Circos, and example circulating temperament

I was thinking about two claims, one that 17th and 18th century
musicians were very mathematically sophisticated, and knew all the
math they needed, and the other that circulating temperaments should
not have any fifths sharper than a pure fifth. The temperament below
was easily calculated as an example relevant to both, and is a part
of an infinite family.

I think musicians back then had all the math they needed to have to
solve their problems, but there certainly were things outside of
their scope. One of these was least squares optimaziation. I ran a
least squares to optimize the fifth and major third for each major
triad, with a sinusoidal weighting factor going from 3 for C to 1 for
F#. This serves two purposes: one to show the use of types of math in
such tuning problems which would be beyond the scope of Werckmeister,
the other to show how sharp fifths arise naturally in this kind of
problem. I probably overdid the weighting a bit, but it's interesting
and looks like it could be pretty spicy. It clearly circulates, at
any rate.

! circos.scl
[1, 3] weight range weighted least squares circulating temperament
12
!
89.617502
195.633226
300.984164
391.528643
501.698852
587.325482
697.824592
796.007518
893.677492
1002.402613
1088.884819
1200.000000

> I think musicians back then had all the math they needed to have to
> solve their problems,

It's hard to get in much trouble beyond what you can understand.
Which is why new discoveries always raise new questions.

> I ran a
> least squares to optimize the fifth and major third for each
> major triad, with a sinusoidal weighting factor going from
> 3 for C to 1 for F#. This serves two purposes: one to show
> the use of types of math in such tuning problems which would
> be beyond the scope of Werckmeister, the other to show how
> sharp fifths arise naturally in this kind of problem.

The sinusoidal thing seems like quite an assumption. If
you really intend all keys to be playable, why does it
matter? Presumably one doesn't want too much contrast
between chords one might be using in the same piece of
music... how endemic are sharp fifths among solutions like
this, and how does the weighting factor effect it?

-Carl

Interesting angle (& reminiscent of John Barnes' work in 1979, except
that he set the limit at a Pythagorean third). See below...

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> I was thinking about two claims, one that 17th and 18th century
> musicians were very mathematically sophisticated,

There is no good reason to believe this.

One reason someone might believe it is if they studied a pre-selected
list of sources which contained only mathematical treatments of
tuning, and discarded (or just ignored the existence of) all the other
sources of information about what musicians did.

If all you do is read the chapters in the works that appear to say
something mathematical, you get a totally false impression of probable
trends in musical culture. And you also probably end up
misinterpreting those chapters (as they relate to music-making), due
to the illusion that the authors were talking about mathematical
quantities rather than physical ones.

All one can deduce by looking at the mathematical sources is that a
very few people (i.e. Galle, Rossi, Huyghens, Sauveur, Neidhardt,
anyone else??) *were* mathematically sophisticated, and knew at least
the rudiments of musical theory that allowed them to apply their
mathematics without producing obvious nonsense. But were any of these
guys musicians?

> [17th and 18th century
> musicians] knew all the math they needed,

This though is a perfectly sensible idea.

That amount may well have been zero, since it is possible to learn
what a pure fifth, a pure third, a syntonic comma, etc. etc. are -
without writing down a single numeral!

As in that violin tuning source I quoted from for JI, or Schlick, Aron
and many others, for meantone and modifications thereof, it is a
matter of aural experimentation and the fact that pure intervals are
easily identifiable.

> and the other that circulating temperaments should
> not have any fifths sharper than a pure fifth.

... well, even Neidhardt published a few with wide fifths, though what
musical purpose they served is not clear.

> ran a least squares to optimize the fifth and major third for
> each major triad,

With what relative weighting between the two intervals? (What kind of
meantone would such an optimization give without circularity /
weighting?... quarter-comma almost always wins, I guess)

> with a sinusoidal weighting factor going from 3 for C to 1 for
> F#.

> This serves two purposes: one to show the use of types of math in
> such tuning problems which would be beyond the scope of Werckmeister,

It cuts both ways... Werckmeister's level of experience in tuning
organs by ear, with just hand tools, was beyond the scope of anyone on
this list. And I don't think a list of cent values would be any use to
him, even if he knew what they were.

Besides, if it's an infinite family of tunings (or a family of
families, since one could vary the weighting of thirds relative to
fifths, or add a weighting for minor thirds, etc.) how is anyone going
to find the 'right' one - except by listening? Then you might as well
bypass the theory and start directly investigating what you find
aurally acceptable given the norms of musical composition.

Still, theoretically, it's an interesting experiment.

