The + / - Of 144 Notes Of Separation In The Ancient Scales: 12 - 144, 24 - 288, 36 - 432

"The 11th Grand Cycle was about 12/144,
The 12th Grand Cycle was about 24/288,
and, The 13th Grand Cycle will be about 36/432,
and, the 144 notes of separtation
within the ancient scales of +/- 144 from the base point of 432"
Susan Lynne Schwenger (1987)

"La 11ème Grand Cycle était d'environ 12/144, le 12e Grand Cycle était d'environ 24/288,
et, le 13e Grand Cycle sera d'environ 36/432, et, le 144 notes de separtation
dans l'ancienne échelles de +/- 144 de la base de 432"
~Susan Lynne Schwenger (1987)

"El 11o gran ciclo era de aproximadamente 12/144, La 12ª Grand Ciclo
era de aproximadamente 24/288, y, el 13o gran ciclo será de unos 36/432,
y el 144 notas de separtation dentro de la antigua escalas de +/- 144
desde la base punto de 432"
~ Susan Lynne Schwenger (1987)

 

0 aka ZERO POINT AKA ZEROPOINT AKA 0 POINT
-1 + 0 + 1 = 0
144
288
432
576
720
864
1008
~susan lynne schwenger
13

 

TUNING SYSTEMS, SOUND & FREQUENCIES
INTRODUCTION
You might wonder what this topic is doing on my web site?

Well, in the past couple of years I have developed
an interest in frequency and tuning.

I suppose you do know that music

• and sound in general
• does have an effect on how we feel.

How can we improve the sensation that music gives us?

That was one of many questions that started my search
for more information about sound and music.

Some of my findings I have placed on my web site,
you might provide an answer to some of your questions as well.

Roel


WHAT "MAKES" A TUNING SYSTEM?
A tuning system is Concert Pitch + Musical Interval System
(+ Temperament).

CONCERT PITCH
A "Concert Pitch" is a tone used as REFERENCE
for musical instruments to tune to.

This tone / frequency does NOT tell us anything
about the relationship between this tone
and any other tone that might be used.

TUNING SYSTEM
A "Tuning System" is the combination of: Concert Pitch
(the reference point), the Musical Interval System
(the range of notes within the system /
the number of actual tones available to use)
and the Temperament
(the interval distances between the tones).

NOTE: Even though all (proper) tuning system can be explained
/ described using mathematical formulas,
not every mathematical formula will "generate"
a proper (harmonious) tuning system!

CONCERT PITCH


Concert pitch refers to the pitch reference
to which a group of musical instruments are tuned for a performance.

Concert pitch may vary from ensemble to ensemble,
and has varied widely over musical history.

In the literature this is also called international standard pitch.

The reference note most commonly used is the "A" above middle C.

PITCH INFLATION

During historical periods when instrumental music rose
in prominence (relative to the voice),
there was a continuous tendency for pitch levels to rise.

This "pitch inflation" seemed largely a product of instrumentalists'
competing with each other, each attempting to produce a brighter,
more "brilliant", sound than that of their rivals.

This tendency was also prevalent with wind instrument manufacturers,
who crafted their instruments to play generally
at a higher pitch than those made by the same craftsmen years earlier.

At the beginning of the 17th century,
Michael Praetorius reported in his encyclopedic
"Syntagma Musicum" that pitch levels
had become so high that singers were experiencing severe throat strain
and lutenists and violin players were complaining of snapped strings.

More information: Wikipedia

PITCH REFERENCE TOOLS:

Until 1711, musicians and directors had no clear reference point
to tune their instruments.

This all changed with the invention of the “tuning fork”
in England by Royal trumpeter John Shore in 1711.

MUSIC INTERVAL SYSTEM

By musical interval system is meant a range of notes or musical intervals theoretically available to a composer.

The qualification "theoretically" is important,
as such systems could include notes or intervals
which are not actually audible for human beings.

The most basic of distinctions among such systems
is between open and closed systems,
where a closed system has a finite set of possible musical intervals,
and an open system has an infinite set.
More information: XenHarmonic

TEMPERAMENT

In musical tuning, a temperament is a system
of tuning which slightly compromises the pure intervals
of just intonation in order to meet other requirements
of the system.

