A question for George

Hi George.

I was very interested in your 1/4-comma meantone TX. Do you think you could
trace a bit of some history of the ideas which inspired you for this tuning?
Some time ago, I was keen on making proportionally beating quasi-meantone
tunings and partly I am until now. If you were interested in my systems, you
can tell me.

Thank you in advance.

Petr

PS: Is it true that the ideal case is if C4 = 258Hz? If so, maybe it could
be added to the description or one of the comment lines.

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Hi George.
>
> I was very interested in your 1/4-comma meantone TX. Do you think
you could
> trace a bit of some history of the ideas which inspired you for
this tuning?

Hi Petr,

The history goes back a long way. When I first studied alternate
tunings in the fall of 1963, I experimented with a retuned electronic
organ, which I left in 1/4-comma meantone for several weeks. When I
put it back into 12-ET, I was shocked to find how badly it sounded,
and ever since I have resolved to arrive at a temperament in which as
many triads as possible sound like meantone, while having no triads
with an unacceptable amount of error (e.g., no fifths tempered by
more than 6 or 7 cents). I came up with a temperament for this in
1964, and a couple of improved versions in 1975.

I became interested in proportional-beating temperaments as a result
of discussions in this group, and my first attempt in this direction
was, I recall, a 19-tone circulating temperament in which 9 of the
major triads had 1:1:1 beat ratios (cf. Erv Wilson's meta-meatone
temperament). Realizing that such triads would not be feasible in a
circulating 12-tone temperament, I reasoned the next best thing might
be major triads in which the 5th and major 3rd have the same beat
rate (brat of 4), and it was no long afterward that I announced
development of a 5/23-comma temperament (extra)ordinaire containing
two such triads (with two more having brat of 2):
/tuning/topicId_59689.html#59999

(In case you're not familiar with the term, "brat" (or breat-ratio)
refers to the ratio between the beat-rates of the intervals in a
major or minor triad. When a single number is given, it is the one
obtained by dividing the minor 3rd beat rate by that of the major
3rd.)

A requirement in this (and all of my subsequent) temperaments (extra)
ordinaires is that all of the major triads from Eb to E (i.e., the
good ones in a usual 12-tone meantone tuning) have intonation not
significantly worse than 12-ET.

After this point, Gene Ward Smith provided some very helpful
direction by demonstrating how a temperament constructed with
rational number will provide exact beat-rates, e.g.:
/tuning/topicId_59689.html#61735
Gene also offered advice to use "as many of the 'magic' brats as
possible; load up on 3/2 and 4, and fill in a gap with 2 or 3."
(message #61804)

Around this time I was starting to pay attention to the brats for the
minor triads as well as the major triads. In the course of
attempting to construct a new (extra)ordinaire with less error in the
most consonant triads, I found that a brat of 9 (corresponding to
5/21-comma meantone, which has M3:5th and 5th:m3 beat ratios of 1:3),
also has a nice brat value of 5 for the minor triads. See:
/tuning-math/message/13651

This idea was finalized in my 1/4-comma temperament (extra)ordinaire
on 5 February of this year, but I didn't post it here until this
month:
/tuning/topicId_59689.html#66222
Although it has the 5/21-comma brats of 9 (exact) on the C and G
major triads, the narrowest fifths are tempered approximately 1/4-
comma. It also has very nice (exact) major brats on D (4), E (2), F
(2), Bb (1.5), A (2.75), and Eb (1.2). There are exact minor brats
on A (7), E (5), and Bb (1), and close approximations on B (2) and F#
(4/3).

> Some time ago, I was keen on making proportionally beating quasi-
meantone
> tunings and partly I am until now. If you were interested in my
systems, you
> can tell me.

Yes, I noticed this around the time that you posed it:
/tuning/topicId_62320.html#62320
I didn't respond because I haven't had a lot of time to spare lately,
and I am more interested in circulating temperaments.

> Thank you in advance.
>
> Petr
>
> PS: Is it true that the ideal case is if C4 = 258Hz? If so, maybe
it could
> be added to the description or one of the comment lines.

I don't understand why any particular frequency should be an "ideal
case". The ratios between the beats don't change when you change the
reference frequency. Umm, wait a minute; perhaps you're thinking of
counting beats when tuning by ear, in which case 258 gives a C-G beat
rate of exactly 2 Hz (and also for A-E in my 1/4-comma TX, where C#-
G# is exactly 1/2). Nice! (But it makes A=431.55555..., which is
somewhat lower in pitch than what we're accustomed to.)

Best,

--George