Symmetry is both a "fact" but also a "concept", or better...a
"treatment" in harmony.
There are two ways of thinking, it depends if you focus in "what to
do" or in "what it is"
If you focus in "what to do" you can draw ALL and EVERY interval by
stacking them "symmetrically" (equal intervals), exactly as you do
with major 3rds. So you can make symmetric chords (or lines) out of
ANY interval.
b2 =3D 1 octave in 12 equal parts
2 =3D 1 octave in 6 equal parts
b3 =3D 1 octave in 4 equal parts
3 =3D 1 octave in 3 equal parts
4 =3D 5 octaves in 12 equal parts
#4 =3D 1 octave in 2 equal parts
5 =3D 7 octaves in 12 equal parts
#5 =3D 2 octaves in 3 equal parts
6 =3D 3 octaves in 4 equal parts
b7 =3D 5 octaves in 6 equal parts
7 =3D 11 octaves in 12 equal parts
If you focus on "what it is", you=B4ll notice that as you FOLD any of
these intervals to CLOSE POSITION everything ends being b2nds, 2nds,
b3rds, 3rds or #4ths:
((((Or in other words, if you "flip" your attention to the "amount of
parts" instead of the intervals you only can divide equally by 2, by
3, by 4, by 6, or by 12))))
b2 already in close position
2 already in close position
b3 already in close position
3 already in close position
4ths folded equals b2nds
#4 already in close position
5ths folded equals b2nds
#5 folded equals 3rds
6 folded equals b3rds
b7 folded equals 2nds
7 folded equals b2nds
Hope you enjoy.
Martin ****to
On 27 mar, 07:05, SleepyHead <simonharp...@[EMAIL PROTECTED]
> wrote:
> On 27 Mar, 10:01, SleepyHead <simonharp...@[EMAIL PROTECTED]
> wrote:
>
>
>
> > On 27 Mar, 02:07, "Tom K." <tkor...@[EMAIL PROTECTED]
> wrote:
>
> > > "Alain Naigeon" <anaig...@[EMAIL PROTECTED]
> wrote in message
>
> > >news:47eae637$0$13930$426a74cc@[EMAIL PROTECTED]
>
> > > > Each position of the diminished seventh chord sounds the same
> > > > as the root position, and can be rewritten by enharmony (thus
> > > > allowing modulations)
> > > > I remember having read that this is "because this chord is made of
> > > > equal" intervals - all minor thirds.
>
> > > > Recently, it made me think about a chord made of identical major
> > > > thirds, for instance C E G#...
> > > > First, it turns out that we can't build a seventh, since B# is C !
> > > > (Indeed I should have guessed that an augmented 7th would
> > > > give an octave ;-) )
>
> > > > Then :
> > > > E G# C is the same as E G# B#, and sounds like the root position.
> > > > G# B# E is the same as G# B# D## obviously the same kind
> > > > of chord once again, but this time built on G#.
>
> > > > Up to this point I was still thinking that the argument "...
because=
> > > > made of equal intervals" was a sensible one.
>
> > > > But, but..., after all, why would the third be the only interval
we
> > > > would be allowed to have a look at ?
> > > > Thus I took fourths, for instance, and I failed to find anything
> > > > special with the inversions of such "chords" made of fourths !
>
> > > > Now my question isn't about these fourth chords, it's about
> > > > the real reason to these miracles for chords B D F bA
> > > > and C E G# ; since the reason given (equal intervals) is wrong
> > > > because there is at least one counter example, what can be the
> > > > *real* reason ?
> > > > I more or less see that enharmonies are possible with chords
> > > > made of fifths and equal (minor/major) thirds, but not with
> > > > those made of fourths : it's quite obvious that starting from
> > > > C F bB and considering F bB C we can't cheat enough
> > > > with the C to re-build a fourth :-o =A0But... why, then, are
thirds
> > > > behaving in such a special and miraculous way, why only the
> > > > thirds?
> > > > (ok you might find another interval having this property, but
> > > > what I'm looking for is a general explanation why some have
> > > > it, and some don't)
>
> > > It's the 2,3,4 and 6 semitone intervals which form symmetrical
structu=
res.
>
> > Right idea, execution slightly off. Here's a corrected version:
>
> > Min. 2 forms the chromatic scale - 1 semitone, 12 possibilities (12 /
> > 1 =3D 12)
> > Maj. 2 forms the whole-tone scale - 2 semitones, 6 possibilities -
> > (12 / 2 =3D 6)
> > Min. 3/Aug. 2 forms the o7 - 3 semitones, 4 possibilities - (12 / 3
=3D
> > 4)
> > Maj. 3 forms the Aug. triad - 4 semitones, 3 possibilities - (12 / 4
=3D=
> > 3)
> > Aug. 4/Dim.5 forms the tritone - 6 semitones, 2 possibilities - (12 /
> > 6 =3D 2)
> > Octave - 12 semitones, 1 possibility (12 / 12 =3D 1)- Hide quoted text
-=
>
> > - Show quoted text -
>
> Something I forgot - the fact that a perfect 5th (consisting of 7
> semitones) and its inversion - the 4th (consisting of 5 semitones) -
> isn't a factor of 12 is the reason why 5ths/4ths form a circle of
> 5ths / 4ths.
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