Higher-dimensional Pitch Spaces
Other theorists, such as Leonhard Euler (1739), Hermann von Helmholtz (1863/1885), Arthur von Oettingen (1866), Hugo Riemann (who should not be confused with the famous mathematician Bernhard Riemann), and Christopher Longuet-Higgins (1978) have modeled pitch relationships using two-dimensional (or higher-dimensional) lattices, under the name of Tonnetz. In these models, one dimension typically corresponds to acoustically-pure "perfect fifths" while the other corresponds to "major thirds." (Variations are possible in which one axis corresponds to acoustically pure minor thirds.) Additional dimensions can be used to represent additional intervals including—most typically—the octave.
| A#3 | — | E#4 | — | B#4 | — | FX5 | — | CX6 | — | GX6 |
| | | | | | | | | | | | | |||||
| F#3 | — | C#4 | — | G#4 | — | D#5 | — | A#5 | — | E#6 |
| | | | | | | | | | | | | |||||
| D3 | — | A3 | — | E4 | — | B4 | — | F#5 | — | C#6 |
| | | | | | | | | | | | | |||||
| Bb2 | — | F3 | — | C4 | — | G4 | — | D5 | — | A5 |
| | | | | | | | | | | | | |||||
| Gb2 | — | Db3 | — | Ab3 | — | Eb4 | — | Bb4 | — | F5 |
| | | | | | | | | | | | | |||||
| Ebb2 | — | Bbb2 | — | Fb3 | — | Cb4 | — | Gb4 | — | Db5 |
All of these models attempt to capture the fact that intervals separated by acoustically pure intervals such as octaves, perfect fifths, and major thirds are thought to be perceptually closely related. However, proximity in these spaces need not represent physical proximity on musical instruments: by moving one's hands a very short distance on a violin string, one can move arbitrarily far in these multiple-dimensional models. For this reason, it is hard to assess the psychological relevance of distance as measured by these lattices.
Read more about this topic: Pitch Space
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