Yahoo Tuning Groups Ultimate Backup tuning 'Total error' in well-temps, new scoring proposal

I don't know the reasoning behind George's measure of how 'good' a
major triad sounds (total of the deviations of fifth and third from
pure) but it has some curious features.

If we restrict ourselves to 'well'-temperaments that have no wide
fifth or flat thirds, the total error just measures the minor third.
It's independent of the pitch of the root.

Proof: let the pitches be R, T and F in cents. The assumption is that
the fifth is flat and the third sharp. Then the fifth error is (pure
5th) - (F-R) and the third error is (T-R) - (pure M3).

The total error is (pure m3) - (F-T) : the root drops out!

Also, the total error treats the deviation in the 5th on an equal
footing to the M3. This seems to imply that a fifth which is flat by
(say) half a comma is equally as bad as a third that is sharp by half
a comma.

Of course, this depends on personal taste, but the history of
circulating temps. (e.g. Werckmeister 1691, Neidhardt) suggests that
the flattest only-just-tolerable fifth was about 1/3 or 1/4 comma
whereas the sharpest tolerable third was about Pythagorean. So, in my
opinion, a given error in the fifth ought to be counted as about 3 or
4 times worse than the same error in the third.

One measure that does this is the deviation from the meantone relation
T = (Pythagorean M3) - 4(pure 5th-F).

That is, I propose a score equal to

T - (Pythagorean M3) + 4(pure 5th-F)

which can also be written as

(sharpness of M3) - SC + 4(flatness of 5th).

Any regular tuning has score 0. Positive score means that the third is
'worse than expected' from a regular tuning based on the fifth,
negative score means the third is 'better than expected'. In my
experience positive score means the triad sounds 'tenser' and negative
means more 'relaxed'. Typically sharp keys have positive score and
flat keys have negative.

For example D major in Werckmeister III has M3 sharp by 1/2 comma and
5 flat by 1/4 comma (fudging it to the syntonic comma). Then the score is

SC/2 - SC + 4*(SC/4) or +SC/2

so Dmaj is noticeably more tense than in any meantone. +SC/2 is about
my limit of toleration.

Of course this 'score' is not the whole story, and I also impose
limits that the M3 should be below Pythagorean, and the 5th should not
be flatter than 1/4 comma.