Most instruments in modern Western music are tuned in the
equal temperament system.

More information: Wikipedia
Note: For 432-Tuning the Pythagorean Temperament is used.

REFERENCES:

• http://en.wikipedia.org/wiki/Concert_pitch
• http://en.wikipedia.org/wiki/Tuning_fork
• http://en.wikipedia.org/wiki/Musical_temperament
• http://en.wikipedia.org/wiki/Pythagorean_temperament

TUNING & FREQUENCY

http://www.roelhollander.eu/en/tuning-frequency/

 

Monday, November 17, 2014

432


432 & THE "FACTOR 9"

The Factor 9 concept has been explained in a video
called "Sonic Geometry:
The Language of Frequency and Form" (by Eric Rankin).

The number grid shown in that video contains some major mistakes.

One tone is missing from the scale and some frequencies
are not calculated right.

More about this later!

The Factor 9 "formula" generates a 15-tone Temperament.

D4=288Hz is a pretty familiar frequency
for those who have been exploring 432-Tuning.

We find the D at 288Hz with various 12-Tone Temperaments,
such as the Pythagorean Temperament
(in combination with Concert Pitch C4=256Hz or A4=432Hz)
and Maria Renold's "Scale of Fifths".

I will try to "visualize" the difference between the "Factor 9 Temperament"
and 15-Tone Equal Temperament in this article.

After all, the Temperament (system of ratios)
used is the most important part of the tuning system.

If you do not know what a tuning system is made of,
then please do read this article on my blog: Tuning Basics.
(POSTED ABOVE IN #3)

FROM: MATHEMAGICAL MUSIC
(DERRICK SCOTT VAN HEERDEN, WRITER)

"All of this seems to revolve around the number 9
or the frequency of 9 Hz which is a very low D in the factor 9 scale,

The other D's or octaves of 9 Hz are 18 Hz - 36 Hz - 72 Hz - 144 Hz - 288 Hz etc.

You can calculate the harmonic overtone series for any octave
of 9 Hz and it will work as an overlay with this factor 9 scale,
however if you choose another number for the grid,
for example 81 Hz or 90 Hz then the overlay does not work as well.

This means that the factor 9 scale will have a wide range
of good sounding overtone based notes when played in D
while it will have more "off" sounding notes
that are not harmonic with each other in other keys.

The chart below works in octaves,
so all the notes in the row marked "D" are D's,
all the notes marked E are E's and so on.

One octave = any frequency X 2
(on a piano this is C to C or D to D etc)."

Source: http://mathemagicalmusic.weebly.com/

FACTOR 9 vs. 15-TET (15-Tone Equal Temperament)

Eric Rankin and Derrick Scott van Heerden have not been
the only ones creating a 15-tone scale (including the octave).

The 15-TET or 15 EDO
(Equal Division of the Octave) Temperament
is one of a many 15-Tone Temperaments.

Why would I compare the Factor 9 Temperament
with Equal Temperament (the most commonly used Temperament)?

Well, to show you that the "Factor 9" system
turns out to have a Temperament of it's own when we look at the ratios.

Specially ratios (the system of intervals or Temperament)
is the most important part of tuning system (not the Concert Pitch).

ABOUT 15-TET or 15 EDO

"In music, 15 equal temperament, called 15-TET, 15-EDO,
or 15-ET, is the tempered scale derived by dividing the octave
into 15 equal steps
(equal frequency ratios).

Each step represents a frequency ratio of 2^1/15, or 80 cents.

Because 15 factors into 3 times 5,
it can be seen as being made up of
three scales of 5 equal divisions of the octave,
each of which resembles the Slendro scale in Indonesian gamelan.
15 equal temperament is not a meantone system."

Source: Wikipedia

"15-EDO offers some minor improvements
over 12-TET in ratios of 5 (particularly in 6/5 and 5/3),
and, has a much better approximation
to the 7th and 11th harmonics,
but its approximation to the 3rd harmonic is rather off.

