Ganassi's 12-tone well temperament

I hope I'm doing the right thing by posting across lists here.
It seemed to me that a general post of this nature belongs
on the main list rather than at spiritual_tuning.

--- In spiritual_tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:

/spiritual_tuning/topicId_unknown.html#170

> Alas, 17 and 19 combine with powers of 2 and 3 to generate
> intervals nearly indistinguishable from 12-tet. A pretty good
> approximation to 12-tet may be made from a repeated series
> of pitches related as 16:17:18:19 ( I have references
> somewhere to Integer Ratio Chromatic Scales constructed just
> this way).

Sylvestro Ganassi's treatise (1543, _Regula Rubertina_, chapter 4)
is the one I know which presents the chromatic scale this way.

While at first glance it does seem to give "a pretty good
approximation to 12-tet", a deeper look into this tuning
reveals that it is an interesting well-temeperament.

I've made a webpage about this tuning, reproducing what
I write here, and also including an interval matrix
and graph of deviations from JI triads, and links which
play mp3s of each of the major and minor triads:
http://www.ixpres.com/interval/monzo/ganassi/ganassi.htm

Here is a table illustrating the scale, with cents given above
the nominals, and string-length proportions below:

0 89 182 281 386 498 597 702 791 884 983 1088 0

C C#/Db D D#/Eb E F F#/Gb G G#/Ab A A#/Bb B C

120 114 108 102 96 90 85 80 76 72 68 64 60

30 27 24 20 19 18 17 16 15
24 18 17 16 12
20 19 18 17 16 15 12 10

So you can see that it is two fully chromatic tetrachords
with the proportions 20:19:18:17:16:15, connected by a
"tone of disjunction" which is divided 18:17:16.

The cents values show that it is a well-temperament,
giving exact Pythagorean "4ths" and "5ths" and 5-limit JI
"major 2nd/3rd/6th/7ths". The "minor" variety of these
intervals are narrower than either Pythagorean or
5-limit, in a few cases suggesting the 7-limit flavor
of "minor 3rd/7th".

From the interval matrix I have constructed a table showing
deviation from the "perfect 5th" and from Pythagorean and
5-limit JI "3rds" (and from 7:6), useful in analyzing the
properties of this tuning in relation to "common-practice"
(Eurocentric, c. 1600-1900) triadic harmony.

Deviations from JI intervals (in cents):

"root" 7:6 32:27 6:5 5:4 81:64 3:2

B +27 0 -22 + 7 -15 + 7
A#/Bb +39 +11 -10 +13 - 9 +13
A +49 +22 0 +18 - 3 0
G#/Ab +31 + 3 -18 +23 + 1 -11
G +14 -13 -34 0 -22 -22
F#/Gb +20 - 7 -28 0 -22 -10
F +26 - 1 -23 0 -22 0
E +49 +22 0 +18 - 3 0
D#/Eb +49 +22 0 +34 +13 0
D +49 +22 0 +28 + 7 0
C#/Db +31 + 3 -18 +23 + 1 0
C +14 -13 -34 0 -22 0

This table shows that the only "good" triads,
which provide exact proportions of 4:5:6 for "major"
and 1/(4:5:6) for "minor", are "C" and "F" major, and
"D", "D#", "E", and "A" minor.

Some notes on the rest of the "major" triads:

"G" major gives a 5:4 "major 3rd" but the 22-cent-flat
"wolf 5th". "F#/Gb" major gives a 5:4 "major 3rd" with
a "5th" much closer to 3:2.

"A", "E", and "C#/Db" major all give exact 3:2s for the
"5th" and an extremely close approximation to the 81:64
Pythagorean "major 3rd", and "D" major is similar but
sounds quite out-of-tune with a slightly higher "3rd".

"B" major gives a "major 3rd" and a "5th" that are
both ~7 cents wider than 4:5:6, and "A#/Bb" major gives
deviations of ~13 cents wider for both.

"D#/Eb" major is similar to "D" major in that it gives
an exact 3:2 "5th" but an even higher "major 3rd", and
"G#/Ab" major gives a very good approximation to the
Pythagorean "major 3rd" but the "5th" is ~11 cents
narrower than 3:2.

And for the rest of the "minor" triads:

Of the remaining 3 "minor" triads which have a perfect
3:2 "5th", "F" and "C#/Db" minor give a good approximation
to the Pythagorean 32:27 "minor 3rd", and "C" minor gives
a dark narrow "3rd" which is about halfway between the
7:6 and the Pythagorean "minor 3rd".

"G" minor has this same low "minor 3rd", but its "5th"
is the very flat "wolf 5th".

"F#/Gb" and "G#/Ab" minor both have decent approximations
to the Pythagorean "minor 3rd" and a "5th" narrower by
~10 and ~11 cents respectively from 3:2, and "B" minor
gives an exact Pythagorean "minor 3rd" with a "5th" that
is ~7 cents wider than 3:2.

Lastly, "A#/Bb" minor gives almost the same error for
its wide Pythagorean "minor 3rd" and "5th"; its "minor 3rd"
is actually about midway between the Pythagorean and 6:5.

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