Incidentally George was wondering why I would want to improve (the
third of) F#maj at the expense of Dmaj, or Abmaj at the expense of
Emaj. The answer is that I want a real circulating temperament in
which *every* key sounds pretty good even on the overtone-rich
harpsichord. I've had a lot of practice in designing these, but not
tried to introduce proportional beating until now.

~~~T~~~

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> I don't know the reasoning behind George's measure of how 'good' a
> major triad sounds (total of the deviations of fifth and third from
> pure) but it has some curious features.
>
> If we restrict ourselves to 'well'-temperaments that have no wide
> fifth or flat thirds, the total error just measures the minor third.
> It's independent of the pitch of the root.
>
> Proof: let the pitches be R, T and F in cents. The assumption is
that
> the fifth is flat and the third sharp. Then the fifth error is (pure
> 5th) - (F-R) and the third error is (T-R) - (pure M3).
>
> The total error is (pure m3) - (F-T) : the root drops out!
>
> Also, the total error treats the deviation in the 5th on an equal
> footing to the M3. This seems to imply that a fifth which is flat by
> (say) half a comma is equally as bad as a third that is sharp by
half
> a comma.
>
> Of course, this depends on personal taste, but the history of
> circulating temps. (e.g. Werckmeister 1691, Neidhardt) suggests that
> the flattest only-just-tolerable fifth was about 1/3 or 1/4 comma
> whereas the sharpest tolerable third was about Pythagorean. So, in
my
> opinion, a given error in the fifth ought to be counted as about 3
or
> 4 times worse than the same error in the third.

I suggest that this is false reasoning. Perhaps flatter (or sharper)
fifths would have been used but for their disutility when it comes to
meantone-type and closed-12-note tuning systems and the overall
quality of their thirds. The fifth alone isn't the whole story.

I agree that the fifth is more sensitive to mistuning, but I don't
think it's that many times more sensitive. The historical sample is
significantly constrained because of the basically-meantone, and
later closed-12-note, character of Western music since the
Renaissance.

There are many non-meantone (non-Western, you could say) where the
fifth is detuned quite a bit more yet still sounds great. Have you
listened to Blackwood's Suite for 15-equal Guitar?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>

Please re-read my definition:

<< A triad's total absolute error is the sum of the absolute values
of the errors in the three intervals of the triad. >>

There are *three* intervals in a triad: 5th, major 3rd, and *minor*
3rd (or inversions of these, depending on the position and spacing of
the triad). The total absolute error will therefore be double the
absolute error of the interval having the greatest absolute error; in
a well-temperament (no 5ths or minor 3rds wider than just or major
3rds narrower than just) this will be twice the absolute error of the
minor 3rd.

If you didn't count the minor 3rd in your analysis, you would
conclude that the best 4:5:6 in 22-equal is better than that in 19-
equal, whereas a listening test demonstrates the reverse to be true.

> ...
> Also, the total error treats the deviation in the 5th on an equal
> footing to the M3. This seems to imply that a fifth which is flat by
> (say) half a comma is equally as bad as a third that is sharp by
half
> a comma.
>
> Of course, this depends on personal taste, but the history of
> circulating temps. (e.g. Werckmeister 1691, Neidhardt) suggests that
> the flattest only-just-tolerable fifth was about 1/3 or 1/4 comma
> whereas the sharpest tolerable third was about Pythagorean. So, in
my
> opinion, a given error in the fifth ought to be counted as about 3
or
> 4 times worse than the same error in the third.

Yes, your contention that the error of the 5th should be weighted
against that of the 3rd (or 3rds, as I would have it) is quite
valid. I used total absolute error because it's easier to calculate
and seems to be accurate enough to quantify the general impression
one has when listening to the triads.

Paul Erlich also presents a weighted approach to evaluating
temperaments (to include intervals containing higher primes) in his
(by-now famous, long-awaited, and soon-to-be-published, we
hope) "middle-path" paper -- I don't recall exactly how these are
weighted, but I hope Paul will chime in. (I see he already has.)

BTW, I've found 2/5-comma 5ths and major 3rds somewhat wider than
Pythagorean to be not only tolerable, but downright *exciting* under
certain circumstances (e.g., in passing, in other than the tonic
triad; but that's a matter of opinion, and a separate issue that
really doesn't contradict your point). Under these circumstances a
major 3rd as wide as 11:14 might be tolerable, or if (like Margo
Schulter) you're into medieval music (where 3rds and 6ths are not
regarded as being truly consonant) you might just find it your cup of
tea -- preferable to Pythagorean. (But I digress.)

> One measure that does this is the deviation from the meantone
relation
> T = (Pythagorean M3) - 4(pure 5th-F).
>
> That is, I propose a score equal to
>
> T - (Pythagorean M3) + 4(pure 5th-F)
>
> which can also be written as
>
> (sharpness of M3) - SC + 4(flatness of 5th).
>
> Any regular tuning has score 0.

I don't understand this. What do you mean by "regular tuning"? I
understand this (in the present context) to mean a tuning in which
all of the fifths in a chain are the same size. So Pythagorean, 12-
ET, and 1/n-comma meantone would all be regular tunings -- but a
triad from each one of these would have the same score?

> Positive score means that the third is
> 'worse than expected' from a regular tuning based on the fifth,
> negative score means the third is 'better than expected'. In my
> experience positive score means the triad sounds 'tenser' and
negative
> means more 'relaxed'. Typically sharp keys have positive score and
> flat keys have negative.
>
> For example D major in Werckmeister III has M3 sharp by 1/2 comma
and
> 5 flat by 1/4 comma (fudging it to the syntonic comma). Then the
score is
>
> SC/2 - SC + 4*(SC/4) or +SC/2
>
> so Dmaj is noticeably more tense than in any meantone. +SC/2 is
about
> my limit of toleration.
>
> Of course this 'score' is not the whole story, and I also impose
> limits that the M3 should be below Pythagorean, and the 5th should
not
> be flatter than 1/4 comma.

I gather that you're making this statement in reference to a low-
contrast well-temperament, in which any key may (arguably) serve as a
tonic, as opposed to something more divergent from 12-ET, such as a
_temperament ordinaire_, in which all triads are usable, but some are
(arguably) not too suitable as tonics. I say "arguably" because the
issue of how much temperament in a major third (or major triad) is
acceptable (as well as how much temperament is acceptable in the best
keys) is largely subjective and fluid. Two cases in point from my
own experience:

1) When I first tried 1/4-comma meantone temperament (over 40 years
ago) I put it on an electronic organ and played it that way for
several weeks. Upon putting the instrument back into 12-ET, I was
absolutely shocked by how dissonant and out-of-tune it sounded. It
took me a few days to recover to the point where it began to sound
even marginally acceptable, and after a short time, I set about
seeking an alternative (unequal) tuning that would give me some
respite from the constant restless mood I experienced thereafter. In
such a circumstance, you would not have been able to sell me on a
well-temperament with 5/23-comma in the narrowest 5ths; I tried
Valotti-Young and found it wanting, because it didn't deliver the
consonance of the (1/4-comma) meantone triads that I desired. I
eventually ended up designing my own temperament.

2) I tuned my own (acoustic) piano to my own #2 well-temperament, in
which the 5 worst major triads have total absolute error equal to
that of Pythagorean triads. At first I avoided playing anything in
any of those keys, but after a matter of weeks I became accustomed to
their sound, eventually valuing them for the "exciting" mood they
conveyed, in contrast to the calmer meantone-like sound of the most
consonant triads. I kept the piano tuned this way for over 5 years
(until I moved cross-country and sold the instrument).

I don't think one can safely assume that a low-contrast well-
temperament (in which the error of the worst triads is minimized)
will be acceptable to everyone, nor that a well-designed higher-
contrast _temperament ordinaire_ will not be acceptable to anyone in
the remote keys. It's largely a matter of conditioning.

> Incidentally George was wondering why I would want to improve (the
> third of) F#maj at the expense of Dmaj, or Abmaj at the expense of
> Emaj. The answer is that I want a real circulating temperament in
> which *every* key sounds pretty good even on the overtone-rich
> harpsichord.

That depends on what one means by "pretty" and "good", again, a
matter of opinion (see above :-). For me high contrast between the
best keys and 12-ET is "pretty" and "good", but for you that would be
low contrast between the worst keys and 12-ET.

I wasn't wondering *why* you would want to improve F# major at the
expense of D major. I was only giving reasons for *not doing so* in
my #1 temperament, to which I would add another: The F# and C# major
triads presently have an equal amount of total absolute error and B
and G#/Ab virtually the same amount; reducing the error of F# would
impair the balance in the progression of error around the circle of
5ths.

> I've had a lot of practice in designing these, but not
> tried to introduce proportional beating until now.

Same here. I'd say that it adds a completely different dimension to
the task!

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> > That is, I propose a score equal to
> >
> > T - (Pythagorean M3) + 4(pure 5th-F)
>
> Another objection -- this seems to disregard the purity of the minor
> third, which I've found can be a very important factor, all else being
> equal.

There are only two independent variables in a triad. How can you vary
the m3 if 'all else is equal'?

You can rewrite my expression in terms of the minor third and the
fifth if you really want to:

T = F - t

Pure m3 = pure 5th - Pythag. M3 + SC

The 'score' is then

(pure m3-t) - SC + 3(pure 5th-F)

which is the same as

(flatness of m3) + 3(flatness of 5th) - SC

Arithmetically you have the same thing, but it now looks like the
major third is being ignored! So the charge that my expression
neglects the m3 is, to coin a phrase, false reasoning.

What the expression *really* measures is the deviation of the whole
chord from a regular tuning. As I said, this is not the whole story,
but I find it useful for detecting potential problems. Also the sum of
the score over all 12 major triads is zero, which is an arithmetically
nice feature.

I don't think I have any 'false reasoning' in the context of examining
tunings meant to be useful for the post-1700 repertoire, to be played
on appropriate instruments. My principal reference is Page 2 of
Werckmeister 1691 where he says that the perfect consonances can bear
much less tempering than the imperfect ones - and even that the M3 can
bear less tempering than the m3 and M6.

And I did say that this score reflects *my own personal taste and
judgement*. You can invent your own score if you want.

George's use of the total absolute error also reflects his taste and
judgement. He says that most historical circulating temps are 'biased'
towards the flat keys - which, yes, is the case if you evaluate them
based only on the total error or the minor third.

But historically, starting with Werckmeister and Sorge, the *major*
third was the principal method of evaluating a temperament. Using this
method, most historical circular temps favour the sharps and flats
equally (Vallotti, Young I) or even favour the sharps (Young I,
ordinaire). This is what people then thought preferable or ideally
balanced, based on their methods of evaluation.

The only way to judge a temperament is aesthetically, which requires
input of individual taste. Any formulae for 'scoring' temps are only
ways of formalizing personal preferences. Although there are objective
*descriptions* of temperaments, there are no objective evaluations
which tell us beyond dispute that one is better than another.

I said immediately that my proposal has an element of personal taste.
So does George's total error, or any other formula. If you don't like
it, make your own.

