8edo: Difference between revisions
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Wendy gunk (talk | contribs) Added a segment for 8edo's relationship to the father comma. |
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== Theory == | == Theory == | ||
[[File:8edo scale.mp3|thumb|A chromatic 8edo scale on C.]] | [[File:8edo scale.mp3|thumb|A chromatic 8edo scale on C.]] | ||
=== Approximation to JI === | |||
8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]]s 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo. | 8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]]s 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo. | ||
Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out. | Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out. | ||
=== Relationship with the father comma === | |||
When 8edo is treated as a very inaccurate 5-limit system, it ends up tempering out the [[Father]] comma, [[16/15]]. In fact, it is the largest edo that tempers this comma. What this means is that intervals 16/15 apart in 8edo map to the same note, such as [[4/3]] being mapped to the same note as [[5/4]]. | |||
Some other odd equivalencies include: | |||
'''0-2-5''': which can be seen as a minor triad (10:12:15), a sus2 triad (9:8:12), or a major triad in first inversion (5:6:8). | |||
'''0-3-5''': which can be seen as a major triad (4:5:6), a sus4 triad (6:8:9), or a minor chord in second inversion (15:20:24). | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 36: | Line 47: | ||
! rowspan="2" | Cents | ! rowspan="2" | Cents | ||
! colspan="3" | JI approximation | ! colspan="3" | JI approximation | ||
!Other | |||
|- | |- | ||
! 2.11/3.13/5.19* | ! 2.11/3.13/5.19* | ||
! 2.5/3.11/3.13/5* | ! 2.5/3.11/3.13/5* | ||
! 10:11:12:13:14* | ! 10:11:12:13:14* | ||
!Patent val ⟨8 13 19] | |||
|- | |- | ||
| 0 | | 0 | ||
| Line 46: | Line 59: | ||
| 1/1 | | 1/1 | ||
| 1/1 | | 1/1 | ||
|1/1, 16/15 | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 52: | Line 66: | ||
| 12/11 | | 12/11 | ||
| 12/11, 11/10 | | 12/11, 11/10 | ||
|10/9, 25/24 | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 58: | Line 73: | ||
| 6/5 | | 6/5 | ||
| 6/5 | | 6/5 | ||
|6/5, 9/8 | |||
|- | |- | ||
| 3 | | 3 | ||
| Line 64: | Line 80: | ||
| 13/10 | | 13/10 | ||
| 13/10 | | 13/10 | ||
|4/3, 5/4 | |||
|- | |- | ||
| 4 | | 4 | ||
| Line 70: | Line 87: | ||
| | | | ||
| 7/5, 10/7 | | 7/5, 10/7 | ||
|27/20, 25/18, 36/25 | |||
|- | |- | ||
| 5 | | 5 | ||
| Line 76: | Line 94: | ||
| 20/13 | | 20/13 | ||
| 20/13 | | 20/13 | ||
|3/2, 8/5 | |||
|- | |- | ||
| 6 | | 6 | ||
| Line 82: | Line 101: | ||
| 5/3 | | 5/3 | ||
| 5/3 | | 5/3 | ||
|5/3, 16/9 | |||
|- | |- | ||
| 7 | | 7 | ||
| Line 88: | Line 108: | ||
| 11/6 | | 11/6 | ||
| 20/11, 11/6 | | 20/11, 11/6 | ||
|9/5, 48/50 | |||
|- | |- | ||
| 8 | | 8 | ||
| Line 94: | Line 115: | ||
| 2/1 | | 2/1 | ||
| 2/1 | | 2/1 | ||
|2/1, 15/8 | |||
|} | |} | ||
<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible | <nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible | ||