Pythagorean theory and practice

<<
Pythagorean tuning has been used for Chinese music for millenia. Its
importance in
the Hellenic, ancient Hindu, and medieval Arabic worlds seems to have been
great.
Wasn't Pythagorean tuning the standard tuning for Western music from before
900
through about 1450 A.D., giving it a longer reign here than any other tuning
system?
>>

If you'll notice, I said "practical use." I am aware that Pythagorean tuning
has been the "official" standard in much of the world for a great length of
time, but I feel that that convention was honored greatly in the breach, much
like the 12et convention of modern times. For the 5-tone scales of
Pythagorean times, it is indeed possible to apply the Pythagorean 3-limit
idea, but when you start trying to crowd in more tones per octave, you run
into problems. Below are the four miminum-lcm 7-tone 3-limit scales:

Scale: 1

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

Minimum Chords:

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

Scale: 2

LCM: 2^9*3^6

7 768 16/ 9 Grave Minor Seventh
6 729 27/ 16 Pyth Major Sixth
5 648 3/ 2 Perfect Fifth
4 576 4/ 3 Perfect Fourth
3 512 32/ 27 Pyth Minor Third
2 486 9/ 8 Major Tone
1 432 1/ 1 Unison

Minimum Chords:

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

Scale: 3

LCM: 2^9*3^6

7 864 16/ 9 Grave Minor Seventh
6 768 128/ 81
5 729 3/ 2 Perfect Fifth
4 648 4/ 3 Perfect Fourth
3 576 32/ 27 Pyth Minor Third
2 512 256/ 243
1 486 1/ 1 Unison

Minimum Chords:

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

Scale: 4

LCM: 2^9*3^6

7 972 243/ 128
6 864 27/ 16 Pyth Major Sixth
5 768 3/ 2 Perfect Fifth
4 729 729/ 512
3 648 81/ 64 Pyth Major Third
2 576 9/ 8 Major Tone
1 512 1/ 1 Unison

Minimum Chords:

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

For example, let's take the 81/64 and 27/16 intervals from those scales:

3-limit dec 5-lim dec 3-limit to 5 limit ratio
81/64 1.2656 5/4 1.2500 1.0125
27/16 1.6875 5/3 1.6667 1.0125

Notice that the common 5 limit ratios and the 3-limit ratios are separated by
only 1.25%. What was the tuning accuracy of musical instruments during the
specified historical period? How many of those musicians and singers could
really train themselves to produce the Pythagorean interval instead of the
5-limit interval? At one time, I went to considerable trouble to hear the
difference between the two on a pure frequency divider instrument of great
accuracy that I constructed myself. You can hear the difference, but you
have to listen closely. Even most of our modern day electronic musical
instruments don't have the ability to exactly reproduce these intervals
without special low-level programming.

Notice that the lcm of the 81/64 interval is 5184, while the lcm of the 5/4
interval is 20. In my opinion, it is much more likely that singers or
musicians playing microtunable instruments would produce the simpler
interval. I don't think there have ever been any harmony police doing any
checking. Note that most instruments allow the musician some small control
over the pitch in real time.

Discussions of tuning theory are quite interesting, but it is also important
to realize that the theory must be put into practice to have any real meaning.

Marion

Marion wrote:

> Notice that the common 5 limit ratios and the 3-limit ratios are separated by
> only 1.25%. What was the tuning accuracy of musical instruments during the
> specified historical period? How many of those musicians and singers could
> really train themselves to produce the Pythagorean interval instead of the
> 5-limit interval? At one time, I went to considerable trouble to hear the
> difference between the two on a pure frequency divider instrument of great
> accuracy that I constructed myself. You can hear the difference, but you
> have to listen closely. Even most of our modern day electronic musical
> instruments don't have the ability to exactly reproduce these intervals
> without special low-level programming.
>
> Notice that the lcm of the 81/64 interval is 5184, while the lcm of the 5/4
> interval is 20. In my opinion, it is much more likely that singers or
> musicians playing microtunable instruments would produce the simpler
> interval. I don't think there have ever been any harmony police doing any
> checking. Note that most instruments allow the musician some small control
> over the pitch in real time.

Marion, I think your mistake is thinking that all intervals are measured from the 1/1. I
think Joe Monzo is making that mistake as well. In Chinese music (according to Yasser) and
in Western music from 900-1200 AD, the harmony consists of fifths and fourths. So the 81/64
would not be heard against the 1/1, it would be heard against the 27/16, for example. And
the 27/16 would not be tuned 5/3 because it would be heard against the 9/8, which would not
be tuned 10/9 because it would be heard against the 3/2. A chain of fifths, i.e.,
Pythagorean tuning, is clearly the best way to tune such music. Sure, vocalists could use
both 5/4 and 81/64 in the course of a piece of music, depending on immediate harmonic
context, but if we had to choose one for these musics, it would certainly be 81/64.
Melodically, also, the Pythagorean versions of the pentatonic and heptatonic scale are very
smooth and easily learned.

