Yahoo Tuning Groups Ultimate Backup tuning Re: 3-limit harmony

Hi Brett,

At least I'll make sure to get the name right this time:).

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

How about 65536/59040 or 256/243. Face it, limits are pretty useless at
predicting the consonance of individual intervals. As a matter of fact, I
think it can be shown that if you don't care how big the numbers you use are,
you can come as close to any arbitrary ratio as you want just using factors
of two and three.

Considered as a parameter of scales, or even some of the more complex chords,
the limit concept has a definite utility, but I still think it pales next to
the utility of LCM.

<<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, ...>>

One important property of a scale is the number of consonant intervals you
can generate as a function of the numbers of notes. I think the fact that 3
limit scales do poorly on this measure when compared to 5-limit scales is one
of the major reasons the 5-limit concept has prevailed.

Below is a typical 7-tone, 3-limit scale with all the triads it will allow
with LCMs of 360 or less:

LCM: 2^9*3^6

7 729 243/ 128
6 648 27/ 16 Pyth Major Sixth
5 576 3/ 2 Perfect Fifth
4 512 4/ 3 Perfect Fourth
3 486 81/ 64 Pyth Major Third
2 432 9/ 8 Major Tone
1 384 1/ 1 Unison

Chord List:

1 72 6: 8: 9 1- 4- 5
2 72 6: 8: 9 2- 5- 6
3 72 6: 8: 9 3- 6- 7
4 72 8: 9: 12 1- 2- 5
5 72 8: 9: 12 2- 3- 6

Now for a 7-tone, 5-limit scale with all the triads it will produce with LCMS
of 360 or less

LCM: 2^4*3^2*5^2

7 90 15/ 8 Major Seventh
6 80 5/ 3 Major Sixth
5 75 25/ 16
4 72 3/ 2 Perfect Fifth
3 60 5/ 4 Major Third
2 50 25/ 24 Minor Semitone
1 48 1/ 1 Unison

Chord List:

1 60 4: 5: 6 1- 3- 4
2 60 4: 5: 6 3- 5- 7
3 60 10: 12: 15 2- 3- 5
4 60 10: 12: 15 3- 4- 7
5 60 12: 15: 20 1- 3- 6
6 72 6: 8: 9 3- 6- 7
7 90 5: 6: 9 2- 3- 7
8 90 6: 9: 10 1- 4- 6
9 90 10: 15: 18 2- 5- 7
10 120 5: 6: 8 2- 3- 6
11 120 8: 10: 15 1- 3- 7
12 120 8: 12: 15 1- 4- 7
13 180 15: 18: 20 3- 4- 6
14 240 10: 15: 16 2- 5- 6
15 240 12: 15: 16 3- 5- 6
16 360 5: 8: 9 2- 6- 7
17 360 24: 40: 45 1- 6- 7
18 360 36: 40: 45 4- 6- 7

No wonder the Greeks didn't like harmony.

<<
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.
>>

On the contrary, everything about a scale is important. It is certainly your
right to choose your own scales, and your own criteria for choosing scales,
but I see it as unlikely that the rest of the world with see things exactly
as you do. It could be that those who consistently ignore what you are
trying to say above do so for reasons that you haven't grasped yet. Of
course it could also be that the rest of us don't understand something that
you do, but as of now, I am leaning toward the first hypothesis.

I sense an analogy between the ways the English language are taught and the
differences in viewpoint we are experiencing here. One way is the standard
prescriptive grammar method that we all struggled through in high school.
Some grammar book has all the answers and it is up to the student to conform.
The other way is linguistics. The speaker is functioning in his society,
and it is up the scientist to describe what he is doing. I personally favor
the linguistic approach.

In music that would mean that a scale is a means of describing what a
musician is doing, not a means of telling him what to do.

Marion

> Hi Brett,
>
> At least I'll make sure to get the name right this time:).

You didn't, through no fault of your own. I haven't signed my messages, but my name
is Paul Erlich. Though I was previously posting only from my office, I wanted to
participate from home as well, so I signed up through my roommate's computer.

> <<OK, Marion, give us an example of a dissonant interval whose odd limit is
> 3.>>
>
> How about 65536/59040

That reduces to 2048/1845, which has an odd limit of 1845, not 3.

> or 256/243.

That has an odd limit of 243, not 3.

> Face it, limits are pretty useless at
> predicting the consonance of individual intervals. As a matter of fact, I
> think it can be shown that if you don't care how big the numbers you use are,
> you can come as close to any arbitrary ratio as you want just using factors
> of two and three.

That's a fair indictment of prime limits, one that I have used myself. But in your
previous message you spoke against both prime and odd limits. Did you not have two
separate things in mind?

