Wolf Tuning 1125 - "In praise of meantone..."

re: Wolf posting - In Praise of Meantone.

Daniel Wolf's observations on the arrival of horological accuracy,
and the declining use of meantones tunings, with moves towards
well temperaments and eventually in favor of 12tET,
does have a certain irony.
(see end of post *)

Assuming that he is correct (I have no reason to doubt him on this),
and the fashion moved from meantone into well temperaments, and
eventually 12tET, I wonder why meantone tunings were abandoned.

Could it have been due to the limitations imposed on composers and
performers, when using twelve fixed pitches per octave?

The difficulty of quickly retuning meantones would have severely
restricted the harmonic possibilities, yet well temperaments
immediately offered keyboard players both tuning and intervallic
novelties, in different keys.

So the musical innovators of the time gravitated towards
well temperaments or to providing more notes per octave with
instrument developments, resulting in 19tET, 31tET, 53tET etc.

Could 12tET's ubiquity be blamed on the need to simplify the
well-tempered subtleties for subsequent less tonally sophisticated
generations?

Meanwhile the JI advocates were still plugging away with an obsolete
system, which had already been thrashed to death by earlier generations,
and have regularly temporarily resuscitated it for later generations,
as historical curiosities.

I appreciate that there is a strong bias against meantones tunings,
by many tuning list subscribers, who still praise, rank, map, and compare
their JI and ET tunings to "beatless perfection".

I seem to be seeing/hearing a revival of interest in meantones since
their possibilities now made available by recent technological
developments.

Despite their unfortunately ugly name, "meantone tunings" to my
biased mind have many redeeming characteristics which can now be
used practically.


1) Modulation and transposition is easy and limitless.
2) The tunings are clearly and easily defined by:
a) size of Large interval (or difference between
Fourth (IVth) and Fifth (Vth) interval) plus
b) Octave Ratio.
3) All spiral and most circular tunings can also be described in
these terms. Eg. Pythagorean, and almost all ET systems.
4) Harmonic and intervallic relationships are clear.
5) Scales can easily be unambiguously coded.
6) Degrees (or levels) of consonance and dissonance are obvious.
7) Current technology allows rapid retuning with conventional
fingering.
8) All intervals can be described in terms of Large (L) and
small (s) intervals to know precision.

9) The notation system is conventional and well-established.

BTW Has anyone else looked carefully at Bill and Anne Collins
Collinsian Concepts notation system (May 1995)?

So maybe it's time to take a second and deeper look at the
"neglected" meantone potentials.

Charles Lucy
lucy@hour.com
http://www.wonderlandinorbit.com/projects/lullaby

(*Ironic in connection with John "Longitude" Harrison's ideas on time
measurement and tunings. One "world beating" the other neglected.)

For download of his writings go to:
http://www.wonderlandinorbit.com/lucytuning/harrison/harrison.htm

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>>If you allow for the octave equivalence of intervals, all triadic
>>inversions contain the same intervals.
>How so? The way I larnt it, a simple C major triad in root position consists
>of a minor third stacked on top of a major third. In first inversion, it
>consists of a perfect fourth stacked on top of the minor third.

Well, this premise is certainly clear you instead think of a root
position major triad as a P4 atop a m3 atop a M3, and first inversion an M3
atop a P4 atop a m3, and so forth.

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>From the discussion:

>''>>If you allow for the octave equivalence of intervals, all triadic
>>>inversions contain the same intervals.
>>How so? The way I larnt it, a simple C major triad in root position
consists
>>of a minor third stacked on top of a major third. In first inversion, it
>>consists of a perfect fourth stacked on top of the minor third.

> Well, this premise is certainly clear you instead think of a root
>position major triad as a P4 atop a m3 atop a M3, and first inversion anM3
>atop a P4 atop a m3, and so forth.''


The three inversions of a triad (that is, three tones without octave
duplications) are intervallically distinct: (in 12tet interval classes:)
4,3,7; 3,5,8 ; 5,4,9. Throw in an additional tone, doubling the lowest
pitch at the octave, and you get: 4,3,5,7,8,12; 3,5,4,8,9,12; 5,4,3,9,7,12,
which are also distinct, shuffling the two-out-of-three combinations of
(7,8,9). Triadic inversions are equivalent (a) in terms of pitch class content, or(b) under some octave modulus operation where inversions are equivalent. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
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David Finnamore wrote:

>How so? The way I larnt it, a simple C major triad in root position consists
>of a minor third stacked on top of a major third. In first inversion, it
>consists of a perfect fourth stacked on top of the minor third. In second
>inversion it's that perfect fourth with the major third stacked above it.
> From what perspective are those the same set of intervals?

Um, yes. Technically there are six intervals in a major root position
triad: plus a major third, plus a minor third, plus a perfect fifth,
minus a major third, minus a minor third and minus a perfect fifth.
In octave equivalent terms, minus a fifth is the same as plus a fourth.
In these terms, then, all inversions of a major triad contain the same
intervals. I had overlooked the obvious simplification of only taking
positive intervals and this does, indeed, distinguish simple inversions.
However, in the specific method David mentions the chords C-E-G-C, E-G-C-E
and G-C-E-G have identical dissonance. I consider this to be a problem.
Taking all positive intervals, the first chord can be distinguished from
the last two (it's all in the sixths) which is less of a problem.

My mention of utonal/utonal degeneracy betrays some muddy thinking
on my part. What I meant is that a major triad has the same
intervals as a minor triad. This is a completely different issue
to inversions and octave equivalence, and I shouldn't have mentioned
it in that context.

What I mean by a "privileged root" is taking one note from the chord,
calling it the root, and treating it in a privileged way when you
analyse the chord. In traditional harmony, the root of an x major
or minor chord is x. It is considered that a chord is more
consonant if the root is the lowest note and the root is the most
common note in the chord. The problem with privileged roots in
general is that you have to specify why one note in particular should
be the root, and why it should have that privilege conferred upon it.

The best way, IMHO, of having octave invariance without inversional
invariance is to take the lowest note in the chord to be the root,
and measuring only positive intervals that include it. Hence,
C-E-G-C contains a major third, perfect fifth and perfect unison
(octave invariant octave). This is a sensible simplification,
distinguishes all simple inversions of major and minor triads and
tetrads, and is in good agreement with traditional harmony. A
fundamental theory, though, has to explain why this approximation
works. Although I don't know of such a theory, it would have to be
octave specific.

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