> the other to show how sharp fifths arise naturally in this kind of
> problem. I probably overdid the weighting a bit, but it's
interesting (...)

... Now for the donkey work.

> ! circos.scl
> [1, 3] weight range weighted least squares circulating temperament
> 12
> !
> 697.8
> 195.6
> 893.7
> 391.5
> 1088.9
> 587.3
> 89.6
> 796.0
> 301.0
> 1002.4
> 501.7
> 1200

I rounded the numbers to make it more easily comprehensible - the
exact values are meaningless, since the weight values [1,3] were
rather arbitrary. Then put it into circle-of-fifths order. Temperings are:

C -4.2 G -4.2 D -3.9 A -4.2 E -4.6 B -3.6 F#
+0.3 C# +4.4 G# +3.0 Eb -0.6 Bb -2.7 F -3.7 C

total -24.

As for the thirds (taking 386 as pure):

C-E +5.5
G-B +5.1
D-F# +5.7
A-C# +9.9
E-G# +18.5
B-Eb +26.1
F#-Bb +29.1
C#-F +26.1
G#-C +18.0
Eb-G +10.8
Bb-D +7.2
F-A +6.0

Practically, this shows a lot of similarity to various modified
meantone instructions (in particular Schlick!!) ... The quasi-regular
thirds are about 1/4 comma sharp, giving an 'average' tempering about
3/16 comma around those seven fifths.

But I'd be surprised if this is the true least-squares fit: it makes G
and D major as good as, or marginally better than, C. (Who knows what
lurks in the mathematics of 12 partially-degenerate variables...)

If correct, it seems the constraints of circularity prevent the
existence of a tuning which manifestly reflects the weighting scheme.

Also, it makes A-C# and Eb-G noticeably better than ET, which is a
surprise, since these keys have exactly average weighting, and of
course the 'purity' of intonation is on (unweighted!) average just the
same as ET.

And a further surprise is that the fifths are so regular from F
sharpwards round to F#. What happens if the weighting of fifths is
reduced?

As for sinusoidal tunings, I invented one some time ago, which might
be adjusted to the present situation. The method is extremely simple:
just construct a sinusoidal variation in the tempering of the fifth,
offset by a major second relative to the thirds which one wants to
influence. Thus for example

C -4 G -5 D -5 A -4 E -2.5 B -1.5 F#
0 C# +1 G# +1 Eb 0 Bb -1.5 F -2.5 C

giving a variation from C-E +4 through A-C# +14 to F#-Bb +24 ... now
why is this a worse fit than Gene's?

It must arise from the squaring of deviations - i.e. one should not
expect a sinusoidal distribution of thirds. Rather, the *square* of
the deviation of thirds might go approximately as one over the weight...

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> All one can deduce by looking at the mathematical sources is that a
> very few people (i.e. Galle, Rossi, Huyghens, Sauveur, Neidhardt,
> anyone else??) *were* mathematically sophisticated, and knew at
least
> the rudiments of musical theory that allowed them to apply their
> mathematics without producing obvious nonsense.

I'd add Stevin, Mersenne, and Euler.

> > ran a least squares to optimize the fifth and major third for
> > each major triad,
>
> With what relative weighting between the two intervals?

Equal, with no weight given to minor thirds. What would you suggest?

(What kind of
> meantone would such an optimization give without circularity /
> weighting?... quarter-comma almost always wins, I guess)

4/17-comma, actually.

> It cuts both ways... Werckmeister's level of experience in tuning
> organs by ear, with just hand tools, was beyond the scope of anyone
on
> this list. And I don't think a list of cent values would be any use
to
> him, even if he knew what they were.

Yes, but the claim was for great mathematicial sophsitication by the
likes of Werckmeister, whereas even Euler didn't know about least
squares (though in an alternate history he could have easily invented
it.)

> But I'd be surprised if this is the true least-squares fit: it
makes G
> and D major as good as, or marginally better than, C.

That kind of irreguarity is typical of least aquares optimizations--
four fifths give a five, starting from a given point, so thirds and
fifths are offset.
> And a further surprise is that the fifths are so regular from F
> sharpwards round to F#. What happens if the weighting of fifths is
> reduced?

Lets stick to weighting fifths, major thirds, and minor thirds in the
twelve major triads. What would make sense?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> > very few people (i.e. Galle, Rossi, Huyghens, Sauveur, Neidhardt,
> > anyone else??) (...)
>
> I'd add Stevin, Mersenne, and Euler.

OK then. Though I don't know what Mersenne added to Galle's work
(except publicity).

> > > ran a least squares to optimize the fifth and major third for
> > > each major triad,
> >
> > With what relative weighting between the two intervals?
>
> Equal, with no weight given to minor thirds. What would you suggest?

