What are the main features of the scale triangles

Are there geometric symmetries in musical harmonics?

Richard

If the Barrel net is laid out as follows:

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

Although there are symmetries related to diatonic harmony (and diatonic scales etc) - which are interesting and worth discussing! -, the symmetries you show and ask for are really just a by-product of your choice around yours To lay out the barrel net .


In this regard, note that the consonant triads - major and minor - both consist of a small third and a large third. (The major triad has the major third on the bottom and the minor third on top, and the minor triad reverses this.) In addition, the remaining interval can be thought of as the perfect quarter.

The small third is made up of 3 semitones, the big third is made up of 4, and the perfect fourth is made up of 5. So if you get your geometric space right, it's pretty easy to walk away with a 3: 4: 5 triangle.

L. Rodgers

I have an open conclusion here: by-product or deeper feature? It's a bit like amplituhedra or adinkras in physics. Quantum physicists see them as an indication of the deeper physical properties of the media (quantum fields) and as abbreviations for understanding physical phenomena. Your comrades in string theory are heavily involved with acoustic models, FWIW.

Richard

@ BertLee I misunderstood your original question. see edit.

L. Rodgers

I should also mention that the phase of shifting the scale also follows a right triangle, a hypotenuse for Tritones vs. Maj. 3rds. Despite apophenia or affirmative bias, is something more elegant being overlooked?

Richard

@ Dave Thanks; I meant the perfect fourth place which has now been fixed.

Scott Wallace

By-product, not a deeper characteristic. The overtone series is a simple geometry, no proportions of the number of semitones, but tuned. This Barrel net While it's a convenient way of looking at chords, it doesn't tell you anything about how the sound works.

Dave

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

No. I am drawing on your use of (right) triangles and the idea of ​​the hypotenuse, which implies sizes that are in quadrature ala add the Pythagorean theorem.

Our sense of tone is closely related to sounds, which are made up (good approximations of) of an overtone series that are integral multiples of a fundamental, and our sense of harmony relates only to these types of sounds. No quadrature is added here, so no triangles.

L. Rodgers

Maybe, but I noticed that Elaine Chew encountered similar relationships in her helical diagram. The logarithmic / geometric course is compressed in every graph that we construct in music (assuming 12 TET). However, the harmonics are inherent in consonant and ablative wave interactions. Whether we find more or less eurythmy on top of each other seems to be inherent in acoustic consonances, including overtone series ... The point is that a graph is a compression scheme, but it also shows a structure that a linear scale does not.