> Considered as a parameter of scales, or even some of the more complex chords,
> the limit concept has a definite utility, but I still think it pales next to
> the utility of LCM.

It's apples and oranges. The prime limit of a tuning system tells you the most
complex interval you need to be able to tune in order to tune up the whole system.
That's useful for what it is, and LCM is useful for something else.

> <<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, ...>>
>
> One important property of a scale is the number of consonant intervals you
> can generate as a function of the numbers of notes. I think the fact that 3
> limit scales do poorly on this measure when compared to 5-limit scales is one
> of the major reasons the 5-limit concept has prevailed.

Marion -- this discussion was about styles in which only 3-(odd) limit intervals
were considered consonant. Of course, once 5-limit intervals were accepted as
consonant, the tuning of the scales needed to be tweaked in order to increase the
total number of consonant intervals. This led to developments such as schismatic
tuning and meantone temperament, which you can read more about in Margo Schulter's
paper (and I recommend you do!) But before that, only fifths and fourths were used
as consonances in the music and other intervals were used as dissonances. To apply
your criterion of maximizing the number of 5-limit consonant intervals would be a
terrible anachronism if applied to Western music c. 900-1200.

> <<
> 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.
> >>
>
> On the contrary, everything about a scale is important. It is certainly your
> right to choose your own scales, and your own criteria for choosing scales,
> but I see it as unlikely that the rest of the world with see things exactly
> as you do. It could be that those who consistently ignore what you are
> trying to say above do so for reasons that you haven't grasped yet. Of
> course it could also be that the rest of us don't understand something that
> you do, but as of now, I am leaning toward the first hypothesis.

I beseech you to consider again what I said, in the context in which I said it.
That context was you claiming certain Pythagorean ratios could be better considered
as 19-limit ratios. If the Pythagorean scale were used against a drone at 1/1, then
indeed the ratio representing each pitch would tell you the harmonic interval it
produced, and I agree with you that 256:243 and 729:512 are far too complex to
really be a "target" for a harmonic interval.

But this whole discussion was in response to your claim that Pythagorean tuning was
never really used, or doesn't represent a valid model for the tuning of any real
music. In Medieval times in Europe, however, music did not consist of a drone
against which all other pitches were played. Music was polyphonic, consisting at
first of lines in parallel fourths and fifths, and later of independent lines where
fouths and fifths (and octaves) functioned as the consonant intervals and were
often approached by contrary motion. In this context, your 19-limit ratios for some
pitches could only worsen the consonant intervals, while having an effect on some
dissonant intervals that would probably be rather unimportant but might at worst
hinder their ability to function as dissonances.

Both of our criteria for choosing scales are valid ones, albeit in different
musical contexts. All I'm trying to demonstrate to you is that there was in fact a
place in musical history/geography where Pythagorean tuning makes the most sense as
a model for what musicians were doing. I was disappointed that after my previous
attempts to do this, you proceeded to analyze the scale in ways which are not
relevant to the style in question.

> In music that would mean that a scale is a means of describing what a
> musician is doing, not a means of telling him what to do.

I fully agree with that philosophy! When 5-limit intervals did start being used as
consonances in the 15th century, theorists were still clinging to the Pythagorean
scale and were out of date. It took them some time to catch up with what musicians
were doing, but when they did, they had useful contributions to make, e.g. in the
field of tuning keyboard instruments.

But prior to that, in the Gothic period, a music flourished that sounds very
strange to modern ears. In that music thirds and sixths were not consonant but
dissonant. Many of the progressions from that era sound backwards to me, a musician
who grew up with and practices triadic harmony. As far as tuning is concerned, our
best hope of understanding the music of this era, and by understanding I mean all
the ineffable types of "knowledge" that are involved in appreciating a work of art,
is to use Pythagorean tuning, since it makes all the intervals that functioned as
consonances as consonant as possible, it makes all other intervals dissonant, in
accord with their function, and it represents the way fixed-pitch instruments were
in fact tuned according to virtually all accounts from the time. We will never be
able to fully divorce ourselves from the triadic environment in which we were
brought up. But using the appropriate tuning may at least help us to partially do
so.

Margo Schulter has suggested 17-tET as a "Xeno-Gothic" tuning. Using this tuning
for Gothic music may help to overcompensate for our modern, 5-limit listening
tendencies. The thirds and sixths generanted by the 17-tET fifth have no
simple-ratio approximations and would function more clearly as dissonances, while
the mistuning of the 17-tET fifth itself is less than 4 cents, i.e., fairly
undetectable.

Re: 3-limit harmony

🔗Mckyyy@xxx.xxx

🔗monz@xxxx.xxx

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>