Q about Law of Octaves

[RflctnOfU]

I just want to add to what you said that there are ascending scales, and descending scales - evolutionary and creative respectively. In either case, the resulting Do is contained within the initial Do and completes the cycle, while beginning a new one potentially.

An interesting observation made by Russel A. Smith in "Cosmic Secrets" is the overall structure of the octave is in miniature within the octave at the missing semitone intervals. Mi-Fa/Fa-Sol and Si-Do/Do-Re. Using the previous frequencies I cited, that would be 30 - 32/32 - 36. (2. 4. ). And 45 - 48/48 - 54. (3. 6. ). The increase doubles, just as succeeding octaves double, although the difference is pronounced.

Kris

It's sounds Asymptote ... If a frog jumps half the distance to a wall with every hop, is it possible for him to ever reach the wall? OR if an instrument continues to play C an octave higher with each subsequent note, is is possible for it to reach silence?

Silence in relation to what?

Kris

 

[Sol Logos]

The vibration of each note an octave higher doubles indefinitely to infinity. In theory an infinite vibration isn't actually vibrating so wouldn't be making a sound (not that sound is detectable past a certain frequency though). Theoretically its would be silent. But that instrument wouldn't get to silence, in the same way that the frog wouldn't reach the wall in that anecdote. That's all I meant.

 

[RflctnOfU]

The vibration of each note an octave higher doubles indefinitely to infinity. In theory an infinite vibration isn't actually vibrating so wouldn't be making a sound (not that sound is detectable past a certain frequency though). Theoretically its would be silent. But that instrument wouldn't get to silence, in the same way that the frog wouldn't reach the wall in that anecdote. That's all I meant.

I thought that was where you were going :)

Kris

 

[MusicMan]

Reading through the comments on this subject I could not find a reference to "the twelfth root of two", which is the mathematical division between each of the notes in an Octave if you include the semitones.
There are twelve in all: A-flat, A, B-flat, B, C, D-flat, D, E-flat, E, F, G-flat, G; and then the next octave starts at the next A-flat and so on.
Some people talk in sharps, so their octave goes like this: (if A-flat is equal to G-sharp)
G-sharp, A, A-sharp, B, C, C-sharp, D, D-sharp, E, F, F-sharp, G.
The flats (or sharps) correspond to the black notes on a piano keyboard.

Check your guitar. From the open string to the twelfth fret makes one octave.
Each fret is separated from the next by the twelfth root of two (1.05946 of the octave)
The thickest string is the E note, so its octave starts there.
The next string is the A string, then the D string, the G string, the B string and the thin E string.
The open string chord on the guitar is the G chord.
On a twelve string guitar the adjacent strings to the normal ones are tuned one octave higher.

I know you all wanted to know this.
Knowledge protects.

 

[John G]

Reading through the comments on this subject I could not find a reference to "the twelfth root of two", which is the mathematical division between each of the notes in an Octave if you include the semitones.
There are twelve in all: A-flat, A, B-flat, B, C, D-flat, D, E-flat, E, F, G-flat, G; and then the next octave starts at the next A-flat and so on.
Some people talk in sharps, so their octave goes like this: (if A-flat is equal to G-sharp)
G-sharp, A, A-sharp, B, C, C-sharp, D, D-sharp, E, F, F-sharp, G.
The flats (or sharps) correspond to the black notes on a piano keyboard.

Check your guitar. From the open string to the twelfth fret makes one octave.
Each fret is separated from the next by the twelfth root of two (1.05946 of the octave)
The thickest string is the E note, so its octave starts there.
The next string is the A string, then the D string, the G string, the B string and the thin E string.
The open string chord on the guitar is the G chord.
On a twelve string guitar the adjacent strings to the normal ones are tuned one octave higher.

I know you all wanted to know this.
Knowledge protects.

You can get into some quite complicated things with that 12-structure:

http://www.tony5m17h.net/musPhys.html

Octonion reflexion has given me a new dearly needed tool
to crack the code of musical harmony using octonions.

Intuition has told me that there is a connection but
I've only had three real clues to back it up.

1) 7 notes, 7 imaginaries
2) Tonika-Sub dominant-Dominant, quaternion triple
3) the octave *repeats* itself. It is fundamentally cyclic.