You can't take the major third (5/4 or 81/64) and analyze it out of the context of the rest
of the scale and the type of harmony it is to be used for!

> Discussions of tuning theory are quite interesting, but it is also important
> to realize that the theory must be put into practice to have any real meaning

And it is also important to realize that this practice will occur in a musical context
governed by particular stylistic constraints. That's where you failed in your attempted
connection of theory and practice. When you wrote:

>For the 5-tone scales of
>Pythagorean times, it is indeed possible to apply the Pythagorean 3-limit
>idea, but when you start trying to crowd in more tones per octave, you run
>into problems.

you were quite guilty of divorcing theory from practice. For you were applying a
simple-ratio principle to scales apart from any consideration of how these scales are to be
used, particularly what harmonic simultaneities will appear and what functions they will
have (and to what extent the 7-tone idea is relevant when chromatic alterations appear).
These factors are quite different for heptatonic music of different times and places, and
to assume a single abstract mathematical rule will be relevant to all these musical
situations is to be really adrift in theory-land. Also, the scales of Pythagorean times had
7, not 5, tones!

Marion wrote:

>Below are the four miminum-lcm 7-tone 3-limit scales:

While (as we discussed) lcm makes some sense for comparing chords (when used
to compare repetition periods within a given register), the lcm of an entire
scale doesn't correspond to any useful information.

Even if the scale is played as a giant chord, a diatonic scale will tend to
be heard not in terms of a single repetition period but several. The single
repetition period that fits mathematically is far too long to have any
musical or perceptual relevance.

But in most styles, the chords will not consist of the entire scale but of
very specific subsets. It is for these subsets that the lcm should be
evaluated.

P.S. If it were true that the entire scale was heard in terms of a single
repetition period, then you could never have a scale with a perfect fourth
above the tonic, since then the tonic would be analyzed as a "partial", not
as a "fundamental". But clearly the presence of a perfect fourth above the
tonic in the scale does not impede the tonic's function; most "tonal" scales
do have a perfect fourth above the tonic. Only in a chord does a perfect
fourth above the tonic "destroy" the sense of tonicity.

[Barbaro:]
> the scales of Pythagorean times had 7, not 5, tones!

Excuse me, but aren't we talking here about European music
from around 900 AD to 1450? (your dates)

The scales in use during that time period in Europe
were HEXachords - 6-note, not 7.

And before that it was TETRAchords (4-note).

> to assume a single abstract mathematical rule will be
> relevant to all these musical situations is to be
> really adrift in theory-land

On the contrary, what I'm saying is that musicians used
lots of different tunings that were only partially based
on the Pythagorean.

> your mistake is thinking that all intervals are measured
> from the 1/1. I think Joe Monzo is making that mistake as well.

Please quote me,
and show me exactly where I'm making that mistake.

Of course I assume all ratios are compared to 1/1 when
I construct a lattice, but the actual notes I place on
it, if I'm studying an historical tuning, are exactly
the ones specified in whatever treatise I'm researching.
And I make a real effort to get as close to the source
as I can.

Of course, there's no way at this late date that we can
know the precise tuning of instruments or, even less
likely, vocal music, from that far in the past. But
I weigh as carefully as possible everything that I read
in historical theory treatises, and try to place that
theory in perspective and in context.

All I'm saying definitively is that my research has given
me clues that musicians were using 5-limit ratios in European
harmonic vocal music long - *really* long - before the
theorists said they were.

I don't doubt that Pythagorean tuning ever was,
and still is, widely used in practice. It's still the
basis behind the all-prevasive 12-eq.

-monzo
http://www.ixpres.com/interval/monzo/homepage.html

|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |

--------------------------------

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Joe Monzo wrote:

> [Barbaro:]
> > the scales of Pythagorean times had 7, not 5, tones!
>
> Excuse me, but aren't we talking here about European music
> from around 900 AD to 1450? (your dates)

Marion was referring, I believe, to the times in which Pythagoras lived. Scales
generally had seven notes even then.

> The scales in use during that time period in Europe
> were HEXachords - 6-note, not 7.
>
> And before that it was TETRAchords (4-note).

In all of these systems there were seven notes per octave. Hexachords were
important when one of the seven notes was mutable; and tetrachords spanned a
perfect fourth and were used two at a time to fill an octave.

> > to assume a single abstract mathematical rule will be
> > relevant to all these musical situations is to be
> > really adrift in theory-land

> On the contrary, what I'm saying is that musicians used
> lots of different tunings that were only partially based
> on the Pythagorean.

I was responding directly to Marion so I don't know why you thought I was talking
about you.

> I don't doubt that Pythagorean tuning ever was,
> and still is, widely used in practice. It's still the
> basis behind the all-prevasive 12-eq.

Historically, meantone intervened between Pythagorean and 12-eq for a few centuries
and had 12-eq not been an acceptable approximation of meantone, it would not be our
standard tuning today, notwithstanding 12-eq's even better approximation of
Pythagorean. If you like, you can substitute "5-limit JI" for "meantone" and the
above is still essentially true. So yes, Pythagorean is _like_ 12-eq, but no, it's
not the basis behind 12-eq.

