equal temperaments and pitch classes

The discussion on 19 equal has so far ignored the possibility of
19 pitch classes tuned according to the desired harmony. For
example:

> I remember when I first became interested in more than 12 tones in the
> octave, I liked the _idea_ of as many tones within the octave as
> possible, and tended to consider 31 as the absolute minimum. But
> experience has taught me that the only viable alternative to the 12-tone
> equal is the 19 system.

19 notes, maybe. But why do they have to be equally tempered?

> All tuning systems which are not mere noise, reduce approximately either
> to the 12-tone or the 19-tone equal, melodically considered. This
> follows inescapably from the principle of the melodic limen.

And what is the criterion for a tuning system _being_ mere noise?
That it doesn't reduce to 12 or 19 equal? I still think melodically
in terms of 12 pitch classes, with diesis or comma inflections. What
does this prove?

> 31-tone
> equal for example reduces melodically to the 12-tone equal (with a few
> exceptions where the augmented tone occurs, where it has rather a flavor
> of the 19-equal)

31 equal sounds nothing like either 12 or 19 equal. Melodies in 12
or 19 pitch classes can, however, be played in 31 equal.

> Between C and D, for example the 31-tone equal has Dbb, C#, Db
> and Cx ascending. The first of these is melodically confounded with C,
> the second two are melodically confounded with each other, and the last
> is confounded with D.

So, the melody C-D-Dbb-Cx-Dbb-Cx-Dbb-D is the same as C-D-C-D-C-D-C-D?
You don't hear the interval Dbb-Cx as a semitone?

> The 31-tone equal is in fact a monstrously
> complex, inefficient way of returning us to our melodic starting-point
> of 12-tone equal. It is much smoother indeed, but melodically it is a
> dead-end.

Come on, 31 equal steps aren't _that_ complex. If you agree that it's
smoother, why not use it for its harmony? It's not that difficult to
convert 19 pitch classes into 19 notes from 31 equal. Usually, a string
of 19 fifths will do, especially if you're working within the 5-limit.
One day, I'll get a computer to do it automatically.


> The 22-tone equal suffers from two grave defects however, either one of
> which would be sufficient to condemn it to the category of a curiosity.
> Its tuning degree is at the outermost limit of what can be reliably
> distinguished in melody.

This is a defect? Think of the tricks you can perform!

> second defect of the 22-tone equal involves its failure to close the
> cycle of fifths. I am aware that some will say to themselves - so what?
> But a musical system that does not close the cycle of fifths has at a
> stroke isolated itself from 99% of the music not merely of the western
> 19th century, but from virtually the whole of the western tradition, and
> from many other musical traditions as well. Such musical systems, like
> just intonation, are mere curiosities, and are far more impoverished in
> usable, aurally distinct resources even than the 12-tone equal.

Yes, 22 equal certainly takes you outside of the Western tradition.
It all depends how hung up you are on Western classical music. Are
you saying that all other music is a mere curiosity? One of the
characteristic features of this Western tradition is that everything
has to be given a name, and everything without a name is considered
musically unimportant. I assume be "aurally distinct resources" you
mean intervals that can be given a name, and that people can be trained
to recognise.

What exactly does "close the cycle of fifths" mean? 1/4 comma meantone
has an open cycle of fifths. If 22 equal has a cycle of fifths, it
closes after 22 steps. Do you mean, rather, that there is only one
size of perfect fifth? This seems to raise the diatonic notation to
a principle that I do not consider valid.

A system of 22 pitch classes for 5-limit JI runs into trouble because
the intervals are so unequal. So, there is an argument here for
equal temperament. I haven't investigated Paul Erlich's decatonic
modes enough to decide if they can be treated as 10 pitch classes.
I don't like the theory because it clings to the idea of diatonic
harmony -- that all notes used in harmony must belong to the same key.
I'd rather work out a chord sequence, and fit a melody to it. Or,
define a melodic mode, write a tune in it, and harmonise it
chromatically.

> Your refer to the 7 and 11 limits. The latter is a mere fantasm of the
> just intonationists, and is not audible to the ear as anything other
> than dissonance. The septimal limit is more interesting. But it is
> dissonant, however this may trouble those who aspire to forever extend
> the boundaries of consonance, until someday I presume, we shall find
> everything consonant, and music need trouble itself no longer with any
> rules or constraints whatever... It will also no longer need to trouble
> itself with pleasing an audience, for it will have none. Actually
> however, the 19-tone equal is the _only_ temperament that gives the
> septimal intervals in such a form that they can be melodically
> distinguished from adjacent intervals. This is a consequence of the fact
> that in this system only are the septimal intervals far enough away from
> adjacent intervals to preserve their own unique melodic character.

If you're listening to septimal intervals in 19 equal, no wonder you
think they're dissonant! I outlined a listening experiment on the
list before which confirmed the consonance of the 7-limit. If you
disagree, can you give specific comments on it?