~~~T~~~

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > Paul Erlich also presents a weighted approach to evaluating
> > temperaments (to include intervals containing higher primes) in
his
> > (by-now famous, long-awaited, and soon-to-be-published, we
> > hope) "middle-path" paper -- I don't recall exactly how these are
> > weighted, but I hope Paul will chime in. (I see he already has.)
>
> Since we are talking about an octave-equivalent context here, I'd
> jump slightly away from what's in the paper and say for that ratios
> of 3 (such as the perfect fifth and fourth) a given mistuning is
> about log(5)/log(3) = 1.465 as bad as it would be for ratios of 5
> (such as major and minor thirds and sixths). Judging by the
> concordance of tunings like 15-equal on classical guitar, and many
> other tuning systems I've tried, the multiplier of 4 that Tom
> suggested is clearly way too large.

I agree. One must also consider that the beat rates that occur with
a given number of cents of tempering are much faster for 3rds and
6ths than they are for 5ths and 4ths, even though the latter are
generally more prominent.

> When the errors get *really* large, like 50 cents, ratios of 3 are
> actually *more* permissive, since they don't fall into the fields
of
> attraction of other, equally simple or simpler ratios as a result
of
> the mistuning (while ratios of 5 do).

But I don't think you intended to suggest that errors of that
magnitude are within the "tolerable" range for traditional harmonic
music. :-)

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > > That is, I propose a score equal to
> > >
> > > T - (Pythagorean M3) + 4(pure 5th-F)
> >
> > Another objection -- this seems to disregard the purity of the
minor
> > third, which I've found can be a very important factor, all else
being
> > equal.
>
> There are only two independent variables in a triad. How can you
vary
> the m3 if 'all else is equal'?

We are talking about absolute errors, right? (You may have left out
absolute value symbols in your formula above). At least George Secor
was. So it's quite possible to have four different triads where the
major third and perfect fifth have the same *absolute* error, but the
minor third takes on three different values:

0 400 716
0 400 688
0 372 716
0 372 688

Two of these chords sound better than the others, despite the
equality of the absolute errors of the fifths and major thirds in all
four chords. Listen and see if you agree.

> You can rewrite my expression in terms of the minor third and the
> fifth if you really want to:
>
> T = F - t
>
> Pure m3 = pure 5th - Pythag. M3 + SC
>
> The 'score' is then
>
> (pure m3-t) - SC + 3(pure 5th-F)
>
> which is the same as
>
> (flatness of m3) + 3(flatness of 5th) - SC
>
> Arithmetically you have the same thing, but it now looks like the
> major third is being ignored! So the charge that my expression
> neglects the m3 is, to coin a phrase, false reasoning.

I thought you had implied absolute values in your expression, but I
guess you didn't! My mistake. Clearly, then, this measure begins to
fail as a useful score when the 'flatness' values become negative
(because these intervals are sharp). Right?

> What the expression *really* measures is the deviation of the whole
> chord from a regular tuning.

How are you defining "regular tuning?" Others here may have different
definitions of that.

> As I said, this is not the whole story,
> but I find it useful for detecting potential problems. Also the sum
of
> the score over all 12 major triads is zero, which is an
arithmetically
> nice feature.
>
> I don't think I have any 'false reasoning' in the context of
examining
> tunings meant to be useful for the post-1700 repertoire, to be
played
> on appropriate instruments. My principal reference is Page 2 of
> Werckmeister 1691 where he says that the perfect consonances can
bear
> much less tempering than the imperfect ones - and even that the M3
can
> bear less tempering than the m3 and M6.

Historically, advocates of 12-note circulating temperaments generally
seem to make claims like this, but to my mind this is little more
than self-justification or even advertising. You'll find a whole
range of remarks on this subject, differing radically from writer to
writer and according to time period. Jorgenson is a good reference
for this sort of thing. Robert Smith, for example, says something
quite the opposite to this. I'm just trying to bring the issue up to
date in the context of a more general microtonal endeavor. I probably
overstepped the bounds of this discussion in doing so. I apologize.
Though if I got anyone to listen to sounds or music they otherwise
wouldn't have listened to, I have to say it was worth it.

But Tom, a more general formula for scoring triads can always be put
to use in the context with constraints on the tunings such as those
that meantone or closed 12-note systems, or or any of the other 'non-
traditional' systems, impose. So there's no need, in my mind, to
develop 'specialized' formulas that only work in limited contexts.
George Secor proposed a formula that could work in any context, and
it's not all that difficult to compute. If it ranks triads
differently than your ears would, then you should let us know where
it fails.

