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?
- 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.
- 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
- The Circle of Fifths is a planar projection of a helix;
- The Circle of Fifths is defined by the hypotenuse of Maj. & minor 3rds;
- The scale phase, or sweep of scales, is defined by the hypotenuse of a Tritone & Major 3rd;
- Pitch classes form one axis (here the Y axis),
- Diatonic Keys the other (here the X axis);
- Rotated 90 degr. clockwise & the pitches are aligned to the standard equilateral triadic matrix, in piano orientation;
- Rotated 45 degr. clockwise & this schema resembles the Balzano 3rds space.
- 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
via the www.triangle-calculator.com OL calculator:##
*** Green Triangle 3:4:5 ***
Calculating Right Scalene Pythogorean 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
- C: (0 ,0)
- A: (-10,2)