(no subject)

Hi Greg,

<<
But the major sixth was considered a dissonance, so tuning it to a consonance
would
be contrary to its musical function. Rather than being a missed harmonic
opportunity, the dissonant sixth gave the music impetus to move forward and
resolve.
>>

You're saying that the Greeks sounded this ratio in their music, but didn't
consider it consonant? Does that mean that the only "consonant" ratios were
3/2 and 4/3?

<<
I don't know what you mean. There's plenty of technology that can measure such
things to sufficient accuracy.
>>

It's not clear to me that the above statement is entirely accurate. It's
true that it is relatively simple to measure the frequency of a simple
periodic waveform that does not vary over time, but signals of that degree of
simplicity rarely occur in music. Even a simple periodic waveform whose
harmonic content is changing as a function of time is difficult to measure,
especially if a small amount of the
spectral energy is expressed in the fundamental. That can be done, to some
degree of accuracy, but it it's quite difficult to determine what the
accuracy actually is, depending on the system you are using for your
measurement.

When you get into directly measuring the frequencies of notes that are
sounded together, you are in a whole different universe, and one that is not
thoroughly analyzed. At least that was the state of the art a few years ago,
when I had a lengthy discussion on the subject with a lot of knowledgeable
people. That may have been on the Mills College Tuning List, or a music
discussion group on Bix. I suppose if it was the Mills College List, then
some of the present correspondents might remember.

Perhaps the state of the art has been recently fundamentally improved?

I suppose you could put musicians in isolation from on another and
electronically control the distribution of sounds, and thus isolate all
single channel musical instruments so that pitch analysis would be easier.
That would still be not help for multiple voice acoustical instruments like
guitars or violins. Does anyone know of such a project?

<<
> The limitations of the chain of fifths idea have been discussed in this
forum
> many times. It seems to me that lattices are just an extension of this
> concept and suffer from the same limitations. These concepts, a kind of
> musical parallelism, are useful, but I find other methods better, especially
> in the context of discretely-tuned electronic musical instruments.

Oh?
>>

I am often faced with the problem of getting accurately-tuned music from a
discretely tuned electronic musical instrument. For example, wavetable
synthesis can be viewed mathematically as a frequency divider. To assess
what JI scales you can accurately produce on a frequency divider, you only to
know the maximum LCM the frequency divider can accommodate. Unfortunately,
present day musical instrument manufacturers will not divulge such
information, but if you have it, you can rather simply determine what scales
you can use, for example with my Aliq program.

With a lattice approach you might, for example use the familiar:

6 9
5 8 15
4 6 16

to generate the following scale:

1/1 16/15 9/8 6/5 5/4 4/3 64/45 3/2 8/5 5/3 16/9 15/8

The LCM of this scale is 1,036,800. Using Aliq, or the LCMscal program from
FasTrak, it is fairly simple to ascertain that there are eight 14-tone scales
with this LCM, and a total of 69 scales with 12 or more tones. To get this
information with a lattice process is tedious, at best. Just multiplying a
number of lattice elements together is prone to producing scales that have a
low LCM to tone ratio. In other words, scales which will not fully utilize
the tuning capacity of a discretely-tuned electronic musical instrument.

Of course, these considerations only apply to exact JI. If you allow
approximation to come into the picture, then you are in a different universe,
which in my opinion is not a JI universe but a tempered universe.

In a more general sense, I find LCM useful because it describes the pattern
of waveform interference when a chord is sounded. Here, I am speaking of the
LCMs of the periods of the sounds, not the frequency, though in practice, not
a lot is lost by using the LCMs of the frequencies. I am not claiming LCM is
a completely accurate description of consonance, but I think it certainly
tracks a lot better than limits, be they prime or odd. It is certainly
possible to generate 3-limit intervals that nearly everyone would find
dissonant.

Not all high LCM chords are perceived as dissonant. A good example is 12et
in which all intervals except the octave have an infinite LCM. On the other
hand, all low LCM chords are perceived as consonant, as demonstrated by the
perfect fifth with an LCM of 6, or the 5:3 with an LCM of 15, or the 5:4 with
an LCM of 20. Moreover, it can be demonstrated that those high limit chords
that are perceived as consonant are in some sense a superset of some low LCM
chord. Take for example, the 12et fifth of 1.4983 etc. The real LCM is so
large that it is completely outside the pattern-recognition ability of the
ear to integrate into the realm of harmony, so what the ear perceives is a
fifth with changing phase and frequency relationships resulting in the famous
"beat".

The lattice approach is efficient for generating scales which contain
specified parallelism. That is the above lattice is guaranteed to generate
scales which have 4:5:6 chords. If some sense, the lattice elements are
matrix factors of the scale. You can use the aliq program to detect what
parallelism exists in a scale by using the set select function from lattice
mode.

<<
P.S. I think Margo Schulter addressed your comments in a much more polite but
also
much more historically informed way. How would you respond to her?
>>

It seems to me that both you an Margo, as well as others on the list are much
better informed on music history than I am, and I imagine things will stay
that way as I am much more interested in the future than the past. When I am
studying the past I find it both interesting and informative, but my focus is
on creating new technology. While I have more time to devote to music now, I
also have several other things going which demand my time and energy.

I have to admit that I haven't been paying a lot of attention to the list for
the last couple of years while I was down in San Jose making some money.
When you questioned my assertion about pentatonic scales, I didn't believe at
first, and I did some web surfing to check. I came across a paper by Margo,
which I took as authoritative, but at that time I was unaware of her
contributions to the list.

I did clip the Pythagorean Modes from her paper, and do some analysis from an
LCM point of view. It seems that Pythagorus somehow managed to include all
seven of the scales with LCMs less than 2^10*3^6, or 746,496. I'm sure the
Greeks were aware of LCM as an abstract concept, but it is unclear to me how
they related it to music.

Also, I became interested in the problem of what smaller LCM approximations
are available for Pythagorean ratios, so I took some ratios from the
pythagorean modes and did some investigation.

The 16/9 ratio has an LCM of 144, but it can be approximated by 7/4 with an
LCM of 28 and a 1.59% error.

The 27/16 ratio has an LCM of 432, but it can be approximated by 5/3 with an
LCM of 15 and a 1.25% error.

The 32/27 ratio has an LCM of 864, but it can be approximated by 13/11 with
an LCM of 143 and a 0.28% error. It can also be approximated by 6/5 with an
LCM of 30 and an error of 1.23%.

The 81/64 ratio has an LCM of 5184, but it can be approximated by 19/15 with
an LCM of 285, and an error of 0.08%. It can also be approximated by 5/4
with an LCM of 20 and an error of 1.25%.

The 128/81 ratio has an LCM of 10,368, but it can be approximated by 19/12
with an LCM of 228 and an error of 0.19%. It can also be approximated by 8/5
with an LCM of 40 and an error of 1.23%.

The 243/128 ratio has an LCM of 31,104, but it can be approximated by 19/10
with an LCM of 190 with an error of 0.08%.

The 729/512 ratio has an LCM of 373,248, but it can be approximated by 10/7
with an LCM of 70 and an error of 0.33%.

Is it possible that these musicians were trying to play 3-limit but were
really playing 19 limit music? We'll probably never know, but we may someday
be able to accurately describe what contemporary musicians are doing with a
reasonable degree of accuracy.

It seems clear to me that the 729/512 ratio at least was almost never used in
practice. At 440hz the pattern created by this interval would take 850
seconds, or 14 minutes to repeat-longer than most songs. Even the 32/27
pattern would take almost 2 seconds at 440HZ. Some of these claims of tuning
accuracy seem a little shaky to me.

Marion