optimal tuning for diatonic scale

Assume all notes form a single chain of identical fifths, so you don't need two versions for the second note of the major scale, and so you can change keys without multiple versions of notes. Intervallically, the three 5-limit consonances should all be considered, and each should be evaluated in terms of its field of attraction (Harry Partch's Observation One: the field of attraction is inversely proportional to the limit of the interval). The worst error in some previously recommended tunings:

Tuning Worst Error (cents*limit)
50-Equal 18.2
2/7-comma meantone 18.4
Kornerup's Golden 21.4
LucyTuning 21.7
1/4-comma meantone 26.9
31-equal 29.8
1/3- & 2/9-comma meantone 35.8
19-equal 36.8
1/5-comma meantone 43.0
43-equal 43.3
55-equal 50.9
1/6-comma meantone 53.8
12-equal 78.2
26-equal 85.4
Pythagorean 107.5

The minimax principle tells us to find the tuning with the smallest worst error. The answer is 5/18-comma meantone temperament*, where the fifths are flattened by 5.97 cents (5/18 of a syntonic comma) relative to Pythagorean. Then the major thirds are 2.39 cents sharp and the minor thirds are 3.58 cents flat relative to Just Intonation.

The worst errors (that of the fifth and minor third) both have cents*limit = 17.9 in 5/18-comma meantone. 50-equal is certainly close enough.

APPENDIX
*Here we find the tuning with the smallest worst error. It will be where the minor third, relative to its field of attraction, is tied with the fifth, relative to its field of attraction (the major third will be better), because then improving one of them will worsen the other. So 1/5 the error of the perfect fifth equals 1/3 the error of the minor third. Since the minor third is generated by 3 fifths, its error is a comma minus three times the error in the fifth, so we have 1/5 the error in the fifth equals 1/3 of (a comma minus 3 times the error in the fifth), or

x/5=1/3*(comma-3x)
x/5=1/3*comma-x
x*6/5=1/3*comma
x=5/18*comma

> (Harry Partch's Observation One: the field of attraction is inversely
proportional
> to the limit of the interval).

Unfortunately the tolerance for mistuning works the other way. By your
reasoning one
could temper the octave by an even larger amount than the fifth, but
clearly that will
sound awful.
Not that 5/18-comma is so bad but it's so-called "meaner than meantone"
(major third and
fifth both smaller than pure).

Manuel Op de Coul coul@ezh.nl

> From: manuel.op.de.coul@ezh.nl
>
> > (Harry Partch's Observation One: the field of attraction is inversely
> proportional
> > to the limit of the interval).
>
> Unfortunately the tolerance for mistuning works the other way.

If that's so, then the equation becomes

x/3=1/5*(comma-3x)
x/3=1/5*comma-3/5*x
x*(1/3+3/5)=1/5*comma
x*14/15=1/5*comma
x=3/14*comma

In 3/14-comma meantone temperament,
the fifths are 4.6 cents flat,
the major thirds are 3.1 cents sharp,
and the minor thirds are 7.7 cents flat.

The ranking changes to this:

Tuning Worst Error (cents/limit)
3/14-comma meantone 1.54
2/9-comma meantone 1.59
1/5-comma meantone 1.72
31-equal 1.73
43-equal 1.73
1/4-comma meantone 1.79
Kornerup's Golden 1.91
50-Equal 1.99
55-equal 2.04
2/7-comma meantone 2.05
1/6-comma meantone 2.15
LucyTuning 2.15
1/3-comma meantone 2.39
19-equal 2.41
12-equal 3.13
26-equal 3.42
Pythagorean 4.30

But if that is so, that would mean that intervals like 11/10 and 12/11,
which Harry Partch called consonant, can be mistuned more than twice as much
as thirds. If minor thirds can be tolerated with an error of 7 cents, then
by Manuel Op de Coul's reckoning 11/10 can be tolerated if tuned to 12/11
and 12/11 can be tolerated if tuned to 11/10. But if they are consonant,
shouldn't there be a dissonant region between them?

