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Press Release
Dr. Allen has been studying so hard to learn how to play the blues
for you! “I said if I listen to these guitar tunings they will talked to me,
and the tunings told me to follow the numbers in the tablature.”
“It’s a good system to copy blues guitar licks off a record,” says Dr.
Terry Allen of Spokane Washington, “because you can follow where
notes go in 3 directions by arithmetic, which you can’t follow by ear.
“Using vectors, which are like geometric arrows, I changed the
tuning on my guitar. That’s how I discovered music is a Hilbert Space,”
says Allen. I found 3 vectors that could change the guitar tuning to 3
different states on guitar, that makes a triangle between the 3 tuning
states. (Key of E Standard Tuning/Key of D Drop D Tuning/Key of E Drop
D Tuning, and back to Standard, See Figure 1).
“I was surprised”, said Allen, “to find 3 perpendicular vectors in 6
dimensions! How could vectors make a triangle with 3 right angles?”
“I compared the music triangle I discovered to a similar triangle
that Euler, the Swiss mathematician, drew in 1739. I discovered Euler
drew the original pitch-position triangle incorrectly. He thought the
triangle was flat (co-planar). This is an odd error because it was the
Pythagoreans who created the music triangle, making Hilbert space
when they used the square root of 2 to temper the scale, making the
steps on the hypotenuse equal number of steps the sides: this turns the
Pythagorean formula for distance on its head. The triangle can’t be flat
because the hypotenuse is bowed. The Greeks made music a Hilbert
space! That’s the whole idea of tempering, make everything have the
same uniform metric.
Ever since Euler everybody just follows the primrose path of Euler’s
mistake, without re-thinking the problem. Allen says Euler’s ‘Donut’ is
the mathematician’s egg. Since then, we have the common tone period
where music is seen as a union of harmonic and chromatic scale
functions. “Mathematically I discovered a number ring over the real
closed field. That defines Hilbert Space”, says Allen.
“If Euler had used my triangle”, says Allen, “Euler would have
assembled a sphere, not a donut. Euler’s 2-center system has no room
for 6 guitar strings. In my system, says Allen, you have as many centers
as you want, but only 1 fundamental can exist in any music system.
“So music isn’t just the two harmonic and chromatic scales,’
continues Allen, who’s played guitar for 50 years, it’s also the product
of the harmonic and chromatic functions as the dependent variables,
and the chromatic and harmonic functions also combine to make a
third direction of tone movement possible (a composite function of the
2 scales). Mathematicians call this SO3 symmetry.
“Using this system,” says Allen, “I own Open G and D, at least a
dozen more tunings. To do this, you got to know how!”
Experimental Demonstration:
Pitch-Position moves by pitch, position,
and pitch-position as a composite
Demonstrating 3-Fold Path of Tonal Movement
On Piano:
1.
The chromatic path by note
2.
The harmonic path by key
3.
And on the transposing piano a third path by mechanical lever (algebraic
operator) that moves the instrument position relative to strings.
Notes on the Music Staff
1.
By Position on Staff
2.
By Clef
3.
By Concert Pitch
On Guitar
1.
By Strings so that String Number and Pitch Value Rise Together (by tuning)
2.
By Frets so that Fret Number and Pitch Value rise together
3.
By String and Fret as a composite function where String Number rise or fall but
pitch values do not (iso-pitch line ciphers pitch not position)
A String
1.
Mass
2.
Tension
3.
Length
Tablature
1.
By Lower Note
2.
By Tuning
3.
By Capo
By Transposition
1.
To change position literally
2.
Also to change pitch
Pitch-Position can move as a composite function.
3 Different Guitar Tuning States Related
by 3 Perpendicular Vectors Make An
Equilateral Right Triangle
Key of E
Standard
0 -2 -2 -2 -2 -2
3 points
3 equal sides
3 right angles
6 dimensions
-2 0 0 0 0 0
Key of D
Drop D
222222
Key of E
Drop D
Vectors are perpendicular if the sum is 0, or if they do not influence each other.
The vectors (0 -2 -2 -2 -2 -2) and (-2 0 0 0 0 0) change tuning but are
perpendicular; summing these with (2 2 2 2 2 2) equals zero: therefore all 3
vectors are perpendicular. Note vectors add and do not multiply because pitch =
log frequency. This is the same triangle that Euler misconstrued as planar with 2
centers, but instead must have 1 center.