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).

Helical tonnetz, pitch/staff orientation

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?


  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

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)
  • Any additional utility to this tonnetz schema is arguable, although its geo-spatially orthagonal to pitch & key. [Equivalence to Cube Dance matrix transforms][1] [1]: drive.google.com/file/d/0B14eEoqHqa5RMzBYWm9VZGoydHM/…
    – L. Rodgers
    May 8, 2017 at 16:01
  • 1
    What is "scale phase"?
    – Dave
    May 8, 2017 at 16:52
  • 1
    Are you mathematically certain that the red triangle is a right triangle?
    – Richard
    May 8, 2017 at 17:00
  • I don't understand the geometry in this book, but you might like it: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory) by Dmitri Tymoczko. a.co/6DyCpbK May 8, 2017 at 18:07
  • TY Michael, I've delved into some of Tymoczko's tomes. Amazing stuff, but the 3D tetrahedral reflection of the equilateral triangles are too hard to use visually. But the way he tied it all together is completely amazing, esp. the relationship of the tone-pair helix, to the moebius, torus & 3D tonnetz.
    – L. Rodgers
    May 9, 2017 at 14:44

2 Answers 2


When the Tonnetz is laid out like this:

enter image description here

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.

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.

  • 1
    I'm left with an open ended conclusion here: Byproduct, or deeper feature? It's a bit like the matter of Amplituhedra or Adinkras in physics. Quantum physicists see them as indicative of the deeper physical properties of the media (quantum fields), as well as providing shortcuts to understanding physical phenomena. Their comrades in string theory are heavily into acoustic models, FWIW.
    – L. Rodgers
    May 8, 2017 at 16:08
  • @BertLee I misunderstood your original question; see edit.
    – Richard
    May 8, 2017 at 16:18
  • I should also mention that the shifting scale phase also follows a right triangle, a hypotenuse to Tritones vs. Maj. 3rds. Apophenia or confirmation bias notwithstanding, is something more elegant being overlooked?
    – L. Rodgers
    May 8, 2017 at 16:40
  • Right but the 3:4:5 is based on the 5ths (7 semitones) arising as the hypotenuse of the Maj & min 3rds. (TY for interjecting there Dave, Richard had me going there for a sec.)
    – L. Rodgers
    May 8, 2017 at 16:43
  • 1
    Byproduct, not deeper feature. The overtone series is simple geometry, not ratios of numbers of semitones however tuned. This Tonnetz may be a convenient way of looking at chords, but it tells you nothing about how sound works. May 8, 2017 at 18:09

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.

  • Perhaps, but I've noted that Elaine Chew has encountered similar relationships in her helical graph. The logarthmic / geometric progression is compressed in any graph we construct in music (assuming 12 TET). But the harmonics are inherent to consonant & ablative wave interactions. Whether we find one over the other more or less eurythmic seems inherent to acoustical consonances, incl. overtone series... Point being that a graph is a compression scheme, but it also reveals structure that a linear scale won't show.
    – L. Rodgers
    May 8, 2017 at 17:11

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