 55-tET. Check the archives for more detail on this.
Paul, how
> could you?
>
> Johnny Reinhard

Sorge, Johnny. The reference is Sorge. I don't know if Sorge was
wrong or right about Telemann. Is there something in the archives
about the Sorge reference?

But certainly it fits well historically, given Tosi, Quantz, L.
Mozart, etc.

Sorge. Don't you remember, Paul, looking at the notation by Telemann, written on his deathbed? The notes went past the octave and counted up to 55. This was the wrong notation! The actual different notes is much less. And there is nothing about equal divisions.

I'm not aware that Sorge knew Telemann. Did he? Did you?

Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:

> Sorge. Don't you remember, Paul, looking at the notation by
Telemann, written on his deathbed?

Yes, I remember. I didn't know about the deathbed part. Can you
translate as much of the accompanying text as possible?

> The notes went past the octave

Yes.

> and counted up to 55.

No, it was 56:

Cbbb Dbbb Ebbb Fbbb Gbbb Abbb Bbbb Cbbb
Cbb Dbb Ebb Fbb Gbb Abb Bbb Cbb
Cb Db Eb Fb Gb Ab Bb Cb
C D E F G A B C
C# D# E# F# G# A# B# C#
C## D## E## F## G## A## B## C##
C### D### E### F### G### A### B### C###

So there are only 49 different notations here.

>This was the wrong notation!

The wrong notation for what?

> The actual different notes is much less.

Did Telemann specify the equivalencies? Or do you mean the actual
number of notes found in Telemann's compositions is less?

> And there is nothing about equal divisions.

Is Sorge specifically referring to this one passage?

> I'm not aware that Sorge knew Telemann. Did he?

I don't know, but shouldn't we look at what Sorge actually wrote,
rather than letting it all ride on a tidbit from Telemann which
happened to have 56 non-distinct note names, and the coincidental
nearness of 55 to 56?

> Did you?

Nope!

Anyway, for anyone who's confused, 55-tET is virtually identical to a
favorite tuning of the Baroque and Classical eras, 1/6-comma meantone
(it's shaded a bit toward 1/5-comma). In that era, musical notation
was understood to refer to a meantone system (which was not called
meantone at the time, but simply "correct intonation").

Clearly an entire gamut of 55 equal steps per octave is of no
practical relevance for music of the time. But it was a convenient
method of reckoning, uses by many string and wind teachers in the
18th century. The whole tone was reckoned as 9 steps, the diatonic
semitone 5 steps, and the chromatic semitone 4 steps. Observing these
specifications with some approximate accuracy would ensure "correct
intonation" for the music of the time (though Beethoven and even a
few Mozart passages with enharmonic modulation become problematic).

The 9, 5, and 4-step intervals would yield the usual notated notes
with the following spacing:

C

C#
Db

D

D#
Eb

E
Fb

E#
F

F#
Gb

G

G#
Ab

A

A#
Bb

B
Cb

B#
C

Filling in the gaps by using the same rules, extended to double
sharps and flats, yields:

C
Dbb

B##
C#
Db

C##
D
Ebb

D#
Eb
Fbb

D##
E
Fb

E#
F
Gbb

E##
F#
Gb

F##
G
Abb

G#
Ab

G##
A
Bbb

A#
Bb
Cbb

A##
B
Cb

B#
C

Adding triple sharps and triple flats:

C
Dbb

B##
C#
Db
Ebbb
B###
C##
D
Ebb
Fbbb
C###
D#
Eb
Fbb

D##
E
Fb
Gbbb
D###
E#
F
Gbb

E##
F#
Gb
Abbb
E###
F##
G
Abb

F###
G#
Ab
Bbbb

G##
A
Bbb
Cbbb
G###
A#
Bb
Cbb

A##
B
Cb
Dbbb
A###
B#
C

This is how Telemann's 49 notated notes would lie according to the "9
parts per tone, 5 parts per diatonic semitone" specification. Notice
that there are 6 "holes", which brings the total number of divisions
to 55. I'm not saying that Telemann's deathbed inscription in any way
proves that he was thinking 55 -- only that there's isn't yet any
sort of contradiction in my mind against Sorge's assertion.