To state my own opinion, which is the only one I trust, a just 4:6:7
chord is more consonant than a major triad in 12 equal. Extending
consonance to the 7-limit means just that -- no further. You can't
disprove an argument by extending it to ridiculous extremes.

An example of a 31 equal mode with septimal harmony is Ab-A-C-C#-Eb-
F-F#-Ab. The chord C#-F-Ab could be retuned Db-F-Ab, or left as it
is. If the former, this mode is 7 from 12 pitch classes. However,
it certainly sounds like a different mode in 12 equal. In 19 equal,
it is recognisable, but has quite a mournful sound. It does work
melodically in LucyTuning, but the harmonies are still way out.

To state that 19 equal is the only temperament to do anything is
clearly absurd. Sharpen one note my a millicent, and nothing
important changes. Perharps you discounting out of hand all the
meantone temperaments between 19 and 31 equal. From this
distance, it's very difficult to tell.


Now, to some details:

> I agree also that the fifths major thirds of 31-tone equal are slightly
> but noticeably smoother and more pleasing than those of the 19-tone
> equal. However, if the 19-tone octave is stretched by 2-3 cents, its
> fifths become quite as good as those of the 31-tone.

The biggest problem harmonically with 13 equal is that it's fifths
aren't very good. Stretching the octave for 19 equal works because
both the 3rd and 5th overtones are approximated flat. I'd still much
rather use 31 equal, with pure octaves and almost pure major thirds.

> 55-60 cents as the limen of intervallic perception is well-established
> in the literature. Pratt for example in his Meaning of Music, as I
> recall, found a value "just wider than a quarter-tone", and Seashore &
> Jenner found a value between 53 and 61 cents.
> Naturally no one should accept these results without making trial for
> himself. I find however that my own ears are fully in agreement with
> these results.

So there's no problem with the 54.54 cents of 22 equal, then? I'll do
the experiment myself when I understand what the terms mean. I reckon
one step of 41 equal is about the smallest interval that can be used
melodically, but I still tend to avoid it.


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> > (By the way, the seeming beat-spotlighting character of the long consonances
>>in slow movements is mostly an illusion: performers at that time would
>>improvise filler and ornaments in such occasions.)
> On the keyboard sonatas? I really don't agree there, as the development

I wasn't really talking about keyboard music, but if you want to get into
specifics, it would depend a great deal on the period, the style, the
composer, and so on. If, for example, I was faced with a very plain
keyboard sarabande from circa 1700, you bet I would improvise lots of
melodic filler. Bach himself tends to leave less room for this sort of
thing, as did most composers as you get into the late 18th century.
>
>>In this context, the practical differences between temperaments like some
>>of Neidhardt's and Marpurg's and ET are angels dancing on the head of pin.
> On this I must disagree strongly. Modern day ears can easily hear the
>differences in a tuning that provides key character and ET ,which does not.

Perhaps I was not clear enough in my qualifications. I had in mind some
temperaments such as Neidhardt's and Marpurg's divisions of the ditonic
comma. Looking them up in Barbour (who at least comes in handy when
comparing things to 12TET), I see that Marpurg's 1/3 comma temperament is
out at most 6 cents from ET, with a mean deviation of 3 cents. Neidhardt's
1/4 comma is at most 4 cents away, and Marpurg's 1/6 comma has a mean
deviation of only .7 cents!

Now, would I be able to tell the difference between such temperaments and
ET on a synthesizer or a harpsichord tuned with electronic help?
Absolutely. Perhaps Ed thought that I was saying the opposite, which I am
not.

Would I be able to tell the difference between such temperaments if tuned
by ear counting beats of simultaneous intervals by a skilled tuner who took
his time? Again, yes, I think so.

Would I be able to tell the difference between such temperaments if tuned
by ear of successive intervals, not checking thirds, by a keyboard player
who had a half hour before a rehearsal. Hmm, in that situation I think the
margin for error Marpurg, Neidhardt, and ET will start to overlap.

But let's take a more extreme case, as you suggest, of pure versus ET
intervals. Certainly such differences were quite significant to practical
musicians of that time when tuning harpsichords. But the original question
was about the use of fretted strings in ensembles with harpsichords.

The question then becomes, do those differences create enough dissonance to
be objectionable? Since we have plenty of documentary evidence that lutes,
archlutes, theorbos, and viols often played with harpsichords -- especially
in the 17th century -- the answer has to be either "no" or that
harpsichords were tempered to be at least closer to the fretted strings.

My own feeling is that the objectionability depends on many factors,
including the size of the ensemble and the speed of the music. Certainly
lutes were often used in large ensembles, especially in France and Italy in
the seventeenth century, but I heard a fine performance the other night on
PBS of Purcell trio sonatas and songs accompanied by a harpsichord and an
archlute. While I don't know how they tuned the harpsichord, the result was
lovely.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^


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Subject: 31-tone equal melodic progression
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