Re: Reply to Marion

> Hi Greg,

Hi (who's 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?

That is true not of the Greeks, but of the Europeans from 900-1200 (and to some extent through 1450). You
should read Margo Schulter's stuff again.

The ancient Greeks did not like harmony in their music at all.

> 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.

OK, Marion, give us an example of a dissonant interval whose odd limit is 3.

> 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.

Again, Marion, you are making the mistake of assuming that all these pitches will be measured against the
1/1. That is not so. 729/512 is so tuned in order to be consonant with the 243/128, which is so tuned in
order to be consonant with the 81/64, which is so tuned in order to be consonant with the 27/16, which is so
tuned in order to be consonant with the 9/8, which is so tuned in order to be consonant with the 3/2, which
is so tuned in order to be consonant with the 1/1. And that is a complete account of the relationships that
would have been considered consonant in Medieval times. 3:2 and 4:3 in all cases. No other relationships
were considered consonant. The properties of 729/512 sounding against 1/1 are completely irrelevant (except
for the important property that it is not consonant).

People,

Just because ratios are being used to measure pitches relative to a fixed 1/1, doesn't mean the harmonic
properties of those ratios have ANYTHING to do with music constructed with those pitches. It is only the
ratios BETWEEN pitches that are actually sounded together in the music that matter. We have to be careful to
distinguish between ratios used to represent pitch and ratios used to represent intervals. The best way I've
seen is to use a slash (e.g., 729/512) when representing pitches and to use a colon (e.g., 3:2) when
representing intervals. But a notational convention will only help if we are thinking carefully about what
is going on.

Marion wrote:

> With a lattice approach you might, for example use the familiar:
>
> 6 9
> 5 8 15
> 4 6 16

What is that? It doen't remind me of any lattice approach I've ever seen. A lattice is an infinite
arrangement of points such that the configuration of every point relative to its neighbors is the same as
that of any other point relative to the second point's neighbors.

> 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.

What does electronics have to do with it?

> 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.

Approximations exist in JI as well. It is unavoidable that one will hear a very complex ratio as a simple
one if it the former is very near to the latter, e.g., 30001:20001 will be heard as 3:2.

> 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.

Again, I don't know what lattice approach you're talking about. The triangular lattice approach that has
been used on this list for some time now (by me, Paul Hahn, Graham Breed, Carl Lumma, Dave Keenan, bram,
etc.) is efficient for finding scales where many consonant structures can be found, parellelism or no.

Furthermore, the rectangular lattice and the LCM idea are intimately tied together, as shown by Euler's work
centuries ago. Tenney's complexity measure for intervals on the rectangular lattice is just the log of the
LCM.

> 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.

Do you mean that the fact that she made contributions to the list made you stop thinking her paper was
authoritative?