Yahoo Tuning Groups Ultimate Backup tuning Simple key colour metric for 12-note 5-limit tunings

In sets of 12 pitches considered as tunings for the 5-limit, there is
a simple way of classifying them with respect to how they deal with
the (major and minor) thirds.

Namely, how many of each type of thirds are better or worse than equal
temperament, as approximations to 5:4 and 6:5?

Meantone as usually defined results in 8 major and 9 minor thirds
being better and 4 and 3 resp. being worse.
Pythagorean tuning results in 8 major and 9 minor thirds being worse
and 4 and 3 resp. being better (in fact virtually pure).

Hence, if a 12-note circulating temperament has a majority of thirds
better than ET, it can be labelled 'M'; if it has a majority of thirds
worse than ET, it can be labelled 'P'.

(Or if they are evenly distributed about the ET value, we can label it
NO, for 'no preference'...)

Clearly, the type of third that is in the majority must also be closer
to equal than the type that is in the minority.

In other words, 'M' tunings have a majority of thirds that are better
than ET by X, and a minority that are worse than ET by Y, with on
average Y>X. 'P' tunings are the other way round.

Werckmeister V is an 'M' tuning, being similar to 1/8 comma meantone;
whereas 'Kirnberger III' is a 'P' tuning, being closer to Pythagorean
intonation.

There are clearly degrees of 'M'- and 'P'-ness depending on how large
the majority is. For example Werckmeister III has 4 minor thirds worse
than ET and 3 better - not a great difference.

When it comes to key colour, both types have disadvantages. 'M'
tunings tend to give the same, relatively comfortable, colour to a lot
of central keys, but a few outlying keys become strikingly
uncomfortable. (Although they may still be bearable, depending on the
degree of inequality.)

'P' tunings give strikingly pure thirds in a few central keys, but the
intonation in most other keys is relatively uncomfortable and
unvarying (e.g. the chronically Pythagorean intonation of Werckmeister
III in many keys).

To obtain optimal key colour variation, meaning that within given
limits on the size of thirds, there should be as little as possible of
the circle of fifths where the purity of thirds is not audibly
varying, it seems to me that one ought to get close to the 'NO'
temperaments where the thirds deviate symmetrically either side of ET.

Almost all Neidhardt tunings, and Kellner's, and Lehman's, are mildly
'P', whereas Werckmeister's 1698 continuo instructions give an 'M'
tuning.

Young and Vallotti, being absolutely symmetrical, are 'NO' - but Young
1 and Vallotti have the flaw with respect to key colour that there are
strings of 3 major, and 4 minor thirds with no change in intonation.
So 'NO' is not a sufficient condition.

Barnes is an improvement in being 'NO' and only having strings of 2
major and 3 minor thirds of the same size. Young II is still better.

Of course, key colour variation is only one thing to be considered,
besides the absolute purity of the 'best' and 'worst' thirds and
fifths, the balance between sharps and flats, etc...

For example if you don't want any Pythagorean major thirds, then you
might try replacing Young's PC/12 fractions with SC/12 and sticking
the extra schisma at F#-C# for

C -2 -2 -2 A -2 -1 -1 F# -1 0 0 Eb 0 -1 -1 C

~~~T~~~

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> Meantone as usually defined results in 8 major and 9 minor thirds
> being better and 4 and 3 resp. being worse.
> Pythagorean tuning results in 8 major and 9 minor thirds being worse
> and 4 and 3 resp. being better (in fact virtually pure).

I don't much like this analysis, because it is a pure 5-limit analysis.
A meantone tuning in an unweighted optimal 5-limit range around 81-et
will also have nearly pure 7/6s for the "bad" minor thirds which use
augmented seconds. The trouble with calling this a "bad" minor third is
that it isn't clear to me that it is actually any worse than an
ordinary minor triad in terms of consonance.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> What do you think of using flat octaves and thus doing
> better than ET on average, according to some ways of
> calculating consonance?
>
> -Carl
>

I don't think it would be effective on 'historical' stringed keyboard
instruments. At least, whenever I've played an audibly flat octave,
I've found it unacceptable.

If the effect on thirds were strong enough, one might think it was
worth it. But the arithmetic is wrong. Make the octave flat by X and
you've improved the major third (on average) by X/3 and the minor
third by X/4.

Whereas in meantone the tempering of the fifth is multiplied by 3 or 4
for the amount the thirds are better than Pythagorean.

~~~T~~~

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > Meantone as usually defined results in 8 major and 9 minor thirds
> > being better and 4 and 3 resp. being worse.
> > Pythagorean tuning results in 8 major and 9 minor thirds being worse
> > and 4 and 3 resp. being better (in fact virtually pure).
>
> I don't much like this analysis, because it is a pure 5-limit analysis.
> A meantone tuning in an unweighted optimal 5-limit range around 81-et
> will also have nearly pure 7/6s for the "bad" minor thirds which use
> augmented seconds. The trouble with calling this a "bad" minor third is
> that it isn't clear to me that it is actually any worse than an
> ordinary minor triad in terms of consonance.
>

I knew someone would say that. Meantone and Pythagorean are limiting
cases, and the metric isn't very enlightening for them, in that it
just tells you that meantone is 'meantone-like' and Pythagorean is
'pythagorean-like' with respect to their distribution of key colour.

If you read the rest of the message, what I was using it for was to
evaluate certain aspects of key colour variation in *circulating*
tunings that interpolate between the two: such circulating tunings are
generally such as not to have any 'thirds' close to 7:6.

In any case, historically, the goal of constructing circulating
12-note tunings was to produce approximations to the 5-limit. All the
sources I have seen say that very narrow minor thirds, whether or not
they happened to be anywhere near 7:6, were particularly undesirable.