Brett Barbaro <barbaro@noiselabs.com> wrote:

>Assume all notes form a single chain of identical fifths, so you don't
need two versions for the second note of the major scale, and so you can
change keys without multiple versions of notes. Intervallically, the three
5-limit consonances should all be considered, and each should be evaluated
in terms of its field of attraction (Harry Partch's Observation One: the
field of attraction is inversely proportional to the limit of the interval)...
>
>The minimax principle tells us to find the tuning with the smallest worst
error. The answer is 5/18-comma meantone temperament*, where the fifths are
flattened by 5.97 cents (5/18 of a syntonic comma) relative to Pythagorean.
Then the major thirds are 2.39 cents sharp and the minor thirds are 3.58
cents flat relative to Just Intonation.
>
>The worst errors (that of the fifth and minor third) both have cents*limit
= 17.9 in 5/18-comma meantone. 50-equal is certainly close enough.

I think it's clear that historically, this region has not been favoured
anywhere near as much as the 1/4-comma / 31-equal region and those closer
to Pythagorean.

I think the mistake above is in assuming that "tolerance of mistuning" is
proportional to "field of attraction". Partch was concerned with ratios up
to 11, and for high limit ratios I think this assumption is true. However
by this assumption, the intervals for which we would tolerate the greatest
mistuning, are the unison and the octave. This is clearly not true,
although they do indeed have the greatest field of attraction.

When one looks at a dyadic dissonance curve (whether Erlich or Sethares,
and probably others) one sees that the notches corresponding to low limit
ratios are much deeper and have much steeper sides. This means that a small
mistuning, while it will not lead to the interval losing its identity and
becoming ambiguous (as it would for a high limit ratio), it *will* lead to
a larger absolute increase in dissonance. And this also is intolerable.

My feeling is that tolerance of mistuning is a minimum at the unison/octave
and increases thru the fifth, fourth, major sixth, major third, and peaks
(roundedly) at around the minor third (5:6) or augmented fourth (5:7)
before starting to fall with increasing limit (or increasing n+d). Note
that more than a 10 cent error in a fifth is a "wolf" while errors of 14
and 16 cents in major and minor thirds (12-equal) are apparently quite
tolerable.

I certainly agree with your optimising of the worst error (i.e. Max
Absolute) as opposed to the RMS or sum-of-squares error favoured by some.
It's mainly your relative weighting of the error in the fifth/fourth that I
disagree with.

Another consideration is that the diatonic may be considered to contain
some 9-limit ratios (but not 7's). If you include them, even with *your*
weighting scheme, what do you get?

Regards,
-- Dave Keenan
http://dkeenan.com

On Thu, 8 Apr 1999, Dave Keenan wrote:

> From: Dave Keenan <d.keenan@uq.net.au>
>
> Brett Barbaro <barbaro@noiselabs.com> wrote:
>
> >Assume all notes form a single chain of identical fifths, so you don't
> need two versions for the second note of the major scale, and so you can
> change keys without multiple versions of notes. Intervallically, the three
> 5-limit consonances should all be considered, and each should be evaluated
> in terms of its field of attraction (Harry Partch's Observation One: the
> field of attraction is inversely proportional to the limit of the interval)...
> >
> >The minimax principle tells us to find the tuning with the smallest worst
> error. The answer is 5/18-comma meantone temperament*, where the fifths are
> flattened by 5.97 cents (5/18 of a syntonic comma) relative to Pythagorean.
> Then the major thirds are 2.39 cents sharp and the minor thirds are 3.58
> cents flat relative to Just Intonation.
> >
> >The worst errors (that of the fifth and minor third) both have cents*limit
> = 17.9 in 5/18-comma meantone. 50-equal is certainly close enough.
>
> I think it's clear that historically, this region has not been favoured
> anywhere near as much as the 1/4-comma / 31-equal region and those closer
> to Pythagorean.
>
> I think the mistake above is in assuming that "tolerance of mistuning" is
> proportional to "field of attraction". Partch was concerned with ratios up
> to 11, and for high limit ratios I think this assumption is true. However
> by this assumption, the intervals for which we would tolerate the greatest
> mistuning, are the unison and the octave. This is clearly not true,
> although they do indeed have the greatest field of attraction.
>
> When one looks at a dyadic dissonance curve (whether Erlich or Sethares,
> and probably others) one sees that the notches corresponding to low limit
> ratios are much deeper and have much steeper sides. This means that a small
> mistuning, while it will not lead to the interval losing its identity and
> becoming ambiguous (as it would for a high limit ratio), it *will* lead to
> a larger absolute increase in dissonance. And this also is intolerable.
>
> My feeling is that tolerance of mistuning is a minimum at the unison/octave
> and increases thru the fifth, fourth, major sixth, major third, and peaks
> (roundedly) at around the minor third (5:6) or augmented fourth (5:7)
> before starting to fall with increasing limit (or increasing n+d). Note
> that more than a 10 cent error in a fifth is a "wolf" while errors of 14
> and 16 cents in major and minor thirds (12-equal) are apparently quite
> tolerable.
>
> I certainly agree with your optimising of the worst error (i.e. Max
> Absolute) as opposed to the RMS or sum-of-squares error favoured by some.
> It's mainly your relative weighting of the error in the fifth/fourth that I
> disagree with.
>
> Another consideration is that the diatonic may be considered to contain
> some 9-limit ratios (but not 7's). If you include them, even with *your*
> weighting scheme, what do you get?
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com
>
> ------------------------------------------------------------------------
**************************************************************************

Gentlemen,

Watching this very interesting n-ologue I thought the following reference
might be of use to somebody in this debate. The paper addresses the issue
of "fudging" that every piano tuner has to go through during their career.
The author comes up with a mathematical model for "optimal" fudging.

REFERENCE: A.A. Goldstein, "Optimal Temperament", Classroom Notes in
Applied Mathematics, SIAM, pp.242-251, ISBN0-89871-204-1,
date? (around 1988-89).

Sincerely,

Can Akkoc
Alabama School of Mathematics and Science
Mobile, Alabama 36604-2519
Phone: (334) 441-2126
Fax : (334) 441-3290

Brett,

Your message got posted on the 7th but if you missed a followup of mine, I
didn't write one because I had little to add to Dave Keenan's response.
Perhaps I should reread what Partch writes about consonance but I think
12/11 and 11/10 can hardly be called consonances.
If the partials in the sound are strong enough to hear them beating, then
with higher numbers the region of tolerance will get smaller again because
beating will be faster. The 11th partial is 3.5 octaves above the
fundamental.
One can speak about region of tolerance only in a broad manner because many
factors play a role like the timbre, register, note duration, etc.

Manuel Op de Coul coul@ezh.nl

manuel.op.de.coul@ezh.nl wrote:

> Your message got posted on the 7th

So it did. My mail reader has some weirdness. Sorry about the confusion.

> If the partials in the sound are strong enough to hear them beating, then
> with higher numbers the region of tolerance will get smaller again because
> beating will be faster. The 11th partial is 3.5 octaves above the
> fundamental.

Why doesn't this apply to lower numbers, giving them slower beating/larger
tolerance?

Dave Keenan wrote:

> Another consideration is that the diatonic may be considered to contain
> some 9-limit ratios (but not 7's). If you include them, even with *your*
> weighting scheme, what do you get?

The only ratios in question are 9:8 and 10:9.
The same major second must approximate both.
The errors are larger that for any of the ratios of 5 or 3.

Therefore, 1/4 comma meantone, where the second is exactly halfway between the 9:8
and 10:9, is what you get, in my original weighting scheme, or equal-weighted.
Using Manuel Op de Coul's weighting scheme, the errors of the fifth and minor
third are worse in 3/14-comma meantone than those of the seconds, so the seconds
don't matter.