Best,
Paul

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> >
> > > Paul Erlich also presents a weighted approach to evaluating
> > > temperaments (to include intervals containing higher primes) in
> his
> > > (by-now famous, long-awaited, and soon-to-be-published, we
> > > hope) "middle-path" paper -- I don't recall exactly how these
are
> > > weighted, but I hope Paul will chime in. (I see he already
has.)
> >
> > Since we are talking about an octave-equivalent context here, I'd
> > jump slightly away from what's in the paper and say for that
ratios
> > of 3 (such as the perfect fifth and fourth) a given mistuning is
> > about log(5)/log(3) = 1.465 as bad as it would be for ratios of 5
> > (such as major and minor thirds and sixths). Judging by the
> > concordance of tunings like 15-equal on classical guitar, and
many
> > other tuning systems I've tried, the multiplier of 4 that Tom
> > suggested is clearly way too large.
>
> I agree. One must also consider that the beat rates that occur
with
> a given number of cents of tempering are much faster for 3rds and
> 6ths than they are for 5ths and 4ths, even though the latter are
> generally more prominent.

Right.

> > When the errors get *really* large, like 50 cents, ratios of 3
are
> > actually *more* permissive, since they don't fall into the fields
> of
> > attraction of other, equally simple or simpler ratios as a result
> of
> > the mistuning (while ratios of 5 do).
>
> But I don't think you intended to suggest that errors of that
> magnitude are within the "tolerable" range for traditional harmonic
> music. :-)

Not normally, but with fast-moving music in certain timbres where
pitch is difficult to clearly discern such as pure sine waves, or
with some inharmonic timbres, they sometimes can be (as a few list
members have been brave enough to admit).

🔗Carl Lumma <clumma@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > I agree that the fifth is more sensitive to mistuning, but I
> > don't think it's that many times more sensitive.
>
> Would you say it's 1.6719500161730103 times more sensitive?

I don't know, maybe, this is kind of close to the number I gave.

> > The historical sample is significantly constrained because of
> > the basically-meantone, and later closed-12-note, character
> > of Western music since the Renaissance.
>
> True, though the popularity of 1/4-comma meantone perhaps
> suggests thirds damage might be weighted over fifths damage.

Not at all. Under many criteria that weight the fifth damage more
(whether you measure 'damage' in straight cents or in one of the
other senses I've defined), 1/4-comma still shows up the as optimal
variety of meantone.

> > There are many non-meantone (non-Western, you could say) where
> > the fifth is detuned quite a bit more yet still sounds great.
> > Have you listened to Blackwood's Suite for 15-equal Guitar?
>
> I love that piece, but I wouldn't say the fifths sound great.
> More like severely disturbing (though this quality does not
> leak into the music's gestalt).

Hmm . . . well, I'd say that the *triads* in the piece don't sound
disturbing, and triads is what we were discussing here. If fifth
error got several times the weight of third/sixth error, the 15-equal
triad should sound several times more disturbing than the 12-equal
triad. To my ears, it doesn't, unless I've saturated my ears with
nothing but 12-equal music for a long period of time.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > You can rewrite my expression (...)
> > The 'score' is then
> >
> > (pure m3-t) - SC + 3(pure 5th-F)
> >
> > which is the same as
> >
> > (flatness of m3) + 3(flatness of 5th) - SC
> >
> > Arithmetically you have the same thing, but it now looks like the
> > major third is being ignored! (...)
>
> (...) this measure begins to
> fail as a useful score when the 'flatness' values become negative
> (because these intervals are sharp). Right?

Not necessarily! You can use the formula for *any* major triad, be it
ordinaire or 2/7 comma or whatever. Whether it gives you results that
are useful is up to your personal taste.

Of course if the m3 becomes sharp, all bets are off... but at that
point you've already made a circular temp. impossible.

But if the M3 is pretty sharp and the fifth is a bit sharp, the chord
can sound (to me) *better* than if the fifth were a bit flat or pure.
The score then decreases, reflecting my judgement that a slightly
sharp fifth can make a triad sound *less* tense if the third is quite
sharp.

I suspect this may correspond with the use of some temperament
ordinaires in flat keys.

> > What the expression *really* measures is the deviation of the whole
> > chord from a regular tuning.
>
> How are you defining "regular tuning?" Others here may have different
> definitions of that.

Really? I would say 'any meantone or Pythagorean'.

> > I don't think I have any 'false reasoning' in the context of
> examining
> > tunings meant to be useful for the post-1700 repertoire, to be
> played
> > on appropriate instruments. My principal reference is Page 2 of
> > Werckmeister 1691 where he says that the perfect consonances can
> bear
> > much less tempering than the imperfect ones - and even that the M3
> can
> > bear less tempering than the m3 and M6.
>
> Historically, advocates of 12-note circulating temperaments generally
> seem to make claims like this, but to my mind this is little more
> than self-justification or even advertising.

That's a pretty low view of their, and my, motivation. How would you
know that Werckmeister and others say weren't actually writing from
their personal taste and judgement, rather than a desire for material
gain and glory? The fact that it differs from your taste and judgement
doesn't make it any less valid. I happen to like 12-note circulating
temperaments. De gustibus.

> You'll find a whole
> range of remarks on this subject, differing radically from writer to
> writer and according to time period. Jorgenson is a good reference
> for this sort of thing. Robert Smith, for example, says something
> quite the opposite to this.

No practical musician took up Smith's 'equal harmony' tuning. By
contrast, Werckmeister became notorious within a generation. The
musicians of the day will judge whether a tuning meets their needs or
not.

> I'm just trying to bring the issue up to
> date in the context of a more general microtonal endeavor.

Well, I admit my score wouldn't be very useful for things which are
not 12-note temps.

> But Tom, a more general formula for scoring triads can always be put
> to use in the context with constraints on the tunings such as those
> that meantone or closed 12-note systems, or or any of the other 'non-
> traditional' systems, impose. So there's no need, in my mind, to
> develop 'specialized' formulas that only work in limited contexts.
> George Secor proposed a formula that could work in any context, and
> it's not all that difficult to compute. If it ranks triads
> differently than your ears would, then you should let us know where
> it fails.

You seem to be saying that unless I'm claiming that George's measure
is wrong, I should shut up. No. The triad has two dimensions and *any*
measure (be it George's or mine) that reduces it to a single number
misses one of them out. So we *need* more than one measure. Also, what
sounds better than what depends on the harmonic content of the
instrument.

