well, well, well......

Hey, gang,

I'm new around here, but I've decided to go ahead and put my head on the chopping block so you can all kick it around a bit.

Recently, I've been tinkering (at least on paper) with what, I think, may be a new family of well-tempered tunings. I'm sure some of the more historically well-read folks on the list will set me straight if I've merely rediscovered something--I'm somewhat of a novice here--but here goes:

I got to thinking about how the historical well temperaments we associate with 18th century music have so many advantages--the utility of equal temperament, at least some of the beauty of 1/4-syntonic mean-tone and 5-limit JI, and that quality whereby each key has its own "color" in tonal music--but one notably annoying quality in that they tend to possess a certain "bumpiness," owing to the stark placement of two different sizes of fifth squarely next to one another in the circle of fifths. This is the situation we have, say, in Werckmeister III, where we find pure fifths next to those with 1/4 Pythagorean comma removed, and Vallotti, where the small fifths have 1/6 of the comma shaved off of them. "Wouldn't it be cool," I thought, "if some of that bumpiness could be smoothed out by using intermediate sizes of fifth?"

Then, one evening, while I was in the bath, an arithmetical idea came to me:

3/12 + 3/12 + 2/12 + 2/12 + 1/12 + 1/12 = 12/12
i.e. 1/4 + 1/4 + 1/6 + 1/6 + 1/12 + 1/12 = 1

following this pattern, I also got:

4/20 + 4/20 + 3/20 + 3/20 + 2/20 + 2/20 + 1/20 + 1/20 = 20/20
i.e. 1/5 + 1/5 + 3/20 + 3/20 + 1/10 + 1/10 + 1/20 + 1/20 = 1

and:

5/30 + 5/30 + 4/30 + 4/30 + 3/30 + 3/30 + 2/30 + 2/30 + 1/30 + 1/30 = 30/30
i.e. 1/6 + 1/6 + 2/15 + 2/15 + 1/10 + 1/10 + 1/15 + 1/15 + 1/30 + 1/30 = 1

(getting the picture?)
and finally........

6/42 + 6/42 + 5/42 + 5/42 + 4/42 + 4/42.....= 42/42
i.e. 1/7 + 1/7 + 5/42 + 5/42 + 2/21 + 2/21 + 1/14 + 1/14 + 1/21 + 1/21 + 1/42 + 1/42 = 1

Now, if we take these to be fractions of the Pythagorean comma (and work outward in pairs from the tonal linchpin of D to provide equitable treatment of sharp and flat keys), this yields the following four temperaments (G# and Ab are the same pitch, and unless indicated, a fifth is assumed to be just):

1. 2. 3. 4.

Ab Ab Ab Ab
[ [ [ [-1/42
Eb Eb Eb Eb
[ [ [-1/30 [-1/21
Bb Bb Bb Bb
[ [-1/20 [-1/15 [-1/14
F F F F
[-1/12 [-1/10 [-1/10 [-2/21
C C C C
[-1/6 [-3/20 [-2/15 [-5/42
G G G G
[-1/4 [-1/5 [-1/6 [-1/7
D D D D
[-1/4 [-1/5 [-1/6 [-1/7
A A A A
[-1/6 [-3/20 [-2/15 [-5/42
E E E E
[-1/12 [-1/10 [-1/10 [-2/21
B B B B
[ [-1/20 [-1/15 [-1/14
F# F# F# F#
[ [ [-1/30 [-1/21
C# C# C# C#
[ [ [ [-1/42
G# G# G# G#

So, what's the verdict? I think the first one has potential for being tryed out on acoustic keyboards, mainly because
A. it gives the purest thirds in the near keys
and B. it would be easy to implement in real life, using only four different sizes of fifth, all of which are fairly familiar (and one, VERY familiar, i.e., the equal-tempered fifth) to those cute little piano-tech-y types we all know and love so well.

If any of you would like to try this temperament out, the frequencies are as follows; I'd love to hear anybody's opinion of it.

C 263.12
C# 277.49
D 294.33
Eb 312.18
E 329.26
F 351.2
F# 369.99
G 393.79
G# 416.24
A 440
Bb 468.27
B 493.32

Thanks for listening,
Dale Scott

Greetings,
Dale writes:
>So, what's the verdict? I think the first one has potential for being
>tryed out on acoustic keyboards, mainly because
> A. it gives the purest thirds in the near keys

Hmm, this is not a new concept.

and B. it would be easy to implement in real life, using only
>four different sizes of fifth, all of which are fairly familiar (and one,
>VERY familiar, i.e., the equal-tempered fifth) to those cute little
piano-tech-y
>types

Oh great, now I am a cute little piano-tech-y type! I suppose that is
better than the wild hairy bushman of dissonance some have decided best
describes me while in tuner mode.
I must take issue with the idea that piano techs are familiar with
anything other than a fifth that is compressed by 1.95 cents or a third that
is tempered more or less than 13.7 cents. Anything other than these is often
viewed as simply "out of tune". This is a cost of ET's hedgemony. I will be
addressing the point this summer in Kansas City, while giving two classes at
the national convention of the Piano Technician's Guild. (There is some
controversy in that organization concerning the use of anything but ET, but I
intend to meet the wizard or see the elephant, if he is there!)
When you get around to producing four different sizes of fifths, on an
acoustic piano, you must have a plan of checks and measures, judgement by
ear of the fifth's beating rates will do a poor job of giving you four
accurate sizes. Describing the temperament via the sizes of the thirds is
more managable.
We have at our disposal, today, the means to investigate tonality beyond
anything that has occured in the past. Whereas earlier theorists had to build
instruments with many levers, we have microchips that can provide all those
tones for us.
Whereas it took years of study to find the past constructions, (Boethius
or Barbour, take your pick), today, we can call up all the past temperaments
at the flick of a button or two, and THEN send them over the internet for
peer review. This is the magic that will power the next evolutionary spasm
in intonation's continuing life.

