Lattice, LCM, and Aliquot Parts

Hi Graham,

On octave invariance, it has just occurred to me that, in LCM
terms, the octave has a property that no other ratio shares.
That is two notes that differ only by octaves can be mixed
without making the length of the resulting "interference" pattern
longer than the waveform of the lowest note.

If you mix two notes of wavelength 1 and 8, the wavelength of the
resulting pattern is still 8, but if you mix two notes of
wavelength 2 and 3, the resulting pattern has a wavelength of 6.

This property makes it much easier to sing in octaves than any
other interval, because the basic perception of the fundamental
frequency of a melody is not disturbed, and is a justification
for the principle of octave invariance.

Another form of invariance, ratio invariance, is important in LCM
analysis. This became clear to me during a discussion I had with
Paul E. some time ago. Thinking back over that, it seems to me
that our differences of opinion on the subject of the length of
LCM patterns was mostly due to different assumptions about ratio
invariance. I was assuming ratio invariance, he was not. I'm
still not absolutely clear on all this and would appreciate any
comments.

Aliquot parts are simply a list of all the possible numbers that
can be derived by multiplying subsets of the prime factors of a
given number. For example, here is a list of the aliquot parts
of 2880:

1 1
2 2
3 3
4 2^2
5 5
6 2*3
8 2^3
9 3^2
10 2*5
12 2^2*3
15 3*5
16 2^4
18 2*3^2
20 2^2*5
24 2^3*3
30 2*3*5
32 2^5
36 2^2*3^2
40 2^3*5
45 3^2*5
48 2^4*3
60 2^2*3*5
64 2^6
72 2^3*3^2
80 2^4*5
90 2*3^2*5
96 2^5*3
120 2^3*3*5
144 2^4*3^2
160 2^5*5
180 2^2*3^2*5
192 2^6*3
240 2^4*3*5
288 2^5*3^2
320 2^6*5
360 2^3*3^2*5
480 2^5*3*5
576 2^6*3^2
720 2^4*3^2*5
960 2^6*3*5
1440 2^5*3^2*5
2880 2^6*3^2*5

This list contains the seven-tone Zarlino scale, and 2880 is the
LCM of that scale. That is not a coincidence. A list of aliquot
parts contains all the scales that can be made from a given LCM.

Most of the utilities distributed with my FasTrak Sequencer use
this principle.

To classify the chords in a given scale, I just sort them by
their LCM's, or by their pattern length, if I do not want to
think in terms of ratio invariance. I find this to be much
simpler and easier to work with than multidimensional ratio
analysis.

But the big advantage, from my exact JI point of view, is that it
gives me an easy way to analyze the musical possibilities of a
given frequency divider network, and all modern electronic
musical instruments use some form of frequency division in the
sense that all the musical output is phase locked to some high
frequency signal in the system, therefore these systems have the
capability of producing exact musical intervals if programmed
properly.

Note that octave inversion of a chord can double its LCM.

Marion

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Edward Remler asks

> 1) Well Temperament arose in the latter half of the 17th century and
> Bach attempted to popularize it circa 1716 with his first WTC.
> Presumably the further a piece�s key is removed from C, the stranger it> will sound to the modern ear, but I have never heard the preludes and
> fugues played on a well tempered instrument. Does such a recording
> exist?

You forget that equal temperament is also a well temperament.
A tuning is a well temperament when there is no wolf fifth and it can
be played in all keys. One precise definition is that there may be no
fifths larger than pure and no major thirds smaller than pure.
Your presumption does not hold for well temperaments. So, I don't know
of any recordings of Das Wohltemperierte Klavier which are not in a
well temperament! On the other hand I neither know records which have
the tuning specified on the cover, but haven't made a real investigation.
There have been interesting discussings on the list before about which
tuning Bach would have had in mind. You might want to read those. It
started in Sept. 1995. The old digests can be downloaded from the Mills
server.
I don't have an answer for your 2nd question.

> Any information
> about the transition to Equal temperament would be of interest to me.

There is plenty of information. If you search for the string "temper"
in the tuning bibliography then you get a match for most languages
that you can narrow down.
ftp://ella.mills.edu/ccm/tuning/papers/bib.html

Manuel Op de Coul coul@ezh.nl

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Ed Remler asked,

>> I have never heard the preludes and
>> fugues played on a well tempered instrument. Does such a recording
>> exist?
>
>
>The only one I have heard is the recording by Anthony Newman. It was
>wonderful; my only complaint was that the harpsichord sounded rather shrill.
>My roommate in college had these on cassette and I have not been able to find
>them since. I know Davitt Moroney has recorded at least some of WTC and I
>suspect that they may be in a historical tuning (he has also done some
>meantone Byrd recordings) but I don't know for sure.

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I have the impression that most, if not all, recent recordings of Das
Wohltemperierte Klavier on 'historical instruments' are in some variety of
historical tuning. In fact, it is such a commonplace that it won't normally
get mentioned in the liner notes - but is immediately audible. Starting
with Dutch early music specialists in the sixties, most professional
harpsichordists and fortepianists have done their own tuning, and usuallywith a single tuning fork (typically using the A440A as Bb to get something
something close to A415). Learning to set meantone, a Kirnberger or a
Werkmeister by ear isn't terribly hard - while tuning 12tet is really
specialist work unless you're using a digital tuner or a strobe.

At both of my Universities in the States, and here at the Frankfurt
Musikhochschule, the early keyboard faculty and students are themselves
responsible for tuning the instruments. There has been some influence on
the piano world as well - Keith Jarrett, enthused by Lou Harrison's
_Concerto for Piano with Selected Orchestra_ keeps a piano tuned (by
himself) in Kirnberger II, and has released recordings using it.

In the organ world, it has become standard practice to build new or to
restore old tracker organs in historical tunings, in some cases with split
keys. The Oregonian composer Douglas Leedy has made a small career out ofcommissions for new pieces for new organs in historical temperaments. Along
these lines, I guess that Ligeti's _Passacaglia Ungarese_ was the first
work in this century to be composed specifically for a meantone instrument
(Harpsichord).

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