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== harmonic augmneted 7th chords based on 8:10:13 ==
== Harmonic augmented 7th chords based on 8:10:13 ==


The central triad in all three chords is 8:10:13,
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 intervals are listed below in terms of cents.
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 inversions:
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 ==


== agumented 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.


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 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]]
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