Augmented chords in just intonation, some: Difference between revisions
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== | == Harmonic augmented 7th chords based on 8:10:13 == | ||
The central triad in all three | The central [[triad]] in all three [[chord]]s is 8:10:13, but each uses a different 7th degree, to come as close as possible to the familiar aug7 (without straying into [[17-limit|high harmonics]]). | ||
but each uses a different 7th degree, | |||
to come as close as possible to the familiar aug7 | |||
(without straying into high harmonics). | |||
Some of the | Some of the [[interval]]s are listed below in terms of [[cents]]. | ||
The chords: | |||
8 : 10 : 13 : 14 | aug 7 | |||
10 : 13 : 16 : 18 | aug 7 | |||
13 : 16 : 20 : 24 | aug [[neutral]] 7 | |||
The thirds: | The thirds: | ||
10 : 8 ~ 386 cents | 10 : 8 ~ 386 cents | ||
13 : 10 ~ 454 cents | 13 : 10 ~ 454 cents | ||
16 : 13 ~ 359 cents | 16 : 13 ~ 359 cents | ||
The sevenths (as reduced fractions): | The sevenths (as reduced fractions): | ||
7 : 4 ~ 969 cents | 7 : 4 ~ 969 cents | ||
9 : 5 ~ 1018 cents | 9 : 5 ~ 1018 cents | ||
24 : 13 ~ 1061 cents | 24 : 13 ~ 1061 cents | ||
== Augmented 7th chords based on 9:11:13 == | |||
Interestingly, none of these chords has a [[5/4|5:4]]. Someone might argue that a chord with a [[neutral third scale|neutral third]] is not properly augmented, but they certainly sound closer to augmented than they do to anything else in Western music. | |||
Some of the intervals are listed below in terms of cents. | Some of the intervals are listed below in terms of cents. | ||
The central triad in all three chords is 7:9:11, | The central triad in all three chords is 7:9:11, but each uses a different 7th degree, to come as close as possible to aug7 (without straying into high harmonics). | ||
but each uses a different 7th degree, | |||
to come as close as possible to aug7 (without straying into high harmonics). | |||
The inversions: | The inversions: | ||
7 : 9 : 11 : 13 ~ aug #7 | 7 : 9 : 11 : 13 ~ aug #7 | ||
9 : 11 : 14 : 16 ~ aug 7 | 9 : 11 : 14 : 16 ~ aug 7 | ||
11 : 14 : 18 : 20 ~ aug (neutral?) 7 | 11 : 14 : 18 : 20 ~ aug (neutral?) 7 | ||
The thirds: | The thirds: | ||
9 : 7 ~ 435 cents | 9 : 7 ~ 435 cents | ||
11 : 9 ~ 347 cents | 11 : 9 ~ 347 cents | ||
14 : 11 ~ 418 cents | 14 : 11 ~ 418 cents | ||
The sevenths: | The sevenths: | ||
13 : 7 ~ 1072 cents | 13 : 7 ~ 1072 cents | ||
16 : 9 ~ 996 cents | 16 : 9 ~ 996 cents | ||
20 : 11 ~ 1035 cents | 20 : 11 ~ 1035 cents | ||
{{todo | intro | discuss title | review | inline=1}} | |||
[[Category:Just intonation chords]] | |||
[[Category:13-limit chords]] | |||
Latest revision as of 22:13, 22 December 2024
Harmonic augmented 7th chords based on 8:10:13
The central triad in all three chords is 8:10:13, but each uses a different 7th degree, to come as close as possible to the familiar aug7 (without straying into high harmonics).
Some of the intervals are listed below in terms of cents.
The chords:
8 : 10 : 13 : 14 | aug 7
10 : 13 : 16 : 18 | aug 7
13 : 16 : 20 : 24 | aug neutral 7
The thirds:
10 : 8 ~ 386 cents
13 : 10 ~ 454 cents
16 : 13 ~ 359 cents
The sevenths (as reduced fractions):
7 : 4 ~ 969 cents
9 : 5 ~ 1018 cents
24 : 13 ~ 1061 cents
Augmented 7th chords based on 9:11:13
Interestingly, none of these chords has a 5:4. Someone might argue that a chord with a neutral third is not properly augmented, but they certainly sound closer to augmented than they do to anything else in Western music.
Some of the intervals are listed below in terms of cents.
The central triad in all three chords is 7:9:11, but each uses a different 7th degree, to come as close as possible to aug7 (without straying into high harmonics).
The inversions:
7 : 9 : 11 : 13 ~ aug #7
9 : 11 : 14 : 16 ~ aug 7
11 : 14 : 18 : 20 ~ aug (neutral?) 7
The thirds:
9 : 7 ~ 435 cents
11 : 9 ~ 347 cents
14 : 11 ~ 418 cents
The sevenths:
13 : 7 ~ 1072 cents
16 : 9 ~ 996 cents
20 : 11 ~ 1035 cents
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Todo: intro , discuss title , review |