LATENT REFERENCES / TAG2
Prime Numbers and Just Intonation
Original title: 素数 純正律
This reference note belongs to Tag2 in Latent References, an archive curated by Keigo Yoshida. Its archive region is Film. The note preserves its source text and links so that readers can trace the material behind the 3D map.
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English translation of the archived note. JP shows the original text. Source links and literal code are retained; the translation does not update or independently verify the source claims.
Following just intonation strictly requires infinitely many scales
We provisionally defined 8 notes in the scale above, but this is insufficient. For example, defining 2:3 and 4:5 relative to B (495Hz) yields 742.5 Hz and 618.75 Hz. Repeating from 742.5 Hz ... actually produces an infinite scale.
This happens because however many times it is repeated, one never returns to the original note. If it returned like today's scale—do, re, mi, ..., do—it would stop there, but unfortunately “2 times,” representing a note an octave above, and “3 times” and “5 times,” producing beautiful chords, are prime numbers, so it never returns however one tries.
純正律を貫くと無限に音階が必要
上の音階ではとりあえず8音を定義しましたが、これでは不足します。たとえば、シ(495Hz)に対して2:3と4:5を定義すると742.5 Hzと618.75 Hzになり、742.5 Hzから。。。と繰り返していくと実は無限に音階が出来上がります。
なぜこうなるかというと、どれだけ繰り返しても元の音には戻らないからです。今の音階のようにド、レ、ミ、…、ドと戻るのであればそこで打ち切られるのですが、残念ながらオクターブ上の音をあらわす「2倍」と美しい和音の「3倍」と「5倍」は素数なので、どうやっても戻らないのです。
Source updated 2023-07-18 · Snapshot 2026-10-08
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