Schisma: Difference between revisions
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Strangely enough, subtracting 9/8 from 8/5 gives you 64/45 even in JI, so mentioning that is not needed. |
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The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). Tempering it out gives a [[5-limit]] microtemperament called [[Schismatic family#Schismatic aka Helmholtz|schismatic, schismic or Helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments. | The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). Tempering it out gives a [[5-limit]] microtemperament called [[Schismatic family#Schismatic aka Helmholtz|schismatic, schismic or Helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments. | ||
=== Mathematical observations === | |||
The schisma is the reason for why the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike. The largest common power of two between these numbers is 2<sup>5</sup>, and when reduced by that, 1049760/1048576 becomes 32805/32768. | |||
== See also == | == See also == | ||
Revision as of 12:27, 19 January 2022
| Interval information |
reduced harmonic
The schisma, 32805/32768, is the difference between the Pythagorean comma and the syntonic comma. It is equal to (9/8)4/(8/5) and to (135/128)/(256/243) and also to (9/8)3/(64/45). Tempering it out gives a 5-limit microtemperament called schismatic, schismic or Helmholtz, which if extended to larger subgroups leads to the schismatic family of temperaments.
Mathematical observations
The schisma is the reason for why the greatly composite numbers 1048576 (220) and 104976 (184) look alike. The largest common power of two between these numbers is 25, and when reduced by that, 1049760/1048576 becomes 32805/32768.