From: mclaren
Subject: "The" Pythagorean third
--
It's common knowledge that the Pythagorean
major third is formed by the difference twixt
four 3/2s up and 2 octaves up. 1.5^4/2^2 =
1.265625 = 407.82 cents.
Everyone knows this, and it's wrong.
There is no one single "Pythagorean major third."
In fact there are an *infinite* number of Pythagorean
thirds--major, minor and neutral.
This post describes a few of the more obvious ones.
Far and away the most major-3rd-like Pythagorean major
third is obtained by taking the difference twixt 8
3/2s down and 3 octaves down: (4/3)^8/2^3 = 1.2485899
= 384.35965 cents.
Is there some reason why we can't call this interval
= 384.35965 cents.
Is there some reason why we can't call this interval
"Pythagorean"?
Was it not obtained by taking the difference between
a chain of 3/2s and a chain of octaves?
Why must we use "the" 407.82-cent Pythagorean third
instead of *this* 384.35965-cent Pythagorean third?
Is this 384.35965-cent major third any less
"Pythagorean" than any other?
Why isn't *this* third called "the" Pythagorean
major third?
Can someone explain this?
Next, let's consider the difference twixt
56 just fifths up and 32 octaves up: this is
(3/2)^56/(2^23) = 1.6910274. The difference
twixt this interval & an octave is 368.2238 cents.
This is a Pythagorean quasi-major third. It sounds
consonant to my ears, albeit somewhat more neutral
than major. Still, if we're going to use a third that's
consonant to my ears, albeit somewhat more neutral
than major. Still, if we're going to use a third that's
off from the 5/4 just major third, why not use a third
that errs toward a neutral sound, rather than a third
whose size significantly *exceeds* the just 5/4 major
third?
Can someone explain this?
The 368.2238-cent Pythagorean third is arrived at
strictly by adding 3/2s and taking the difference
from a string of octaves. There's nothing exotic
involved. Yet *this* Pythagorean third is far more
palatable than "the" Pythagorean third when harmonic
series timbres are employed, sans reverb.
A slightly larger quasi-major Pythagorean third occurs
by taking the difference twixt 191 just fifths up and
112 octaves up: this is (3/2)^192/2^112 = 1.2421162 =
375.36017 cents.
A still larger quasi-major Pythagorean third pops up
when we subtract 551 3/2s up from 322 octaves up:
A still larger quasi-major Pythagorean third pops up
when we subtract 551 3/2s up from 322 octaves up:
i.e., (3/2)^551/(2^322) = 1.2434409 = 377.205 cents.
Of course, you might prefer a larger major third.
No problemo.
There are infinitely many.
Take the difference twixt 80 just fifths down and
33 octaves down: 243.59 cents.
Now take the difference between 71 just fifths up
and 41 octaves up: 638.80506 cents.
Now take the difference between the two intervals:
it's 395.21 cents.
Someone will want to post a lengthy screed explaining
why this interval cannot be used as a major third.
Smoke & mirrors would be helpful, along with plenty
of double-talk.
A slightly larger Pythagorean major third can be
found by taking the difference between 204 just
fifths up and 119 octaves up: that is, (3/2)^204/2^119
found by taking the difference between 204 just
fifths up and 119 octaves up: that is, (3/2)^204/2^119
= 1.2590627 = 398.82018 cents.
Moving on to neutral thirds, take the difference
between 32 just fifths up and 18 octaves up:
this is 337.44002 cents. An interval pleasantly
between the just 6/5 and the just 5/4. Yet this
interval is also Pythagorean.
Sounds fine, works well.
What's the problem?
Why do we "need" to use a 407-cent third in the
Pythagorean tuning system? What law of
nature requires it?
Another Pythagorean neutral third is obtained by
taking the difference between 21 just fifths up
and 14 octaves up. This is 341.055 cents. The
interval proves entirely acceptable in triads:
it sounds consonant to my ears. What's the
difficulty?
it sounds consonant to my ears. What's the
difficulty?
An almost perfect neutral third is obtained by
taking the difference between 180 just fifths
up and 105 octaves up: (3/2)^180/(2^105) =
1.2253978 = 351.90016 cents.
A more elaborate way of obtaining a neutral
Pythagorean third involves first taking the
difference between 92 just fifths up and 53
octaves up: 220.13993 cents. Now take
the difference between 35 just fifths up
and 20 octaves up: 568.42508 cents.
To get the neutral third, subtract the larger
Pythagorean interval from the smaller one:
this gives 348.2851 cents. Since this is
close to the geometric mean between 3/2
and 1/1, the difference tones of a just-fifth
triad with this third will be more in tune
with one another than in a 4:5:6 or a 10:12:15
triad with this third will be more in tune
with one another than in a 4:5:6 or a 10:12:15
triad. Thus this third is superior to the just
5/4 or the 6/5 third by at least one yardstick.
To obtain Pythagorean minor thirds, take
the difference twixt 44 just fifths up and
25 octaves up: this gives 313.97996 cents--
an extremely appealing minor third, quite close
to the 6/5.
Alternatively, you can get another Pythagorean
minor third by taking the difference between
13 just fifths up and 7 octaves up: 317.59501
cents.
So why does anyone talk about the purportedly
"large" size of Pythagorean thirds? The Pythagorean
tuning system is infinite. In it, thirds of *any*
desired size can be obtained. Like all unbounded
just systems, the Pythagorean tuning system
admits of an infinite variety of intervals when
just systems, the Pythagorean tuning system
admits of an infinite variety of intervals when
carried on far enough.
Incidentally, the operation of going up by X
just fifths and Y octaves, taking the difference,
then subtracting it from an octave is identical
to going down by X just fifths and then down by
some other number (< Y) of octaves & taking
the simple difference between the two intervals.
When explicitly calculating a chain of 3/2s going
down, one takes the inversion of 3/2 = 2/3 and
octave-reduces to obtain 4/3, which interval is
then raised to a power equal to the number of
fifths by which you want to go downward. The
difference twixt this interval and a downward
chain of octaves is then calculated by dividing
(4/3)^X by 2^Y.
--mclaren