How to preceed from here was somewhat of a puzzle for me
because the seven keys do not add up trivially into the octonionic rules.

Another problem was the fact that the octave is
just a subset of the 12-tone scale.

That the octave is a subset of the 12-tone scale
I interpret as the octonion being embedded in a larger geometrical structure.

This structure is related to the 24-cell.
I stress that the octonion is still the arena of harmony.
The 12-structure is just the background fundament from which
the harmonic brilliance of the octonion stands out.
Octonion harmony is nevertheless reflected in this structure.

We recall that the octonion has a 3 + 4 or a 3 + 1 + 3 structure.
Having in mind that the octonion is reflexive we should expect
this structure to pop up quite a lot.

The first appearence of this structure is immediately seen
by observing the white keys on the keyboard.
Each white key is separated by a black key except
at two points in the octave in which the white keys intersect.
This divides the octave into a 3 + 4 structure.

But from the imaginaries we see
that an 3 + 1 + 3 structure is also implicit,
namely where the IJK mirrors ijk and E is an Extraordinary number.
Is this also true?
Yes.
The keys corresponding to i,E and I are considered base harmonics.
I is called the Dominant because it "dominates" over E.
The reason is that harmonically IJK resembles ijk.
Just look at the keyboard.
The IJK section looks (and sounds) like a copy of the ijk section.

Finally we see a 3 + 1 + 3 structure if we play triad chords succesively
throughout the octave (i-k-I, j-E-J, k-I-K, E-J-i, I-K-j, J-i-k, K-j-E)

These make up 3 Major chords, 3 minor chords and 1 diminished.

Now over to the 12-structure.
We know that there are 7 white keys and 5 "background" black keys.
We may call it a 7+5 structure.
The tone which harmonizes best with i is i one octave up
(i.e. 12 halftones away).
The tone which harmonizes next best is the Dominant, I.
Guess how many halftones this is away from i.
That's right, 7.
So the chord i+I+i2 makes a 7+5 structure.
A very common pattern in music is INVERSION.
Inverting the 7+5 structure we get 5+7.
The chord which fits this pattern is i+E+i2,
and E as we recall is the Sub-dominant.
But let is stick with the 7-chord,
i.e. i+I.
Suppose these seven half tones are a mirror image
of the seven imaginaries.
We should then expect there to be a 3+4 structure in it.
Is there?
Yes!
The 4+3 = the Major chord (i+k+E), the most harmonic triad in music.

What about its inversion 3+4?
That makes up the Minor chord,
the second most harmonic triad in music.

This discourse contains no evidence, only indications.
Firm evidence requires a thorough understanding of
the 12-structure which the octonion is embedded in.
If this structure can be understood then
I think we will have a theory of music.

Onar.
-------------------------------------------------

TO EMBED
the 7-structure and the 5-structure in an octonionic 12-structure,
consider the Witting polytope in 8 real dimensions.

The Witting polytope also lives in 4 complex dimensions.

The Witting polytope can be constructed from the
4-dim 24-cell by a Golden Ratio expansion
of the 24-cell to a 24+96 = 120 vertex 4-dim polytope, the 600-cell.

It has octonionic structure and lives in 8-dim space.

Since the Witting polytope has 240 = 20x12 vertices,
it has 12-structure.

To see the 7 and 5 structures,
start with the 240-vertex Witting polytope in 8-dim,
and consider two of its projections in 4-dim,
the 600-cell and the 24-cell:

-------------------------------------------------

FIRST USE THE 600-CELL TO SEE THE 5-STRUCTURE:

The 600-cell has 120 vertices.
Each cell, or 3-dim face, is a tetrahedron.
A 3-dim projection of the 600-cell is an icosahedron,
with 12 vertices and 20 faces.
The faces are triangles, 5 at each vertex.

You could also look at the dual figure to the 600-cell,
the 120-cell with 600 vertices.
Each of the 120 cells are dodecahedra,
which have 20 vertices and 12 pentagon faces.