George's measures 'purity vs. impurity of the worst interval': mine
measures (IMO) 'tenser vs. more relaxed'. You can invent other scores
to measure 'sharpness vs. flatness' or whatever, but they'll only be
linear combinations of two things.

Actually, in well-temps, comparisons based on my score are really
quite similar to those based on the total abs. error. I just place a
little more importance on the fifth. Perhaps not a coincidence that my
harpsichord has strong 3rd harmonic content...

If you really want me to disagree with the total absolute error, here
goes. I might prefer

0 406 702

0 402 698.

~~~T~~~

🔗Carl Lumma <clumma@yahoo.com>

> > > I agree that the fifth is more sensitive to mistuning, but I
> > > don't think it's that many times more sensitive.
> >
> > Would you say it's 1.6719500161730103 times more sensitive?
>
> I don't know, maybe, this is kind of close to the number I gave.

Just saw you newer post. I agree that the octave-equivalent
measure is better here than the Tenney-height one I used above.

> > > The historical sample is significantly constrained because of
> > > the basically-meantone, and later closed-12-note, character
> > > of Western music since the Renaissance.
> >
> > True, though the popularity of 1/4-comma meantone perhaps
> > suggests thirds damage might be weighted over fifths damage.
>
> Not at all. Under many criteria that weight the fifth damage more
> (whether you measure 'damage' in straight cents or in one of the
> other senses I've defined), 1/4-comma still shows up the as
> optimal variety of meantone.

Using the factor you just posted I get a 5th of roughly
697.974 cents as the optimal meantone. Isn't this right?

> > > There are many non-meantone (non-Western, you could say) where
> > > the fifth is detuned quite a bit more yet still sounds great.
> > > Have you listened to Blackwood's Suite for 15-equal Guitar?
> >
> > I love that piece, but I wouldn't say the fifths sound great.
> > More like severely disturbing (though this quality does not
> > leak into the music's gestalt).
>
> Hmm . . . well, I'd say that the *triads* in the piece don't sound
> disturbing, and triads is what we were discussing here. If fifth
> error got several times the weight of third/sixth error, the
> 15-equal triad should sound several times more disturbing than
> the 12-equal triad. To my ears, it doesn't, unless I've saturated
> my ears with nothing but 12-equal music for a long period of time.

The triads of 15-equal sound significantly worse to my ears than
those of 12 equal, even if I've listened to the former for a long
period of time. I don't know how many times worse, but certainly
worse.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> > (...) this measure begins to
> > fail as a useful score when the 'flatness' values become negative
> > (because these intervals are sharp). Right?
>
> Not necessarily! You can use the formula for *any* major triad, be
it
> ordinaire or 2/7 comma or whatever. Whether it gives you results
that
> are useful is up to your personal taste.
>
> Of course if the m3 becomes sharp, all bets are off...

Well that's what I meant by 'fail' above. What did you mean by 'not
necessarily'?

> but at that
> point you've already made a circular temp. impossible.

That's a separate discussion, one that George Secor may wish to chime
in about, but not at all my concern here. I'm speaking of a way of
evaluating the approximate 4:5:6 triads regardless of what kind of
temperament you're constrained to work in.

> But if the M3 is pretty sharp and the fifth is a bit sharp, the
chord
> can sound (to me) *better* than if the fifth were a bit flat or
>pure.

Exactly!

> The score then decreases, reflecting my judgement that a slightly
> sharp fifth can make a triad sound *less* tense if the third is
quite
> sharp.

Good. So your score has some of the properties a general score should
have, but falls apart in certain (non-typical for Western tunings)
situations. Is this a fair statement?

> I suspect this may correspond with the use of some temperament
> ordinaires in flat keys.

I don't get it, but I'll keep my eyes peeled for an explanation.

> > > What the expression *really* measures is the deviation of the
whole
> > > chord from a regular tuning.
> >
> > How are you defining "regular tuning?" Others here may have
different
> > definitions of that.
>
> Really? I would say 'any meantone or Pythagorean'.

Gene uses a quite different definition. (Extended) JI, and any
temperament (or tuning, for that matter) with uniform interval sizes,
are regular to Gene.

> Well, I admit my score wouldn't be very useful for things which are
> not 12-note temps.

But I see no reason why one couldn't use a general scoring procedure
to evaluate *any* ~4:5:6 triads, and then apply this to both 12-note
and other temperaments. Conceptually, this seems a lot cleaner and
more satisfying than having the temperament constraints mixed up in
the scoring to begin with.

> > But Tom, a more general formula for scoring triads can always be
put
> > to use in the context with constraints on the tunings such as
those
> > that meantone or closed 12-note systems, or or any of the
other 'non-
> > traditional' systems, impose. So there's no need, in my mind, to
> > develop 'specialized' formulas that only work in limited
contexts.
> > George Secor proposed a formula that could work in any context,
and
> > it's not all that difficult to compute. If it ranks triads
> > differently than your ears would, then you should let us know
where
> > it fails.
>
> You seem to be saying that unless I'm claiming that George's measure
> is wrong, I should shut up. No. The triad has two dimensions and
*any*
> measure (be it George's or mine) that reduces it to a single number
> misses one of them out. So we *need* more than one measure.

If you're trying to find some sort of 'optimum', it seems that
ultimately you have to reduce it down to one measure,
otherwise 'optimum' can't even be defined.

So what are you going to do with your more than one measure once you
have them?

> Also, what
> sounds better than what depends on the harmonic content of the
> instrument.

OK.

> George's measures 'purity vs. impurity of the worst interval':

Worst interval? I thought it was the sum of all three(?)

> mine
> measures (IMO) 'tenser vs. more relaxed'. You can invent other
scores
> to measure 'sharpness vs. flatness' or whatever, but they'll only be
> linear combinations of two things.

George's measure is *not* a linear combination of two things. The
absolute value function is highly nonlinear. Perhaps you meant that