[Barbaro:]
> In all of these systems there were seven notes per octave.
> Hexachords were important when one of the seven notes
> was mutable; and tetrachords spanned a perfect fourth
> and were used two at a time to fill an octave.

OK - I'll give you the first one.
The second one - partially.

But tetrachords didn't always 'fill an octave' in
ancient Greek theory. One of their basic systems
was a 7-tone scale that only spanned a 'minor 7th'.
(see Nicomachus, _Enchiridion_.)

> Historically, meantone intervened between Pythagorean
> and 12-eq for a few centuries and had 12-eq not been
> an acceptable approximation of meantone, it would not be our
> standard tuning today, notwithstanding 12-eq's even better
> approximation of Pythagorean. If you like, you can substitute
> "5-limit JI" for "meantone" and the above is still essentially
> true. So yes, Pythagorean is _like_ 12-eq, but no, it's
> not the basis behind 12-eq.

OK, I can agree that 12-eq would not have been accepted
if it hadn't approximated meantone well, because meantone
*was* well-established during the 'classical' period.

However, I think more weight has to be given to the fact
that 12-eq approximates Pythagorean so well. Maybe I used
the wrong choice of words in calling it the 'basis behind
12-eq, but it's not just a coincidence or an accident.

A *big* part of the reason why 12-eq was finally accepted
(after much debate) is because those Pythagorean intervals
had been such an ingrained part of music - definitely theory,
and probably practice to some extent too - for such a long time.

Another thing that should be kept in mind is that, totally
aside from any intrinsic harmonic approximations, the
interval matrix (or 'square of the intervals', as Partch
called it) of the Pythagorean diatonic scale, if
used only melodically, strongly implies or approximates
12 equal divisions of the 'octave'.

This would naturally lead to thinking of the 'octave' as
being easily divisible by 12, even if the steps are only
quasi-equal.

|\=/|.-"""-. Joseph L. Monzo....................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\
_U//_/-/__/// |"...I had broken thru the lattice barrier..."|
/monz\ ((jgs; | - Erv Wilson |

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[Paul Erlich/Brett Barbaro, TD 142.13:]
> Marion was referring, I believe, to the times in which
> Pythagoras lived. Scales generally had seven notes even then.
>
>>[quoting me, monz:]
>> The scales in use during that time period in Europe
>> [I was referring to 900-1400 AD] were HEXachords -
>> 6-note, not 7.
>>
>> And before that it was TETRAchords (4-note).
>
> [Paul:]
> In all of these systems there were seven notes per octave.
> Hexachords were important when one of the seven notes was
> mutable; and tetrachords spanned a perfect fourth and were
> used two at a time to fill an octave.

These are good points, but there was one I was trying
to make that wasn't expressed clearly.

I had already responded to this:

[me, monz, TD 143.1:]
> OK - I'll give you the first one.
> The second one - partially.
>
> But tetrachords didn't always 'fill an octave' in
> ancient Greek theory. One of their basic systems
> was a 7-tone scale that only spanned a 'minor 7th'.
> (see Nicomachus, _Enchiridion_.)

But I thought more could be said on this point.

Altho ancient Greek theorists recognized the 'octave'
as a primary consonance, it didn't really come into play
as part of their scale constructions.

It is not until after the Romans conquered Greece
that theorists such as Nicomachus and Ptolemy (c. 100 AD)
specifically utilized the 'octave' in their descriptions
of scales. Before that it was all tetrachords,
and in many cases the systems used two *conjunct* tetrachords
which only filled the space of a 'minor 7th'.

I think there are two possible reasons for this
recognition of the 'octave': the use of the organ
(a Greek invention) by the Romans, and the introduction
of 'oriental' musical tendencies after the Roman Empire
expanded into Egypt and Mesopotamia, altho evidence
seems to go against the latter.

But acceptance and use of the organ as a musical instrument
certainly played a (if not *the*) major role in the
development of harmony/polyphony as it happened in Europe.

My theory is that the other major component (in the
development of harmony) was the high-ceilinged 'Gothic'
cathedrals with lots of reverb, which would have emphasized
the overtones in the singing of the choir. Perceptive listeners,
hearing the 5th partial emerging from the whole complex of
sound in the cathedral space, would have noticed that the
64:81 Pythagorean ditone ('major 3rd') did not 'fit' the
overall sound.

The first mention in European theory treatises of singers
altering the pitch of Pythagorean '3rds' and '6ths', so
that they more closely approximate 5-limit JI, is in
the late 1200s/early 1300s (Marchetto was one of them).

This was about 100 years after the invention of the
architectural techniques which enabled the construction of
the 'Gothic'-style cathedrals (such as Notre Dame in Paris).

This was also during the time of the Crusades, when
soldiers returning to Italy and France brought back
with them from 'the Holy Land' the ancient Greek theory
texts (such as Ptolemy's) which described all sorts of
non-Pythagorean ratios.

-monz

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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