I don't know what would be 'better' or 'worse' : probably if one
increases the relative weight of fifths one gets a slightly more equal
and/or regular result, and vice versa. I think it's worth varying the
mixture, just to see what results.

Given the relative insensitivity of 'optimal' meantone to the relative
weight of fifths (see below), you'd probably have to vary it quite a
lot to see any effect.

For minor thirds (those within each major triad), you could just try
increasing their weight up from zero and see if it does anything.
Pushes towards equality, I would guess.

> > (What kind of meantone would such an optimization give ...)
>
> 4/17-comma, actually.

Yeah, I worked it out soon after writing that. General formula

D5 = S * 4/(16 + W5/W3)

where D5 is the deviation of the fifth, S is the comma and W5/W3 is
the relative weighting factor. So if you want 1/5 comma, you have to
go as far as W5/W3 = 4.

> the claim was for great mathematical sophsitication by the
> likes of Werckmeister,

I doubt that can be supported. Of course he could do rationals (modulo
typos) and had to be quite clever to invent the Septenarius (again
modulo typos) ... But 'sophisticated'? He used arithmetical division
of the comma (eg 240:241:242:243) and didn't bother about the schisma.

> > And a further surprise is that the fifths are so regular from F
> > sharpwards round to F#. What happens if the weighting of fifths is
> > reduced?
>
> Lets stick to weighting fifths, major thirds, and minor thirds in the
> twelve major triads. What would make sense?

If we stick to a sinusoidal scheme going round the circle (which fits
the Barnes numbers pretty well!), I would just tweak the relative
weights up and down and see if anything interesting happens. For
example, even if we go to an extremely small weight for fifths, does
it still produce a meantone-like tuning from F through F#?

To deal with 'tweaking' sensibly one really needs graphical output
which makes it obvious what has or hasn't changed in the distribution
of fifths. (Spreadsheet time!) Otherwise one is stuck with huge lists
of cent values to X decimal places.

About the sinusoidal weighing, it makes sense if you are only playing
in major keys; but minor keys need a dominant major chord which is way
over on the sharp side, compared to the relative major. This may be
why modified meantones in the late Baroque period usually have C#
major as the worst chord (see Lindley): the weighting gets shifted a
bit sharpwards, on average.

One interesting, but somewhat offbeat thing, would be to try and
reproduce Schlick. In that case you'd have to tailor the weights a
bit, since his discussion doesn't really indicate circularity. I
propose, based on what he says in the text:

All fifths except C#-G# weight 5
C#-G# weight 1

Thirds F-A, C-E, G-B weight 3
Bb-F, Eb-G, Ab-C, D-F#, A-C# weight 2
E-G# weight 1
All others weight 0.5

~~~T~~~

Tom wrote...

> [Werckmeister] had to be quite clever to invent the
> Septenarius (again modulo typos) ...

I just looked this up and found

http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html

where the guy claims wide fifths are typos. What's the deal
with this tuning?

> > minor thirds in the
> > twelve major triads. What would make sense?

I don't think measuring the deviation of the minor
thirds makes sense in a WT.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Tom wrote...
>
> > [Werckmeister] had to be quite clever to invent the
> > Septenarius (again modulo typos) ...
>
> I just looked this up and found
>
> http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
>
> where the guy claims wide fifths are typos. What's the deal
> with this tuning?

If you read that webpage carefully, it should be clear - cause I wrote it!

What I said was:

"Werckmeister gave a table of the fifths in the tuning (...)
Unfortunately this contained several errors, for example some narrow
fifths are wrongly signalled as wide."

This doesn't say or mean that ALL wide fifths are typos. On the
contrary, G#-D# is consistently wide in the monochord numbers and, so
far as I can recall, in the table.

This particular interval is also wide in 'Werckmeister IV' (along with
D#-Bb there) and 'V', and is one of the ones he invites you to tune
wide in the 1698 tuning instruction.

What's the deal? Read the webpage and links and make your own mind up.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Thirds F-A, C-E, G-B weight 3
> Bb-F, Eb-G, Ab-C, D-F#, A-C# weight 2
> E-G# weight 1
> All others weight 0.5

Including minor thirds?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > Thirds F-A, C-E, G-B weight 3
> > Bb-F, Eb-G, Ab-C, D-F#, A-C# weight 2
> > E-G# weight 1
> > All others weight 0.5
>
> Including minor thirds?
>

Nope. Schlick doesn't ask for minor thirds to be checked so far as I
remember.

The results are at the end of the file I just uploaded. (Sorry if you
don't have Mathematica...)

I had to tweak the weight of Ab-C down a bit, to get Eb-Ab to be sharp
like Schlick says it oughtta be.

The astonishing thing for me is that the program with sinusoidal
weights spits out a row of seven (almost) regularly-tempered narrow
fifths, no matter how you tweak the relative values of fifths and thirds.

Perhaps there's something in this game!

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> The results are at the end of the file I just uploaded. (Sorry if you
> don't have Mathematica...)

I don't think putting Mathematica notebooks in the files area is a good
idea.

> > > [Werckmeister] had to be quite clever to invent the
> > > Septenarius (again modulo typos) ...
> >
> > I just looked this up and found
> >
> > http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
> >
> > where the guy claims wide fifths are typos. What's the deal
> > with this tuning?
>
> If you read that webpage carefully,

I read it carefully enough to know that it doesn't
provide the one piece of information I care about:
the cents values for the pitch clasess of the scale.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>

>
> I read it carefully enough to know that it doesn't
> provide the one piece of information I care about:
> the cents values for the pitch clasess of the scale.
>
> -Carl
>

0: 1/1 0.000 unison, perfect prime
1: 98/93 90.661
2: 28/25 196.198 middle second
3: 196/165 298.065
4: 49/39 395.169
5: 4/3 498.045 perfect fourth
6: 196/139 594.923
7: 196/131 697.544
8: 49/31 792.616
9: 196/117 893.214
10: 98/55 1000.020 quasi-equal minor seventh
11: 49/26 1097.124
12: 2/1 1200.000 octave

EDL is as old as the hills, isn't it?

Cents schments, I'm digging the ratios,

Hmmm....sounds pretty damn good!

Thanks, Tom Dent!

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> I don't think putting Mathematica notebooks in the files area is a good
> idea.

Why not? It's only 58k. I do believe in making mathematical results as
transparent as possible.

Incidentally I seem to get slightly different values than you do,
using equal weight for fifths and major thirds, and sinusoidal
weighting between 3 and 1. My fifths go thusly from C sharpwards:

-3.3, -3.3, -3.2, -3.3, -3.3, -2.5, -0.6, 0.9, 0.5, -0.8, -2.1, -2.9

a similar overall shape, but rather less unequal.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > I don't think putting Mathematica notebooks in the files area is a
good
> > idea.
>
>
> Why not? It's only 58k.

Because few people can use it, and an ascii file of results and methods
is more to the point.

> a similar overall shape, but rather less unequal.

Well, I could try again, but I used 4296-et just for starters, to
ensure I had rational numbers with which to do linear algebra
computations. It should have given essentially identical results.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Incidentally I seem to get slightly different values than you do,
> using equal weight for fifths and major thirds, and sinusoidal
> weighting between 3 and 1. My fifths go thusly from C sharpwards:
>
> -3.3, -3.3, -3.2, -3.3, -3.3, -2.5, -0.6, 0.9, 0.5, -0.8, -2.1, -2.9
>
> a similar overall shape, but rather less unequal.

Did you multiply by the weight and then square, or square and then
multiply? I did the former.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > Incidentally I seem to get slightly different values (...)
>
> Did you multiply by the weight and then square, or square and then
> multiply? I did the former.
>

Oh. OK, my formula is

Total weight = sum_i weight_i (fifthdev_i^2 + thirddev_i^2)

with i running from 1 to 12 and weight_i between 1 and 3.

So I squared and then multiplied.

I guess my justification is that the degree of dissonance goes
approximately with the square of the deviation - consider the
parabolic behaviour near the minima of the harmonic entropy graph, for
example - and the use of each interval varies sinusoidally. Gene's
computation corresponds to minimising quadratic 'dissonance' with a
pattern of interval use that is the square of a sinusoid, which is
more unequal and strongly peaked around the 'central' keys.

I think not squaring the weights is algebraically simpler and has more
intuitive results.

If one has a quantity
(w_1 t_1^2 + w_2 t_2^2)
and minimises it with the constraint that t_1 + t_2 is a constant, one
gets
t_2 * w_2 = t_2 * w_1
- the relation between weights and results is simple.

If one is balancing three thirds against each other, minimising
sum_i w_i t_i^2
subject to the constraint sum_i t_i = diesis
then the solution has
t_1 : t_2 : t_3 = w_2*w_3 : w_3*w_1 : w_1*w_2
the product of the deviation with the weight is still a constant.

Anyway, I will fire up the prog and square the weights to check what
happens. Perhaps it will be another of my failures, what do you think?

~~~T~~~