it can be expressed as a function of two things, which is true.
There's a 2-dimensional space in which 4:5:6 triads can be detuned. I
usually diagram this with 3 axes within this space, at 120-degree
angles from one another, corresponding to the three intervals. In
this way, the constraint that one interval is the sum of the other
two is automatically taken care of by the geometry.

> Actually, in well-temps, comparisons based on my score are really
> quite similar to those based on the total abs. error. I just place a
> little more importance on the fifth. Perhaps not a coincidence that
my
> harpsichord has strong 3rd harmonic content...
>
> If you really want me to disagree with the total absolute error,
here
> goes. I might prefer
>
> 0 406 702
>
> to
>
> 0 402 698.

Well, me too. How about total absolute error of the three intervals
where the weight on the fifth is 1.465 times greater than the weight
on each of the two thirds? Or what if you *square* the three errors
(weighted or unweighted) before adding them?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

> > > > > I agree that the fifth is more sensitive to mistuning, but I
> > > > > don't think it's that many times more sensitive.
> > > >
> > > > Would you say it's 1.6719500161730103 times more sensitive?
> > >
> > > I don't know, maybe, this is kind of close to the number I gave.
> >
> > Just saw you newer post. I agree that the octave-equivalent
> > measure is better here than the Tenney-height one I used above.
>
> The Tenney height (a term Gene invented) of 5:4 is 3.333333333
> times that of 3:2. Maybe you meant Tenney Harmonic Distance?

Oh for Chr... sake, how do you calculate that, then?

> > > >the popularity of 1/4-comma meantone perhaps
> > > >suggests thirds damage might be weighted over fifths damage.
> > >
> > > Not at all. Under many criteria that weight the fifth damage
> > > more (whether you measure 'damage' in straight cents or in
> > > one of the other senses I've defined), 1/4-comma still shows
> > > up the as optimal variety of meantone.
> >
> > Using the factor you just posted I get a 5th of roughly
> > 697.974 cents as the optimal meantone. Isn't this right?
>
> I need to know which exponent (or value of p) you're using, but
> I don't think this is right for any of them, unless you're
> completely ignoring the major sixth / minor third or something.

p=1, and yes I ignored the 5&3 intervals since I was trying
to establish that 1/4-comma meantone favors the major thirds
at the expense of the fifths. But maybe it really is the
5-limit optimal meantone... Gene's last effort on tuning-math
had it as Kees meantone... though it looks impossible to have
all ratios of 5 mistuned 1.465 times as badly as all ratios
of 3 in the conventional meantone map.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > > > I agree that the fifth is more sensitive to mistuning,
but I
> > > > > > don't think it's that many times more sensitive.
> > > > >
> > > > > Would you say it's 1.6719500161730103 times more sensitive?
> > > >
> > > > I don't know, maybe, this is kind of close to the number I
gave.
> > >
> > > Just saw you newer post. I agree that the octave-equivalent
> > > measure is better here than the Tenney-height one I used above.
> >
> > The Tenney height (a term Gene invented) of 5:4 is 3.333333333
> > times that of 3:2. Maybe you meant Tenney Harmonic Distance?
>
> Oh for Chr... sake, how do you calculate that, then?

Tenney Harmonic Distance of a ratio n/d in lowest terms is log(n*d).

(The base you use for the logarithm is irrelevant if you're computing
ratios of Tenney HDs, but in general a lot of people like to use bas
2.)

> > > > >the popularity of 1/4-comma meantone perhaps
> > > > >suggests thirds damage might be weighted over fifths damage.
> > > >
> > > > Not at all. Under many criteria that weight the fifth damage
> > > > more (whether you measure 'damage' in straight cents or in
> > > > one of the other senses I've defined), 1/4-comma still shows
> > > > up the as optimal variety of meantone.
> > >
> > > Using the factor you just posted I get a 5th of roughly
> > > 697.974 cents as the optimal meantone. Isn't this right?
> >
> > I need to know which exponent (or value of p) you're using, but
> > I don't think this is right for any of them, unless you're
> > completely ignoring the major sixth / minor third or something.
>
> p=1,

This means sum-absolute error is what's being minimized.

> and yes I ignored the 5&3 intervals since I was trying
> to establish that 1/4-comma meantone favors the major thirds
> at the expense of the fifths. But maybe it really is the
> 5-limit optimal meantone... Gene's last effort on tuning-math
> had it as Kees meantone... though it looks impossible to have
> all ratios of 5 mistuned 1.465 times as badly as all ratios
> of 3 in the conventional meantone map.

Right.

You may or not be familiar with this table:

Anyway, this shows that for p=1, 1/4-comma meantone is optimal for a
wide range of weightings, wide enough to include what we're talking
about here. It's for other values of p where other meantones show up
as optimal. The first two columns in this table correspond to
p=infinity and p=2, respectively.

🔗Carl Lumma <clumma@yahoo.com>

>>>>I agree that the fifth is more sensitive to mistuning,
>>>>but I don't think it's that many times more sensitive.
>>>
>>>Would you say it's 1.6719500161730103 times more sensitive?
//
>>>Just saw you newer post. I agree that the octave-equivalent
>>>measure is better here than the Tenney-height one I used above.
>>
>>The Tenney height (a term Gene invented) of 5:4 is 3.333333333
>>times that of 3:2. Maybe you meant Tenney Harmonic Distance?
>
>Oh for Chr... sake, how do you calculate that, then?
>
> Tenney Harmonic Distance of a ratio n/d in lowest terms is
> log(n*d).
>
> (The base you use for the logarithm is irrelevant if you're
> computing ratios of Tenney HDs, but in general a lot of people
> like to use bas 2.)

Yes I know, that's what I used (and I suppose I'm like you).
What's Tenney Height?

> > > > > > >the popularity of 1/4-comma meantone perhaps
> > > > > > >suggests thirds damage might be weighted over fifths
> > > > > > >damage.
> > > > >
> > > > > Under many criteria that weight the fifth damage
> > > > > more (whether you measure 'damage' in straight cents or
> > > > > in one of the other senses I've defined), 1/4-comma still
> > > > > shows up the as optimal variety of meantone.
> > > >
> > > > Using the factor you just posted I get a 5th of roughly
> > > > 697.974 cents as the optimal meantone. Isn't this right?
> > >
> > > I need to know which exponent (or value of p) you're using,