>If any of you would like to try this temperament out, the frequencies are
>as follows;
Tuners don't use frequencies to tune pianos, we use cents. The easiest
way for temperamenti (!?) to be communicated is in the cents deviation from
ET. You got any of those numbers in that harmonic bathtub?
Regards,
Ed Foote
Precision Piano Works
Nashville, Tn.

Dale Scott wrote,

>Recently, I've been tinkering (at least on paper)
>with what, I think, may be a new family of
>well-tempered tunings. I'm sure some of the more
>historically well-read folks on the list will set
>me straight if I've merely rediscovered something
>--I'm somewhat of a novice here--but here goes:

>I got to thinking about how the historical well
>temperaments we associate with 18th century music
>have so many advantages--the utility of equal
>temperament, at least some of the beauty of
>1/4-syntonic mean-tone and 5-limit JI, and that
>quality whereby each key has its own "color" in
>tonal music--but one notably annoying quality in
>that they tend to possess a certain "bumpiness,"
>owing to the stark placement of two different
>sizes of fifth squarely next to one another in
>the circle of fifths. This is the situation we
>have, say, in Werckmeister III, where we find
>pure fifths next to those with 1/4 Pythagorean
>comma removed, and Vallotti, where the small
>fifths have 1/6 of the comma shaved off of them.
>"Wouldn't it be cool," I thought, "if some of
>that bumpiness could be smoothed out by using
>intermediate sizes of fifth?"

Yes Dale, this is a very old idea. See, for example,
Owen Jorgenson's book, which details hundreds of
17th, 18th, and 19th century temperament proposals,
or even Barbour's book.

My personal variation on this theme is a literal
interpretation of the term "circulating"
temperament. Namely, the amount each fifth is
tempered is proportional to its "height" on the
circle of fifths -- think of projecting the dots on
a clock face onto a vertical line. The trigonometry
is quite easy in this case since the angles are all
multiples of 30 degrees. The cosines of the angles,
with the fifths to which they are mapped, are:

angle cosine fifth
0� 1 d-a
30� sqrt(3)/2 a-e
60� 1/2 e-b
90� 0 b-f#
120� -1/2 f#-c#
150� -sqrt(3)/2 c#-g#
180� -1 g#-eb
210� -sqrt(3)/2 eb-bb
240� -1/2 bb-f
270� 0 f-c
300� 1/2 c-g
360� sqrt(3)/2 g-d

Now if we make the meantone "wolf", g#-eb,
Pythagorean, the just "wolf", d-a, will be
tempered by 1/6 Pythagorean comma, and all other
fifths will be somewhere in-between. The formula
would be tempering = 1/12*(1+cosine) Pyth. comma.
In cents:

f-c 700
c-g 699.0224996
g-d 698.3069196
d-a 698.0449991
a-e 698.3069196
e-b 699.0224996
b-f# 700
f#-c# 700.9775004
c#-g# 701.6930804
g#-eb 701.9550009
eb-bb 701.6930804
bb-f 700.9775004

Following Jorgenson, I tabulate the major thirds
for this tuning:

f-a 395.3744183
c-e 393.6813379
g-b 393.6813379
d-f# 395.3744183
a-c# 398.3069196
e-g# 401.6930804
b-eb 404.6255817
f#-bb 406.3186621
c#-f 406.3186621
g#-c 404.6255817
eb-g 401.6930804
bb-d 398.3069196

Of course, one could start with the clock rotated
-15�, if one wanted g-d and d-a to be tempered the
same amount so that the tuning followed the natural
symmetry of the keyboard. Then one could assign the
two largest fifths to Pythagorean and temper the
two smallest by 1/6-Pythagorean comma. The fifths
are then:

f-c 699.4761591
c-g 698.56884
g-d 698.0449991
d-a 698.0449991
a-e 698.56884
e-b 699.4761591
b-f# 700.5238409
f#-c# 701.43116
c#-ga 701.9550009
ga-eb 701.9550009
eb-bb 701.43116
bb-f 700.5238409

The resulting major thirds (using Graham Breed's
notation "ga" for g#/ab):

f-a 394.1349974
c-e 393.2276783
g-b 394.1349974
d-f# 396.6138392
a-c# 400
e-ga 403.3861608
b-eb 405.8650026
f#-bb 406.7723217
c#-f 405.8650026
ga-c 403.3861608
eb-g 400
bb-d 396.6138392

Dale, I would be interested to see tables of major
thirds for your temperaments.

Ed Foote wrote:

> The following is a listing of cent deviations from ET that I received
> from one of my tuning mentors, it provides a different shape to the balances
> among thirds as they progress by movements of a fourth or fifth.
> C 4
> C# 1
> D 1
> D# 1
> E -2
> F 5
> F# -1
> G 3
> G# 1
> A 0
> A# 3
> B -2

FIFTHS THIRDS

f-c 699 f-a 395
c-g 699 c-e 394
g-d 698 g-b 395
d-a 699 d-f# 398
a-e 698 a-c# 401
e-b 700 e-g# 403
b-f# 701 b-eb 403
f#-c# 702 f#-bb 404
c#-g# 700 c#-f 404
g#-eb 700 g#-c 403
eb-bb 702 eb-g 402
bb-f 702 bb-d 398

Nice!