Brett Barbaro <barbaro@noiselabs.com> wrote:

>Dave Keenan wrote:
>
>> Another consideration is that the diatonic may be considered to contain
>> some 9-limit ratios (but not 7's). If you include them, even with *your*
>> weighting scheme, what do you get?
>
>The only ratios in question are 9:8 and 10:9.

If you are talking octave equivalent, then yes, but looking at those
numbers one might assume they are unimportant for harmony. They do of
course get used as 9:4 and 9:5, but your results still follow.

>The same major second must approximate both.
>The errors are larger that for any of the ratios of 5 or 3.
>
>Therefore, 1/4 comma meantone, where the second is exactly halfway between
the 9:8
>and 10:9, is what you get, in my original weighting scheme, or
equal-weighted.
>Using Manuel Op de Coul's weighting scheme, the errors of the fifth and minor
>third are worse in 3/14-comma meantone than those of the seconds, so the
seconds
>don't matter.

I'll buy that (3/14 comma being opt diatonic meantone). Thanks.

I know we're talking diatonic here but it should be noted that once you
allow ratios of 7, 1/4 comma looks a lot better than 3/14 comma.

See my http://dkeenan.com/Music/OpenTunings.htm (where I assume
equal weights, but they are easily changed in the accompanying spreadsheet).

I repeat that I think the error-tolerance is neither a linear nor a
reciprocal function of the complexity of the ratio (complexity has so far
been assumed to be given by odd-limit). But it is some function which is
low for both very simple and very complex ratios and is a max somewhere
around minor third or augmented fourth. Anyone suggest a suitable simple
function?

We could just take Min(complexity, 1/complexity) but this seems too pointy.

Regards,
-- Dave Keenan
http://dkeenan.com

At 11:56 AM 4/8/99 +1000, Dave Keenan wrote:
>From: Dave Keenan <d.keenan@uq.net.au>

>I think the mistake above is in assuming that "tolerance of mistuning" is
>proportional to "field of attraction". Partch was concerned with ratios up
>to 11, and for high limit ratios I think this assumption is true. However
>by this assumption, the intervals for which we would tolerate the greatest
>mistuning, are the unison and the octave. This is clearly not true,
>although they do indeed have the greatest field of attraction.

Accordion to Richard Morse <http://accordion.simplenet.com/wetdry.html>, the
wet or musette tuning favored in France and Italy requires free reeds in
unison to be tuned 20 cents apart.

--Charlie Jordan <http://www.rev.net/~aloe/music>

Dave Keenan wrote:

> >I think the mistake above is in assuming that "tolerance of mistuning" is
> >proportional to "field of attraction". Partch was concerned with ratios up
> >to 11, and for high limit ratios I think this assumption is true. However
> >by this assumption, the intervals for which we would tolerate the greatest
> >mistuning, are the unison and the octave. This is clearly not true,
> >although they do indeed have the greatest field of attraction.

Charlie Jordan wrote:

> Accordion to Richard Morse <http://accordion.simplenet.com/wetdry.html>, the
> wet or musette tuning favored in France and Italy requires free reeds in
> unison to be tuned 20 cents apart.

Thanks! That's interesting and helpful. I think the answer for "tolerance of mistuning"
is not some function that increases and then decreases for higher limit ratios. I think
the answer is a very sensitive function of musical context. Manuel's perceptions may be
right in some contexts, field of attraction may be more relevant in other contexts, but
typically different intervals will play different roles and not be strictly comparable
as if the only difference between them were a numerical one.

So, I think I can safely say that for two-voice Renaissance and Baroque music, where
thirds and sixths greatly outnumber fifths (and fourths are considered dissonant),
something close to 2/7-comma meantone would be optimal. In other situations, moving
toward Pythagorean would be more appropriate.