> > > but I don't think this is right for any of them, unless
> > > you're completely ignoring the major sixth / minor third
> > > or something.
> >
> > p=1,
>
> This means sum-absolute error is what's being minimized.

Yes.

I do remember it! Great table.

> Anyway, this shows that for p=1, 1/4-comma meantone is optimal
> for a wide range of weightings, wide enough to include what
> we're talking about here. It's for other values of p where other
> meantones show up as optimal.

Or for M3 & P5 only, p=1.

> The first two columns in this
> table correspond to p=infinity and p=2, respectively.

Yes.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <clumma@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

You guys really should consider writing a joint theoretical book explaining all this new stuff that many cannot comprehend as yet.

Cordially,
Ozan

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 10 Eylül 2005 Cumartesi 8:58
Subject: [tuning] Re: 'Total error' in well-temps, new scoring proposal

> > Yes I know, that's what I used (and I suppose I'm like you).
> > What's Tenney Height?
>
> I called n*d "Tenney Height" because it is an example of what number
> theorists call a "height".

That's fine, I just didn't remember that it lacked a log while
Tennery Harmonic Distance had one.

-Carl

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🔗Tom Dent <stringph@gmail.com>

I meant the *fifth* can be sharp and the score will still make some
sense (to me at least). If the m3 is sharp or the M3 flat, yes, I
wouldn't know what to make of it. But you can still calculate it and
see if it sense to you. Maybe it does, just that I have little
experience in these types of tuning.

> > The score then decreases, reflecting my judgement that a slightly
> > sharp fifth can make a triad sound *less* tense if the third is
> > quite sharp.
>
> Good. So your score has some of the properties a general score should
> have, but falls apart in certain (non-typical for Western tunings)
> situations. Is this a fair statement?

Probably. I invented it as an aid to constructing 12-note circ. temps.
Although, you *can* always calculate it for a 4-5-6 chord just as

(T - pure M3) + 4(pure 5th - F) - SC

and it *will* always tell you how much the third is sharp relative to
a regular tuning based on the given fifth. Whether this is directly
useful depends on whether you are dealing with tunings where the 3rd
is produced by a chain of 3 or 4 comparable fifths and where there is
key-colour.

> > I suspect this may correspond with the use of some temperament
> > ordinaires in flat keys.
>
> I don't get it, but I'll keep my eyes peeled for an explanation.

Meaning... some flat keys in temperaments ordinaires have pretty large
major thirds, which ought to sound bad but aren't so terrible when
combined with the sharper fifths.

> (...) having the temperament constraints mixed up in
> the scoring to begin with.

As I said, the score can be *calculated* for any triad (one can
generalise it to ~12:15:18 too), but its musical usefulness varies in
context. I don't impose any constraints on the calculation itself.

> The triad has two dimensions (...) So we *need* more than one measure.
>
> If you're trying to find some sort of 'optimum', it seems that
> ultimately you have to reduce it down to one measure,
> otherwise 'optimum' can't even be defined.

Ah, but who said that circular (or any irregular) temps. were about
finding a single optimum triad? Unless the optimum is 12tet, you can't
create a 12-note circ. temp and stay at or even very near an optimum.
Irregular temps are precisely about exploiting various different
triads which however all sound reasonably good.

> So what are you going to do with your more than one measure once you
> have them?

Make circ. temps! The main characteristic feature I see of many circ.
temps is that they consist of chunks of *different* regular tunings
split up and stuck together. For example Young I is one chunk each of
1/6 PC and Pythag. plus two little chunks of 12tet. The obvious effect
this has is making the variations of sizes of thirds. So that is my
first score. The M3 and m3 start near pure around C maj and get
progressively worse on either side.

This makes sense to me since often in 18th-19th century music one has
just thirds/6ths without the fifth. So first we require that all
thirds sound OK. But that only gets you so far: it doesn't see the
fifth, and it doesn't allow you to tell the difference between
intonation on the 'sharp' side where the fifths are getting sharper
going round the circle (clockwise) and the 'flat' side where they are
getting flatter.

So I have another score that tells me how adding the fifth affects the
sound. Also, it can differentiate between intonation on the sharp side
and the flat side.

Now, we come to the central point. Circ temps to me are about creating
the widest *variety* of tunings between different keys, but without
any of them being obtrusively out of tune - 'obtrusive' depending on
taste... This means that you take the third-quality from as near pure
as possible (given constraints) to as far from pure as you can take.
*Then* you also try to maximise the distinction between sharp and flat
keys with similar third-sizes. This you can measure with the new
score. However, one should also impose a *limit* on the new score
since triads with a too sharp third and flat fifth are likely to sound
harsh. Within the limits again you go from the minimum to the maximum.

So, you create a trajectory within your preferred allowed region to
enclose the maximum area in the 2d graph and you have a circ. temp
with good-enough intonation all round the circle and the maximum
amount of difference between keys.

> > George's measures 'purity vs. impurity of the worst interval':
>
> Worst interval? I thought it was the sum of all three(?)

I still think that when you add up all three you get twice the worst one.

> > I might prefer
> >
> > 0 406 702
> >
> > to
> >
> > 0 402 698.
>
> Well, me too. How about total absolute error of the three intervals
> where the weight on the fifth is 1.465 times greater than the weight
> on each of the two thirds? Or what if you *square* the three errors
> (weighted or unweighted) before adding them?

All interesting - particularly the business of squaring errors, which
would tend to pull away from say 1/6 comma and towards tunings with
either pure 5ths or nearly-pure 3rds. What would happen with an
evolutionary algorithm applying such a score to an initially random
circ. temp.?? You might get something like Kirnberger 'III' or
'Broadwood Best'...

