Re: 55-tet major improv

Hi there,

Actually, on checking it, the accidentals I'm using are
C#, F#, and G# at + 3 steps, and Eb, and Bb = -4 steps.

The mode is 0 9 18 23 32 41 50 55 in 55-tet, or as steps 9 9 5 9 9 9 5.

So as a twelve tone scale it's: 3 6 5 4 5 3 6 3 6 5 4 5.

Robert Walker

--- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi there,
>
> Actually, on checking it, the accidentals I'm using are
> C#, F#, and G# at + 3 steps, and Eb, and Bb = -4 steps.
>
> The mode is 0 9 18 23 32 41 50 55 in 55-tet, or as steps 9 9 5 9 9
9 5.
>
> So as a twelve tone scale it's: 3 6 5 4 5 3 6 3 6 5 4 5.
>
> Robert Walker

Hi Robert . . .

I just peeked over on the crazy-music list and saw lots of
misinformation and misleadingly presented information (from
the "nutty professor" as usual) this time about 55-tET.

55-tET was discussed rather extensively on the tuning list a few
months ago (you may have missed it). For example, we quoted from

Chesnut, John Hind. 1977.
"Mozart's teaching of intonation",
Journal of the American Musicological Society
vol. 30 no. 2 [summer], pp. 254-271.

"Leopold Mozart refers to Tosi in general terms as an authoritative
source in a letter to Wolfgang from Salzburg dated June 11, 1778.
Tosi, in 1723, considered the correct tuning system to be what we
would today call a form of regular meantone temperament ... according
to Tosi, the large diatonic half step is theoretically equal to five
ninths of a whole step, and the small chromatic half step is
theoretically four-ninths of a whole step. Tosi thereby divides the
octave into fifty-five equal parts. This is equivalent to tempering
the perfect fifth by approximately one-sixth of a 'comma,' ...
Leopold Mozart, in his violin method of 1756 -- which happens to be
the year of Wolfgang's birth -- also describes what we have
called 'extended regular meantone temperament' as the correct
intonation for the violin; he tells us that keyboard instruments of
his time were played with some form of tempered [i.e., well-tempered]
tuning, but that in the "right ratio" [i.e., meantone] tuning that he
recommends for the violin, flats are higher by a comma than
enharmonically equivalent sharps. It can be shown that for whichever
of the standard commas we choose, the perfect fifths in Leopold
Mozart's system were theoretically flattened by about one-sixth of
that comma . . . Leopold Mozart wrote down a couple of scales
specifically intended for practice in intonation, one leading through
the flats, the other through the sharps. In practicing these scales,
the student is supposed to learn to distinguish between the large
diatonic half steps and the small chromatic half steps. It is
important to emphasize that these scales are not abstractions but
exercises to be mastered . . . "

It is for this correspondence with a flavor of meantone temperament
that 55-tET was advocated by Telemann, etc. etc.

Anyway, back to making microtonal music.

Hi Paul,

Thanks for the comments.

This is what I'm planning to say now:

55-tet twelve tone scale is a close approximation to sixth comma
meantone. Origins seem to be from Tosi in 1723. Each step of
55-tet is very close to a syntonic comma. This makes a twelve tone
scale consisiting of two sizes of semitone, one of five, and one
of four approximate syntonic commas. Leopold Mozart (Wolfgang Mozart's
father) was in favour of this system and wrote a couple of scales to be used
as excerices for it. Teleman advocated it for this same reason.

Here is a site by an author who strongly advocates use of sixth
comma meantone for baroque music:

http://www-midischool.cwru.edu/Duffin/Vallotti/56K/page4.html

On page 4 he mentions that the tritone sounds particularly good
in sixth comma meantone, being at the most optimal position in
a certain sense, and this links in with the improvisations -

I remember noticing that the one from B to F sounded particularly
nice and enjoyed using it.
"

(found the link on search through TL)

> Anyway, back to making microtonal music.

Keep it up!

Robert

--- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:
> "
> 55-tet twelve tone scale is a close approximation to sixth comma
> meantone. Origins seem to be from Tosi in 1723. Each step of
> 55-tet is very close to a syntonic comma.

True -- but be careful!

> This makes a twelve tone
> scale consisiting of two sizes of semitone, one of five, and one
> of four approximate syntonic commas.

This is close to being misleading. You see, the syntonic comma is
represented by 0 (ZERO) steps in 55-tET, not 1 step. This is very
important. By way of contrast, the syntonic comma is represented by 1
step in 53-tET. This makes 53-tET act very differently from 55-tET
when it comes to diatonic triadic music.

Hi Paul,

> > "
> > 55-tet twelve tone scale is a close approximation to sixth comma
> > meantone. Origins seem to be from Tosi in 1723. Each step of
> > 55-tet is very close to a syntonic comma.
>
> True -- but be careful!
>
> > This makes a twelve tone
> > scale consisiting of two sizes of semitone, one of five, and one
> > of four approximate syntonic commas.
>
> This is close to being misleading. You see, the syntonic comma is
> represented by 0 (ZERO) steps in 55-tET, not 1 step. This is very
> important. By way of contrast, the syntonic comma is represented by 1
> step in 53-tET. This makes 53-tET act very differently from 55-tET
> when it comes to diatonic triadic music.

As cents, the syntonic comma is 21.506 cents and one step of 51-tet
is 21.818 cents, a little larger, with the
step of n-tet closest to it in size being 56-tet at 21.429 cents.

So in what sense is the syntonic comma represented by 0 steps in
55-tet?

Let me try and solve the puzzle for myself first.

The fifth in 55-tet is 32 steps.

So going up four fifths as in c to e'', means 128 steps. Going up two octaves
is 110 steps. So the difference between the two is 18 steps, which
is the same as the major third of 55-tet.

So, if one were improvising in 55-tet, then one no longer has a syntonic
comma to worry about!

Interesting...

I've just tried it out in the 55-tone major - one can go up four
fifths from the 1/1, all of them identical in size (698.18 cents),
and one gets to the 55-tet major third at 392.73 cents.

It sounds quite interesting actually, going up four sligthly
unstable fifths, then dropping down a major third that is
relatively pure compared to the ones we are used to hearing
in 12-et, as a kind of resolution. Could be the beginnings
of a piece.

So what happens to the other major third in the circle of fifths
- the one that crosses the break in the circle? (down 8 fifths)

Say, F# to Bb. One gets a sharper third at 414.55 cents.

While in PYthagorean, the one going up four fifths is the one
that's sharp at 81/64 = 407.82, and the one going down 8 fifths is flatter,
and near to the just intonation 5/4 at 8192/6561 = 384.36 cents.

This is relevant to actual making of microtonal music by the way
as I'm thinking about an idea for a piece in 55-tet or an improvisation
- I've just posted a thought about it to crazy_music.

Robert