I first became aware of this concept many years on Gary Garrett's blog, where he primarily refers to it as "the lattice." His introduction to the concept gives a brief history:
Learned about the tonnetz (among other models) from the book "Audacious Euphony"... Challenging (and unfortunately too expensive now), but fascinating stuff...
Accordion buttons have a row of roots with major thirds off on a diagonal, but in a shifted way that you don't have minor thirds on the opposite diagonal. E.g. there is no CEG triangle.
If the layout did that, you could hit minor and major triads chords by pressing triangles on the first two rows.
Why did they make it like that? I assume accordion makers were aware of the convenience of the harmonic table layout, and had good reasons for choosing something else.
> the defects of this tuning method became evident and the more flexible “well-tempered” tuning scheme was devised. This ensures that the ratio of pitch between every two adjacent notes is precisely the same.
The author seems to have confused well-temperament and equal temperament.
It is also odd that he calls just intonation a “simplification” of Pythagorean tuning.
As an indie music composer, this kind of unusual tone arrangement is great for creativity, thanks! I noticed that the same triangle will play different chords from time to time, is that on purpose?
I only listened briefly on laptop speakers, but it did sound like sometimes the same note of a chord would play at a different octave, almost like it was trying to do voice leading from the previously played chord.
I first became aware of this concept many years on Gary Garrett's blog, where he primarily refers to it as "the lattice." His introduction to the concept gives a brief history:
https://www.garygarrett.me/?p=342
More introduction to the lattice:
https://www.garygarrett.me/?p=995
https://www.garygarrett.me/?p=1632
https://www.garygarrett.me/?p=1696
Some comparison audio between equal temperament and just intonation:
https://www.garygarrett.me/?p=1812
Some songs with lattice animations:
https://www.garygarrett.me/?p=103
https://www.garygarrett.me/?p=1253
I also like this book which Gary recommends, although it's very challenging and I never made it all the way through:
https://www.amazon.com/Harmonic-Experience-Harmony-Natural-E...
For the background understanding (minus interactive play) the Wikipedia page is a better starter:
https://en.wikipedia.org/wiki/Tonnetz
Also shows the Tonnetz in the configuration where fifths are horizontal.
> Discovered by Euler
Figures...
Learned about the tonnetz (among other models) from the book "Audacious Euphony"... Challenging (and unfortunately too expensive now), but fascinating stuff...
https://www.amazon.com/dp/019977269X
"Almost Tonnetz" fail:
https://en.wikipedia.org/wiki/Stradella_bass_system
Accordion buttons have a row of roots with major thirds off on a diagonal, but in a shifted way that you don't have minor thirds on the opposite diagonal. E.g. there is no CEG triangle.
If the layout did that, you could hit minor and major triads chords by pressing triangles on the first two rows.
Why did they make it like that? I assume accordion makers were aware of the convenience of the harmonic table layout, and had good reasons for choosing something else.
A recent blog post on thatsmaths.com was about Tonenetz, a diagram of harmonic relationships of notes. Tonenetz means 'tone network' in German :-)
Do you have a link to the post in thatsmaths?
https://thatsmaths.com/2025/10/16/music-and-maths-are-inextr...
> the defects of this tuning method became evident and the more flexible “well-tempered” tuning scheme was devised. This ensures that the ratio of pitch between every two adjacent notes is precisely the same.
The author seems to have confused well-temperament and equal temperament.
It is also odd that he calls just intonation a “simplification” of Pythagorean tuning.
As an indie music composer, this kind of unusual tone arrangement is great for creativity, thanks! I noticed that the same triangle will play different chords from time to time, is that on purpose?
I only listened briefly on laptop speakers, but it did sound like sometimes the same note of a chord would play at a different octave, almost like it was trying to do voice leading from the previously played chord.