--- Brett Barbaro <barbaro@noiselabs.com> wrote:

> If you include
> them, even with *your*
> > weighting scheme, what do you get?
>
> The only ratios in question are 9:8 and 10:9.
> The same major second must approximate both.
> The errors are larger that for any of the ratios of
> 5 or 3.
>
> Therefore, 1/4 comma meantone, where the second is
> exactly halfway between the 9:8
> and 10:9, is what you get, in my original weighting
> scheme, or equal-weighted.
> Using Manuel Op de Coul's weighting scheme, the
> errors of the fifth and minor
> third are worse in 3/14-comma meantone than those of
> the seconds, so the seconds
> don't matter.

Excuse my ignorance if this seems rudimentary, but can you explain the
"weighing schemes" in english ? what is the functional role of a
weighing scheme in these approximations ?

_________________________________________________________
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Skyler S. wrote,

> Excuse my ignorance if this seems rudimentary, but can you explain the
> "weighing schemes" in english ? what is the functional role of a
> weighing scheme in these approximations ?

The weighting scheme determines how much weight is given to each interval in determining
the magnitude of the error. My original weighting scheme multiplied the error in each
interval by the interval's "limit", so that errors were measured in cents*limit. This
caused more importance to be put on the tuning of the thirds than on the fifth, Manuel
suggested a weighting scheme in which the errors were measured in cents/limit, which caused
more importance to be put on the tuning of the fifth. Make sense?

Dave Keenan wrote:

>I certainly agree with your optimising of the worst error (i.e. Max
>Absolute) as opposed to the RMS or sum-of-squares error favoured by some.

I use RMS because if two tunings have the same worst error, the second-worst
errors should allow one to judge one tuning better than the other. RMS still
makes the worst errors more important than the second-worst errors, while
MAD (mean absolute deviation) puts equal weight on all errors.

Brett Barbaro wrote:
> Why doesn't this apply to lower numbers, giving them slower
beating/larger
> tolerance?

Because the lower partials are usually much stronger than higher ones.
They are also closer to the first harmonic (fundamental) so they interfere
more.
And with the octave, the second harmonic of the first tone beats with the
fundamental of the second so that surely gives a strong beat (provided the
second harmonic is present of course).

Manuel Op de Coul coul@ezh.nl

{Paul Erlich, TD 140.3:]
> Skyler S. wrote,
>
>> Excuse my ignorance if this seems rudimentary, but can you
>> explain the "weighing schemes" in english ? what is the
>> functional role of a weighing scheme in these approximations ?
>
> The weighting scheme determines how much weight is given to each
> interval in determining the magnitude of the error. My original
> weighting scheme multiplied the error in each interval by the
> interval's "limit", so that errors were measured in cents*limit.
> This caused more importance to be put on the tuning of the thirds
> than on the fifth, Manuel suggested a weighting scheme in which
> the errors were measured in cents/limit, which caused more
> importance to be put on the tuning of the fifth. Make sense?

Makes sense to me, but for most music, it depends
quite a bit on the context of the music in question,
as Paul had said previously:

[Paul Erlich/Brett Barbaro, TD 139.2:]
> I think the answer for "tolerance of mistuning" is not some
> function that increases and then decreases for higher limit
> ratios. I think the answer is a very sensitive function of
> musical context. Manuel's perceptions may be right in some
> contexts, field of attraction may be more relevant in other
> contexts, but typically different intervals will play different
> roles and not be strictly comparable as if the only difference
> between them were a numerical one.

For a concrete example: in jazz, the importance of the
(unaltered, 'perfect') '5th' in a chord is so minimal that
it's generally omitted entirely!

When a '5th' *is* present in a jazz chord, it's usually
altered (sharp/augmented or flat/diminished).