But I don't want to define 'circulating' to exclude tunings with wide
fifths, or thirds that are impurer than Pythagorean; some versions of
ordinaire have both. Just to point out that such thirds were likely to
be pushing the boundaries of acceptability.

~~~T~~~

🔗Carl Lumma <clumma@yahoo.com>

> > What do you think of using flat octaves and thus doing
> > better than ET on average, according to some ways of
> > calculating consonance?
> >
> > -Carl
> >
>
> I don't think it would be effective on 'historical' stringed
> keyboard instruments. At least, whenever I've played an audibly
> flat octave, I've found it unacceptable.
>
> If the effect on thirds were strong enough, one might think it was
> worth it. But the arithmetic is wrong. Make the octave flat by X and
> you've improved the major third (on average) by X/3 and the minor
> third by X/4.

I'm not sure what you mean here, but if you have a look at

http://lumma.org/music/theory/WellTemperamentComparator.xls

you can see that by some measures, you can do better on average
than in ET.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> In any case, historically, the goal of constructing circulating
> 12-note tunings was to produce approximations to the 5-limit. All the
> sources I have seen say that very narrow minor thirds, whether or not
> they happened to be anywhere near 7:6, were particularly undesirable.

Do you have a cite for the undesirability of narrow minor thirds? My
ears do not agree; they think think wide major thirds stand out a lot
more, but the sound of a triad is dominated by I-III, not II-V, so for
minor triads the significance is less.

Also, I think the JI 1-7/6-3/2 sounds quite consonant when compared to
1-6/5-3/2, whereas 1-5/4-3/2 is much sweeter than the steely 1-9/7-3/2.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > In any case, historically, the goal of constructing circulating
> > 12-note tunings was to produce approximations to the 5-limit. All the
> > sources I have seen say that very narrow minor thirds, whether or not
> > they happened to be anywhere near 7:6, were particularly undesirable.
>
> Do you have a cite for the undesirability of narrow minor thirds?

Praetorius defined F-G# as the 'wolf'; Printz (an advocate of
1/4-comma) went so far as to disfavour 1/6 comma because its *regular*
minor thirds were too narrow!

My
> ears do not agree; they think think wide major thirds stand out a lot
> more,

I didn't say that narrow minor thirds were disfavoured *more* than
wide major thirds - you didn't mention wide major thirds up till now.
Printz certainly worries a good deal about B-Eb etc. in meantone, but
concludes that you just have to live with it (if you can't afford
extra keys...)

Of course in any circulating temperament every very narrow minor third
must coexist with at least two wide major thirds! If you have a triad
approximating 7/6 3/2, simply modulate to the mediant or submediant to
obtain another approximating 9/7 3/2.

> but the sound of a triad is dominated by I-III, not II[I]-V, so for
> minor triads the significance is less.

So if people only played minor chords, we'd be fine! Funeral music in
F minor I guess.

Reminds me of the time Irving Thalberg heard something he disliked in
a movie score: he asked a minion 'What's that awful sound?' 'It's a
minor chord, Mr. Thalberg.' Next morning a notice was posted on the
wall in the studio composers' office: 'From today, no MGM film score
may contain a minor chord.'

Maybe I ought to retitle the thread to reflect the main application
which is to circulating 12-note tunings.

~~~T~~~

🔗Tom Dent <stringph@gmail.com>

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

> > the arithmetic is wrong. Make the octave flat by X and
> > you've improved the major third (on average) by X/3 and the minor
> > third by X/4.
>
> I'm not sure what you mean here, but if you have a look at
>
> http://lumma.org/music/theory/WellTemperamentComparator.xls
>
> you can see that by some measures, you can do better on average
> than in ET.
>
> -Carl
>

What I said about minor thirds was wrong, of course flat octaves make
5-limit m3 *worse*.

Sure, if you put a high enough value on major thirds (not to mention
certain 7-limit intervals...) and a lower value on octaves, you could
get a 'better' result with flat 8ves.

But such a valuation just disagrees with my experience, and with all
17th-18th century sources on keyboard tuning that say tune octaves as
pure as possible (or in one case, barely sharp). Thirds were highly
valued, but octaves were infinitely highly valued.

My (naive 5-limit) point was that even if one does allow narrowing of
octaves, any resulting gain of purity (in cents) to major thirds is a
lot less than the loss to octaves.

What's the justification for considering 7-limit in evaluating
(quasi)-historical, 12-note circulating temperaments? I know I was
raving about septimal aug. and dim. intervals a couple of weeks ago,
but they seem to be almost totally impractical in 12-note circulating
temps. Simply because a 7:5 tritone requires about 5/4 of a
Pythagorean comma to be doled out among the intervening six fifths...

~~~T~~~

🔗Carl Lumma <clumma@yahoo.com>

>What I said about minor thirds was wrong, of course flat
>octaves make 5-limit m3 *worse*.

There's a very wide range of acceptable minor thirds that
all sound about as good as any other between about 290 and 320
cents (or at lesat there are multiple targets in that range).
So my approach is to keep them in that range and otherwise
ignore them.

> Sure, if you put a high enough value on major thirds (not to
> mention certain 7-limit intervals...) and a lower value on
> octaves, you could get a 'better' result with flat 8ves.

I tried two types of value judgements here, without knowing
how they would turn out: unweighted RMS error and
log(n*d)-weighted error.

> My (naive 5-limit) point was that even if one does allow
> narrowing of octaves, any resulting gain of purity (in cents)
> to major thirds is a lot less than the loss to octaves.

Depends on how you weight the octaves. But remember, the
benefits are increased for 10ths.

> What's the justification for considering 7-limit in evaluating
> (quasi)-historical, 12-note circulating temperaments?

If you can beat ET on average, these things can be used for
jazz and contemporary music.

-Carl