Consider the tetrahedron,
the cell (3-dim face) of the 600-cell.
It has 4 triangle faces, and 6 edges.
Note that an edge can be seen as a 2-sided polygon,
just as a triangle is a 3-sided polygon,
so you could say that a tetrahedron has 10 = 4 + 6 "faces".
Also, you can pair up the 4 faces and the 6 edges
(although it is a complicated thing to do with a
tetrahedron because of the lack of central symmetry)
to get from 10 "faces" = 4 triangles + 6 (2-sides) to
5 "faces" = 2 triangles + 3 (2-sides)


This structure is related to the tensegrity transformation
from the tetrahedron to the icosahedron:


-------------------------------------------------

NOW USE THE 24-CELL TO SEE THE 7-STRUCTURE:

The 12-structure of the 4-dimensional polytope,

the 24-cell, can be most clearly understood by noticing two things:
it has 24 vertices and 24 octahedra as 3-dim faces;
and
it can be projected into 3-dim space such that its
central figure is a 14-vertex polytope, the rhombic dodecahedron.

The way to picture the rohombic dodecahedron inside the 24-cell is
to consider the rhombic dodecahedron as containing an interior cube,
with the rhombic dodecahedron being built by adding a 4-sided pyramid
to each of the 6 faces of the interior cube.

Each of the 12 rhombic faces is a collapsed octahedron of the 24-cell,
with the rhombus being a cross-section of the octahedron.

That is,
the 2 outer vertices of the rhombic face are the
top and bottom vertices of the octahedron,
the 2 middle (on the interior cube) vertices are each
front and back side vertices of the octahedron, plus the edge between them,
and the 4 edges of the rhombic face are each
a side triangular face of the octahedron, plus edges.
Further, the rhombic face can be cut into 2 triangles, each of which
is 2 (front and back) triangular faces of the octahedron, plus edges.

That gives the 2 + 2x2 = 6 vertices,
the 2 + 4x2 + 2 = 12 edges, and
the 4 + 2x2 = 8 triangular faces
of the octahedron.

That accounts for 12 of the octahedra of the 24-cell.

Inside the rhombic dodecahedron there are 6 octahedra,
each with one vertex at the center of the rhombic dodecahedron
and one opposite vertex at one of the 6 outer vertices
of the rhombic dodecahedron.

The central square of each of those 6 octahedra is
a square face of the interior cube of the rhombic dodecahedron.

Each of those 6 octahedra is the projection of 2 octahedra of the 24-cell.

This accounts for all 12 + 2x6 = 24 octahedra of the 24-cell.

Since the rhombic docecahedron has 14 vertices,
and they are opposite each other,
you can get 7 vertices.

You can also consider the dual figure to the rhombic dodecahedron,
the cuboctahedron, which has 14 faces (8 triangles and 6 squares).
Since they are opposite to each other,
you can get 7 faces = 4 triangles + 3 squares.

-----------------------------------------------

THEREFORE the 7 and 5 structures in music
correspond to different projections of the Witting polytope
from 8 real dimensions into lower dimensions.

In some sense, the 7-structures are related to the 24-cell
and therefore to the ordinary 4-dim subspace

while

the 5-structures are related to the 600-cell
and therefore to the Golden sqrt(5) 4-dim subspace

of the 8-dim Witting polytope.


From the 4(complex)-dim point of view,

the 7-structures are REAL
and
the 5-structures are IMAGINARY.

The Witting polytope is very interesting in an Enneagram kind of way:

http://vixra.org/pdf/0910.0023v4.pdf

 

[MusicMan]

Wow! I think I opened a 'can of worms' there!
The maths looks fairly technical, I would have to take it off-line and study it to make some sense of it all, but from a cursory look through it, you could imagine that music has become three dimensional, or even four dimensional if you include the 'time/timing' factor.
And when you think about it, that is how we play it and hear it, in four dimensions, while the music is actually written in two dimensions on the score..
Regards
MusicMan

 

[Sol Logos]

I don't know what the math behind this is but a 12 tone chromatic scale doesn't sound very musical. Not to say you can't make music on it but when you play it as a scale that is, it sounds robotic instead of musical. The octave jumps around in terms of a number progression a lot more but for some reason to the ear it sounds eveningly stepped if you know what I mean.

 

[monotonic]

Can you provide an example?

 

[Sol Logos]

Can you provide an example?

If you can get your hands on a guitar and just pluck 12 notes on the lowest E string in a row for example by going up each thret - that's a chromatic scale. If you compare that to do, ra, me... etc. should notice the former sounds less musical for lack of a better term.