~~~T~~~

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> > All interesting - particularly the business of squaring errors, which
> > would tend to pull away from say 1/6 comma and towards tunings with
> > either pure 5ths or nearly-pure 3rds.
>
> I don't know what you mean by that -- here's a graph you might be
> interested in:
>
> /tuning/files/perlich/pop.gif
>
> Squaring corresponds to p=2 on the horizontal axis, and there's a
> purple line on the graph above the numeral '2' which represents it. So
> in a sense, it appears that squaring unequivocally pulls one *away*
> from pure fifths relative to either straight sums (p=1) or maxima
> (p=infinity) for most of the weighting schemes we'd be interested in,
> which would fall between the green line and the blue line . . .

Can you say explicitly what the different coloured lines (ie weighting
schemes) are?

Actually I should have remembered one needs higher powers than 2 to
make a function with more than one minimum. We would need fourth power
at least to get a total weight that 'dipped' around both pure 3rds and
pure 5ths. That would have at least nominal historical motivation in
that the most longlived tunings over history have been Pythagorean and
1/4 comma.

In order to motivate an irregular circ temp by purely formal means,
one would need a score that disfavours 12et relative to both tunings
with purer 3rds and those with purer 5ths. Hence at least a
double-dip. Otherwise 12tet will always come out 'best' given a
constraint of circularity.

That is, unless we introduce yet another score which favours
irregularity - which would require scoring over the whole temperament,
not just triads!!

Or should one weight different keys in the temp by different amounts
reflecting frequency of use (eg Barnes numbers)?? ... which would
automatically introduce inequality.

~~~T~~~

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> > > (...) this measure begins to
> > > fail as a useful score when the 'flatness' values become
negative
> > > (because these intervals are sharp). Right?
> >
> > Not necessarily! You can use the formula for *any* major triad,
be it
> > ordinaire or 2/7 comma or whatever. Whether it gives you results
that
> > are useful is up to your personal taste.
> >
> > Of course if the m3 becomes sharp, all bets are off...
>
> Well that's what I meant by 'fail' above. What did you mean by 'not
> necessarily'?
>
> > but at that
> > point you've already made a circular temp. impossible.
>
> That's a separate discussion, one that George Secor may wish to
chime
> in about, but not at all my concern here.

At that point you've greatly reduced your chances of coming up with a
decent circulating *12-tone* temperament, but if you're seeking a 19-
tone well-temperament, a wide minor 3rd in some of the triads is
virtually inevitable (see following message).

Sorry that it's taken me a few days to reply. I spend my weekends
away from the Internet, and by the time I got back to reading this
thread, it left me in the dust.

> I'm speaking of a way of
> evaluating the approximate 4:5:6 triads regardless of what kind of
> temperament you're constrained to work in.
>
> > But if the M3 is pretty sharp and the fifth is a bit sharp, the
chord
> > can sound (to me) *better* than if the fifth were a bit flat or
> >pure.

If "pretty sharp" means "wider than Pythagorean", then I would expect
the chord to sound better with a slightly wide 5th rather than a
narrow 5th, because that would reduce the total absolute error. As
for situations (such as pure 5th vs. wide 5th in your example) where
the total absolute error between the two triads is the same, there
will certainly be a difference in the sound -- and where such
difference is audible you will probably prefer one or the other.

I'll leave it to you to develop criteria to cover this -- my
intention was only to put up some numbers that make a general
(objective) comparison between individual triads within tunings. If
there's a big difference in the total absolute error of two triads,
then I think you can safely say that one will sound more consonant
than the other, but where the numbers are close, then it's advisable
to do a listening test.

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Hi Tom, talking to you is a pleasure!

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > > All interesting - particularly the business of squaring errors,
which
> > > would tend to pull away from say 1/6 comma and towards tunings
with
> > > either pure 5ths or nearly-pure 3rds.
> >
> > I don't know what you mean by that -- here's a graph you might be
> > interested in:
> >
> > /tuning/files/perlich/pop.gif
> >
> > Squaring corresponds to p=2 on the horizontal axis, and there's a
> > purple line on the graph above the numeral '2' which represents
it. So
> > in a sense, it appears that squaring unequivocally pulls one
*away*
> > from pure fifths relative to either straight sums (p=1) or maxima
> > (p=infinity) for most of the weighting schemes we'd be interested
in,
> > which would fall between the green line and the blue line . . .
>
> Can you say explicitly what the different coloured lines (ie
weighting
> schemes) are?

The colored lines show the size of the fifth in the meantone tuning
that minimizes:

blue: 5*(error in 4:3)^p + 3*(error in 5:3)^p + 3*(error in 5:4)^p
green: (error in 4:3)^p + (error in 5:3)^p + (error in 5:4)^p
red: 3*(error in 4:3)^p + 5*(error in 5:3)^p + 5*(error in 5:4)^p
cyan: (error in 5:3)^p + (error in 5:4)^p

Here's another chart that displays some of the same information but
here it's specific values of p, rather than specific weighting
schemes, which get curves:

/tuning/files/perlich/woptimal.gif

Notice how, for p=1 (i.e., sum of absolute error), the optimal tuning
jumps discontinously from 1/4-comma meantone to Pythagorean when the
weight on the fifth/fourth becomes about 7 times as great as that on
the thirds/sixths.

> Actually I should have remembered one needs higher powers than 2 to
> make a function with more than one minimum.

All those powers are there on the chart, and I don't think any of
them have more than one minimum in this case.

> We would need fourth power
> at least to get a total weight that 'dipped' around both pure 3rds
and
> pure 5ths.

I'm skeptical, but you can give me the function you have in mind and
I can plot it.

> That would have at least nominal historical motivation in
> that the most longlived tunings over history have been Pythagorean
and
> 1/4 comma.

Pythagorean tuning was used in an era when thirds and sixths were not
considered stable consonances that one can resolve to and rest on,
but fourths and fifths were. Thirds and sixths that are "active" and
unstable, as they are in Pythagorean, are the most appropriate for
most Medieval music, and sacrificing the fifth to tune them more
purely wouldn't be appropriate. Margo Schulter agrees:

> Or should one weight different keys in the temp by different amounts
> reflecting frequency of use (eg Barnes numbers)?? ... which would
> automatically introduce inequality.

Right -- that makes sense to me. Getting back to the original point,
though, George's method of rating (using a rating system that can
apply to *any* tuning system) each of the triads in a well-
temperament is a great way of seeing, at a glance, whether all "keys"
are going to be usable, or if some of them would have a tonic triad
too unstable for more than passing use.

Best,
Paul