Are there geometric symmetries in musical harmonics?

Question

This may be more of an acoustical physics question, but is there something fundamental about wave harmonics that leads to geometric symmetries like a 3:4:5 triangle in a Tonnetz triangle?

The image shows a somewhat modified Tonnetz schema (rotated counter-clockwise to staff orientation) whereby each pitch class & key gets its own domain. Like the more standard Tonnetze, this tone net establishes a matrix of the triads, based on the classical consonances of Maj & minor thirds & the Dominant fifths. However its structure is based on the intersection of pitch classes (here the Y axis) and diatonic keys (the X axis).

Is this a known property, that there are right triangles in triadic structures of wave harmonics? Or, in any propagation medium, wouldn't similar relationships between ablative (dissonant) & harmonic (consonant) waveforms arise, perhaps not only in acoustics?

N.b.

1. The 3-4-5 ratio arises from the ratio of the sides. Putting the harmonic intervals 3 - 4 - 7 on an x-y coordinate graph yields a ratio of 3:4:5. That is, the calculated lengths of the x-y segments on the coordinate graph yield a 3:4:5 triangle, irrespective of the named intervals. The actual spatial offsets of the minor 3rd is larger than 3, the Maj 3rd is > 4, and the 7 semitones of the dominant greater than 7. Please see calculation citations at bottom.
2. The "scale phase" is the regular, descending pattern of scale notes (the grayed tone boxes) that form successive keys (downward, going left-to-right). Note it interlocks periodically with the leading-tone/sub-dominant chain.

Tonnetz chart features

1. The Circle of Fifths is a planar projection of a helix;
2. The Circle of Fifths is defined by the hypotenuse of Maj. & minor 3rds;
3. The scale phase, or sweep of scales, is defined by the hypotenuse of a Tritone & Major 3rd;
4. Pitch classes form one axis (here the Y axis),
5. Diatonic Keys the other (here the X axis);
6. Rotated 90 degr. clockwise & the pitches are aligned to the standard equilateral triadic matrix, in piano orientation;
7. Rotated 45 degr. clockwise & this schema resembles the Balzano 3rds space.
8. Other features arise, including
   • the chain of subdominant and leading tones,
   • the shared pentatonic between adjacent keys (the 2nd, 5th & root of F, C & G, respectively),
   • the ability to visualize non-diatonic & non-triadic chords & cadences

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via the www.triangle-calculator.com OL calculator:##

*** Green Triangle 3:4:5 ***

Calculating Right Scalene Pythogorean Triangle

link to calculated Right Pyth. scalene triangle:

note        X,y coord.     actual lengths   factor              base
vectors                                     of 3-4-5 lengths    3:4:5
C → E       0,0 → 4,4       4.243           1.4143333333        3
E → G       4,4 → 7,1       5.657           1.41425             4
C → G       0,0 → 7,1       7.071           1.4142              5

*** Red Triangle ***

Right Scalene Triangle

link to calculated right scalene triangle:

• C: (0 ,0)
• Eb:(-4,-4)
• A: (-10,2)

Answer 1

When the Tonnetz is laid out like this:

we see that these triads are not right triangles, but rather equilateral triangles.

So although there are symmetries related to diatonic harmony (and diatonic scales, etc.)---and they are interesting and worth discussing!---the symmetries you are showing and asking about are really just a byproduct of how you've chosen to lay out your Tonnetz.

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With that in mind, note that the consonant triads---major and minor---are both constructed with one minor third and one major third. (The major triad has the major third on bottom with the minor third on top, and the minor triad inverts this.) Furthermore, the remaining interval can be viewed as a perfect fourth.

The minor third is 3 semitones, the major third is 4, and the perfect fourth is 5. So, assuming you lay out your geometrical space correctly, it's quite easy to walk away with a 3:4:5 right triangle.

Answer 2

Is this a known property, that there are right triangles in triadic structures of wave harmonics?

No. What I'm basing this on is in your use of (right) triangles and the idea of the hypotenuse, which implies quantities that add in quadrature ala the Pythagorean theorem.

Our sense of tone is intimately tied to the sounds that are (good approximations of) comprised of a overtone series that are integer multiples of a fundamental, and our sense of harmony only relates to these kinds of sounds. There is no adding in quadrature here, thus no triangles.