[tuning] Re: New Telemann page - audio files

I've made mp3 files of two musical examples from Telemann's
article, and uploaded them to the Files section.

The first is the two-chord illustration of the comma
from page 270,
/tuning/files/monz/telemann/chordprog.mp3

Eb --> D#
A B
F# F#
C# B

where Eb is a comma higher than D#.

The second is the long (12-measure) 4-voice example
from page 271.
/tuning/files/monz/telemann/musicex55.mp3

This starts in A-minor, modulates to C#-minor, abruptly
jumps back to A-minor, modulates to G#-minor, then ends
suddenly again in A-minor.

There's a typo in the example: at the beginning of the
second line (measures 5 and 6), the alto voice has a "G"
but it should be "G#".

Also, at the end of that line, the top staff in measure 8
is very ambiguous. The soprano and alto have half-notes
at the beginning, then there are an "A#" and "F#" written
as half-notes but spaced like quarter-notes. I have
interpreted these as coming together on the 3rd beat in
both parts. The tie at the end of measure 8 clearly belongs
to the "A#" in the soprano.

Additionally, the "F#" at the end of measure 8 really looks
like it should be "Fx", but I concluded that it should be "F#"
in measure 8 and then "Fx" at the beginning of measure 9,
since the double-sharp sign is written at measure 9 and
the figured bass at that point also indicates "4++", a
doubly-augmented "4th" above "C", which was not indicated
in measure 8. I had originally tried bringing the "Fx" in
at measure 8 but it sounded terrible.

Another possibility concerning the rhythm here is that
perhaps the alto is supposed to have a dotted half-note
on "G" and then a quarter-note on "F#".

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

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, November 11, 2000 12:12 PM
> Subject: [tuning] Re: New Telemann page - audio files
>
>
> I've made mp3 files of two musical examples from Telemann's
> article, and uploaded them to the Files section.
>
> /tuning/files/monz/telemann/chordprog.mp3
> /tuning/files/monz/telemann/musicex55.mp3

I should have made it clear in that post that these
are tuned in 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..., "monz" <joemonz@y...> wrote:
> I've made mp3 files of two musical examples from Telemann's
> article, and uploaded them to the Files section.
>
>
> The first is the two-chord illustration of the comma
> from page 270,
>
/tuning/files/monz/telemann/chordprog.mp3
>
> Eb --> D#
> A B
> F# F#
> C# B
>
> where Eb is a comma higher than D#.

That first chord, classically speaking, is an F# minor chord with a
diminished seventh. I've never heard of a minor chord with a
diminished seventh being used in classical music -- a minor chord
with a major sixth would be common but be notated with a D# on top.
But in 31-tET or 1/4-comma meantone, this chord would approximate
extremely closely the string lengths 4:5:6:7. Perhaps Telemann knew
this??? Anyone else have any thoughts about where Telemann would have
pulled this chord from?

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 12, 2001 5:50 PM
> Subject: [tuning] Re: New Telemann page - audio files
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've made mp3 files of two musical examples from Telemann's
> > article, and uploaded them to the Files section.
> >
> >
> > The first is the two-chord illustration of the comma
> > from page 270,
> >
> /tuning/files/monz/telemann/chordprog.mp3
> >
> > Eb --> D#
> > A B
> > F# F#
> > C# B
> >
> > where Eb is a comma higher than D#.
>
> That first chord, classically speaking, is an F# minor chord with a
> diminished seventh. I've never heard of a minor chord with a
> diminished seventh being used in classical music -- a minor chord
> with a major sixth would be common but be notated with a D# on top.
> But in 31-tET or 1/4-comma meantone, this chord would approximate
> extremely closely the string lengths 4:5:6:7. Perhaps Telemann knew
> this??? Anyone else have any thoughts about where Telemann would have
> pulled this chord from?

Yes, isn't that a strange chord for this time period?

But he definitely didn't have 1/4-comma meantone in mind,
because the comma- and interval-sizes of 1/4-comma don't
match his descriptions.

The best low-integer approximation to the proportions of
the four 55-EDO pitches which I used in the mp3 is
6:16:19:27 -- and this approximation would be an even
better fit for 1/6-comma tuning.

The 55-EDO tuning uses 2^(x/55), x = 4, 27, 41, 14.
In either cycle, the generators are 7, 6, 3, -3.

So going with this low-integer interpretation, that's:

- an 8:3 "11th" (= 4:3 plus "8ve") on the bottom (C#:F#),
- a 19:16 "minor 3rd" in the middle (F#:A), and
- a 27:19 "diminished 5th" on top (A:Eb).

Of course an argument can be made for many other rational
approximations, depending on what level of accuracy is considered.
Here are extremely close and pretty close examples of
rational approximations to the 55-EDO tuning, which are
both much better than the smaller numbers given above:

612:229 or 155:58 "11th" C#:F#
915:767 or 68:57 "minor 3rd" F#:A
713:501 or 37:26 "diminished 5th" A:Eb

As for 1/6-comma meantone tuning...

Two close rational approximations of the F#:A "minor 3rd"
in 1/6-comma are 353:296 and 161:135.

And an excellent rational approximation of the C#:F# "11th"
in 1/6-comma is 2095:784, a slightly less better one is 644:241.

But here's something very interesting:

If tuned precisely in 1/6-comma meantone, the "diminished 5th"
A:Eb happens to be exactly the 64:45 ratio, a typical 5-limit JI
"diminished 5th" spelled with exactly the same letters A and Eb.

