True - as anyone who's written counterpoint will tell you:
ii =3D VII
II =3D vii
iii =3D VI
III =3D vi
IV =3D V
IV+ =3D V=B0
where the term on the right is the inverse of the term on the left, or
vice versa.
On 27 Mar, 13:21, flatnine <martymu...@[EMAIL PROTECTED]
> wrote:
> 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 =A0=3D 1 octave in 12 equal parts
> 2 =A0 =3D =A01 octave in 6 equal parts
> b3 =A0=3D 1 octave in 4 equal parts
> 3 =A0 =3D 1 octave in 3 equal parts
> 4 =A0 =3D 5 octaves in 12 equal parts
> #4 =A0=3D 1 octave in 2 equal parts
> 5 =A0 =3D 7 octaves in 12 equal parts
> #5 =A0=3D 2 octaves in 3 equal parts
> 6 =A0 =3D 3 octaves in 4 equal parts
> b7 =A0=3D 5 octaves in 6 equal parts
> 7 =A0 =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 =A0already 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 "...
becau=
se
> > > > > made of equal intervals" was a sensible one.
>
> > > > > But, but..., after all, why would the third be the only interval
w=
e
> > > > > 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
third=
s
> > > > > 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
struc=
tures.
>
> > > 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.- Hide quoted text -
>
> - Show quoted text -
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