I've even seen published cases where a 'dominant 7th flat 13th'
2^(0/12) : 2^(4/12) : 2^(7/12) : 2^(10/12) : 2^(8/12)
(in whatever arrangement of intervals you choose - that
one is my favorite) where the 'flat 13th' was called an
'augmented 5th', so the chord was referred to as a 'dominant 7th'
with simultaneously both a regular and an augmented '5th'!

Based on things like this, and what I know of jazz theory,
the tuning of the '5th' in jazz seems to be able to withstand
an error of over 100 cents!

So it would seem to me that in jazz the weighting should
definitely place more importance on the '3rd' (and the
ever-present '7th's) than on the '5th', altho the reverse
(Manuel's formula) is probably true for much other music.

Of course, it's quite debatable whether or not jazz
(in its modern form) can in any way be considered 'diatonic',
which was the original subject of this thread.

IMO, it would not have developed without the 12-eq scale,
allowing ring-around-the-rosie circle-of-'4th' progressions,
which is *the* fundamental harmonic procedure in jazz.

[Ara's response, TD 141.6:[]
>
> This seems like a chicken-and-egg sort of thing to me.
> Correct me if i don't understand this right, but your scheme
> would place more importance in tuning the highest "limit"
> presented in a system, whereas the other emphasizes the lowest
> "limit" represented. That is, if the strict error is our
> measuring stick for the moment.
>
> For me, it makes sense to have the higher numbers presented
> more accurately, since i can either sense a fifth or i can't,
> the 3:2 being so fundamentally engraved/engrained/inbrained in
> my head. But then again, if the most "familiar" intervals
> (lowest ones) are off, then there's no hope in getting any
> sense out of a piece of music. So there has to be a balance
> somehow. It seems the attainment of this very balance has been
> the goal of theorists for a while now...<snip>

Ara's response brings up a good point about the cultural
conditioning that also becomes a factor in a 'weighting'
scheme. But as my response above contradicts his second
sentence, I would again agree with Paul and say that it
depends largely, perhaps almost entirely, on musical context.

-monz

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Joseph L. Monzo wrote:
>For a concrete example: in jazz, the importance of the
>(unaltered, 'perfect') '5th' in a chord is so minimal that
>it's generally omitted entirely!
>
>When a '5th' *is* present in a jazz chord, it's usually
>altered (sharp/augmented or flat/diminished).

I suppose you're correct in that the note a 5th above the tonic isn't used a
lot, but perfect 5th intervals are all over the place in jazz. Also, chords
build on the 5th scale interval are used very much.

>[...]
>Based on things like this, and what I know of jazz theory,
>the tuning of the '5th' in jazz seems to be able to withstand
>an error of over 100 cents!

Have they totally phased out the major 7th chords now? Passe, I suppose,
but there are still plenty of them in the Real Book.

-b0b-

Joe Monzo wrote,

>I've even seen published cases where a 'dominant 7th flat 13th'
>2^(0/12) : 2^(4/12) : 2^(7/12) : 2^(10/12) : 2^(8/12)
>(in whatever arrangement of intervals you choose - that
>one is my favorite) where the 'flat 13th' was called an
>'augmented 5th', so the chord was referred to as a 'dominant 7th'
>with simultaneously both a regular and an augmented '5th'!

I would disagree because the scale that goes with this chord (if indeed it
has both a perfect fifth and a flat 13th)is heptatonic and has a flat sixth
degree rather than a sharp fifth. Scale-degree function is important if the
vocabulary is primiarily heptatonic, as it is in jazz.

>Based on things like this, and what I know of jazz theory,
>the tuning of the '5th' in jazz seems to be able to withstand
>an error of over 100 cents!

If that were true, any amount between 0 and 100 cents sharpening would be
acceptible, but clearly no jazz musician would deem that the case. Neither
the augmented fifth nor the flat sixth represent a 3/2 ratio.

>So it would seem to me that in jazz the weighting should
>definitely place more importance on the '3rd' (and the
>ever-present '7th's) than on the '5th', altho the reverse
>(Manuel's formula) is probably true for much other music.