Here's an explanation of that using prime-factor vector addition:
(our reference, generator 0, is C)

2^-4 * 3^4 * 5^-1 = 81/80 = syntonic comma

2^(-4/6) * 3^(4/6) * 5^(-1/6) = (81/80)^(1/6)
= 2^(-2/3) * 3^(2/3) * 5^(-1/6) = 1/6 of a syntonic comma

2^-1 * 3^1 * 5^0 = 3/2
- 2^(-2/3) * 3^(2/3) * 5^(-1/6) = 1/6-comma
-------------------------------
2^(-1/3) * 3^(1/3) * 5^(1/6) = 1/6-comma meantone "5th"

2^(-1/3) * 3^(1/3) * 5^(1/6) = 1/6-comma meantone "5th"
* -3
------------------------------
2^1 * 3^-1 * 5^(-1/2) = -3 generator = "Eb"

Of course, the +3 generator "A" will have exactly the same
absolute exponent-values, with opposite signs:

2^-1 * 3^1 * 5^(1/2) = +3 generator = "A"

Following the cycle strictly as descending "5ths" puts
the "Eb" into the second "8ve" below our reference "C".
So in order to place "Eb" into the reference-"8ve" we have
to go up two "8ves", so we get 2^3 instead of 2^1. In
Telemann's voicing of the chord, the "Eb" is in the "8ve"
above the "A", so we add another "8ve" to get 2^4.

Likewise, following the cycle strictly as ascending "5ths"
puts "A" into the "8ve" above that beginning on our
reference "C", so we subtract one "8ve" and 2^-1 becomes
2^-2.

So now here's our vector addition:

2^4 * 3^-1 * 5^(-1/2) = -3 generator = Eb
- 2^-2 * 3^1 * 5^(1/2) = +3 generator = A
-------------------------------
2^6 * 3^-2 * 5^-1

= 64/45

Since 1/6-comma meantone takes 1/6 of a comma away from
each 3:2 "perfect 5th", any pair of pitches which are
6 generators apart will be an interval of this JI ratio
with no commatic alteration.

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..., "monz" <joemonz@y...> wrote:
> = 64/45
>
> Since 1/6-comma meantone takes 1/6 of a comma away from
> each 3:2 "perfect 5th", any pair of pitches which are
> 6 generators apart will be an interval of this JI ratio
> with no commatic alteration. ^^^^^^^^^^^^^

You mean "this JI-system ratio".

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

> Yes, isn't that a strange chord for this time period?
>
> But he definitely didn't have 1/4-comma meantone in mind,
> because the comma- and interval-sizes of 1/4-comma don't
> match his descriptions.
>
I know, but even 1/6-comma meantone is pretty close.
>
> The best low-integer approximation to the proportions of
> the four 55-EDO pitches which I used in the mp3 is
> 6:16:19:27 -- and this approximation would be an even
> better fit for 1/6-comma tuning.

I beg to differ, if you're saying 16:19:24:27 fits the chord better
than 1/6:1/5:1/4:2/7.

Hi Paul,

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 14, 2001 12:37 PM
> Subject: [tuning] Re: New Telemann page - audio files
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> >
> > The best low-integer approximation to the proportions of
> > the four 55-EDO pitches which I used in the mp3 is
> > 6:16:19:27 -- and this approximation would be an even
> > better fit for 1/6-comma tuning.
>
> I beg to differ, if you're saying 16:19:24:27 fits the chord better
> than 1/6:1/5:1/4:2/7.

Oops... I only considered otonal interpretations.

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..., "monz" <joemonz@y...> wrote:
> Hi Paul,
>
> > From: Paul Erlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Wednesday, November 14, 2001 12:37 PM
> > Subject: [tuning] Re: New Telemann page - audio files
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > >
> > > The best low-integer approximation to the proportions of
> > > the four 55-EDO pitches which I used in the mp3 is
> > > 6:16:19:27 -- and this approximation would be an even
> > > better fit for 1/6-comma tuning.
> >
> > I beg to differ, if you're saying 16:19:24:27 fits the chord
better
> > than 1/6:1/5:1/4:2/7.
>
>
> Oops... I only considered otonal interpretations.

How about 10:12:15:17?

Hi Paul,

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, November 15, 2001 10:52 AM
> Subject: [tuning] Re: New Telemann page - audio files
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > Hi Paul,
> >
> > > From: Paul Erlich <paul@s...>
> > > To: <tuning@y...>
> > > Sent: Wednesday, November 14, 2001 12:37 PM
> > > Subject: [tuning] Re: New Telemann page - audio files
> > >
> > >
> > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > >
> > > > The best low-integer approximation to the proportions of
> > > > the four 55-EDO pitches which I used in the mp3 is
> > > > 6:16:19:27 -- and this approximation would be an even
> > > > better fit for 1/6-comma tuning.
> > >
> > > I beg to differ, if you're saying 16:19:24:27 fits the chord
> > > better than 1/6:1/5:1/4:2/7.
> >
> >
> > Oops... I only considered otonal interpretations.
>
> How about 10:12:15:17?

Well... sure, those proportions are a good fit for the
chord in *root* position, but that's not the inversion
Telemann uses here. This particular rational interpretation
of his chord, in the correct inversion, is 15:40:48:68.

Those numbers are much larger than the ones I gave, 6:16:19:27.

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