Well, fourths are important in a lot of jazz harmony, in addition to fifths
that occur in places other than between the root and '5th' (for example,
between the m3 and m7, and between the M3 and M7). For these reasons, the
ratios of 3 should still have a high weight "in jazz".

>Of course, it's quite debatable whether or not jazz
>(in its modern form) can in any way be considered 'diatonic',
>which was the original subject of this thread.

Right, but . . .

>IMO, it would not have developed without the 12-eq scale,
>allowing ring-around-the-rosie circle-of-'4th' progressions,
>which is *the* fundamental harmonic procedure in jazz.

I disagree completely -- very few jazz tunes circumnavigate the entire
circle of 12 fifths. However, tritone substitution, where the half-octave
functions interchangeably as a 7:5 and as a 10:7, is very common in jazz,
which rules out non-12 diatonic tunings like 19 or 31. Tritone substitution
(aka a xenharmonic bridge of 50:49) is a very non-diatonic trick, and the
only ET where is coexists with the diatonic trick of the vanishing comma
(aka a xenharmonic bridge of 81:80) is 12-eq.

At 05:46 PM 4/25/99 -0400, monz@juno.com wrote:

>Of course, it's quite debatable whether or not jazz
>(in its modern form) can in any way be considered 'diatonic',
>which was the original subject of this thread.
>
>IMO, it would not have developed without the 12-eq scale,
>allowing ring-around-the-rosie circle-of-'4th' progressions,
>which is *the* fundamental harmonic procedure in jazz.

Had the equal temperament involved a different number of intervals, it
stands to reason that jazz would have developed, but perhaps quite
differently in theory and somewhat differently in practice.

Less clear is how blues would have developed with different tuning.

--Charlie Jordan <http://www.rev.net/~aloe/music>

In my first post on this subject, I wrote,

>Assume all notes form a single chain of identical fifths, so you don't need two >versions for the second note of the major scale, and so you can change keys >without multiple versions of notes. Intervallically, the three 5-limit
>consonances should all be considered, and each should be evaluated in terms of >its field of attraction (Harry Partch's Observation One: the field of attraction >is inversely proportional to the limit of the interval). The worst error in
some >previously recommended tunings:
>Tuning Worst Error (cents*limit)
>50-Equal 18.2
>2/7-comma meantone 18.4
>Kornerup's Golden 21.4
>LucyTuning 21.7
>1/4-comma meantone 26.9
>31-equal 29.8
>1/3- & 2/9-comma meantone 35.8
>19-equal 36.8
>1/5-comma meantone 43.0
>43-equal 43.3
>55-equal 50.9
>1/6-comma meantone 53.8
>12-equal 78.2
>26-equal 85.4
>Pythagorean 107.5
>The minimax principle tells us to find the tuning with the smallest worst error. >The answer is 5/18-comma meantone temperament*, where the fifths are flattened by >5.97 cents (5/18 of a syntonic comma) relative to Pythagorean. Then the
major >thirds are 2.39 cents sharp and the minor thirds are 3.58 cents flat relative to >Just Intonation.
>The worst errors (that of the fifth and minor third) both have cents*limit = 17.9 >in 5/18-comma meantone. 50-equal is certainly close enough.

Tonight I finally got a hold of copies of Mandelbaum's dissertation and Jorgenson's tome. The first few pages of both contain information on the proposed tuning system of Robert Smith. According to Mandelbaum, Smith's ideal fifth in his
1759 _Harmonics_ is flat by, you guessed it, 5/18 of a comma. Mandelbaum's description makes it seem as if Smith might have used the same derivation as me. According to Jorgenson, however, Smith's ideal fifth derived from equal-beating
considerations and satisfies the equation 3x^3+4x=16. That implies a fifth 0.0027 cents smaller than that of 5/18-comma meantone. Though this is really splitting paramecial hairs, anyone know who's right about Smith?