However, the particular way in which this approximation
is off is as much a feature as it is a bug,
for it allows the construction of a 5L5s MOS scale
wherein every note of the scale can serve as a root
for a 7-limit otonal or utonal tetrad,
as well as either a 5-limit major or minor 7th chord.

This is known as Blackwood temperament,
named after Easley Blackwood, Jr.,
who is the first to document its existence.

It has also been written on extensively by Igliashon Jones in the paper
"Five is Not an Odd Number".

For an in-depth treatment of harmony
in 15-edo based on this temperament (and its 7- and 11-limit extensions),
see Harmony in 15edo Blacksmith."
Source: Xenharmonic


When we use D4=288Hz as the first degree of the scale
(degree 0 of the Temperament) in combination with 15-TET,
then we find A4 at 416.8Hz, instead of 432Hz.

If you would like to use the 15-TET system
with Concert Pitch A4=432Hz,
all frequencies (listed in the 3rd column)
need to be pitched up with approximately 62 cents.

Below a table with cents, ratio's and more, comparing 15-TET
with FACTOR 9, based on D=9Hz:

Temperament Degree	Tones	Frequencies 15-TET*	Frequencies Factor 9	Cents 15-TET	Cents Factor 9	Ratios 15-TET*	Ratios Factor 9
0	D	288	288		0	1/1	1/1
1	Eb (D#)	301.6	306	80	104.95540950040728	22/21	17/16
2	E	315.9	324	160	203.91000173077487	34/31	9/8
3	F	330.8	342	240	297.5130161323026	85/74	19/16
4	F#↓ (Gb↑)	346.5	360	320	386.3137138648348	77/64	5/4
5	Gb (F#)	362.9	378	400	470.78090733451234	63/50	21/16
6	G	380.0	396	480	551.3179423647566	95/72	11/8
7	Ab (G#)	400	414	560	628.2743472684155	25/18	23/16
8	A	416.8	432	640	701.9550008653874	55/38	3/2
9	A↑	436.5	450	720	772.6274277296696	97/64	25/16
10	Bb (A#)	57.2	468	800	840.5276617693107	27/17	13/8
11	B	478.8	486	880	905.8650025961624	133/80	27/16
12	B↑	501.4	504	960	968.8259064691249	47/27	7/4
13	C	525.2	522	1040	1029.5771941530866	31/17	29/16
14	Db (C#)	550	540	1120	1088.2687147302222	21/11	15/8
15	D	576	576	1200	1200	2/1	2/1

<sub><sup>*I have rounded the frequencies of on 10th behind the dot.
The difference in 100th would be for any "normal" human being too hard to differ.
The reason why I have done so, is that if we have to calculate the fraction of a number
with 14 digits behind the dot, like D# = 301.6207073723412,
we will end up with an extremely big fraction: 19957374/19056131.

That does make understanding the difference rather complicated.

By using only one digit behind the dot of the frequencies,
we end up with a smaller (tempered) fraction, for your convenience fractions with at most 2 digits.

The arrows behind the tones (in column 2)
tell you if the tones are a bit sharper (↑) or flatter (↓),
but less then the tone before or after.</sup></sub>

NOTATION (sheet music)
Notation of a 15-Tone scale using
the traditional 12-Tone notation system
can be a bit "tricky" and would require some time to study
to be able to read it "prima vista".

Below an example of Easley Blackwood's notation system
for 15 equal temperament.

NOTE: there are a few differences between
the "Factor 9" notation and the 15-TET notation listed below,
this is just an example about how different a 15-tone scale
could look using the 12-tone notation system.

"Intervals are notated similarly to those
they approximate and there are different enharmonic equivalents
(e.g., G-up = A-flat-up)."

Source: Wikipedia

DISADVANTAGES of the FACTOR 9 Temperament

The only significant disadvantage of the Factor 9 Temperament
is the usage of instruments, in particular acoustic instruments.

The only acoustic instruments possible to use are: the human voice,
fret-less string instruments (like the Violin family),

Trombone (a wind instrument without valves or tone-holes),

the Harp and percussive instruments.

Naturally one could compose and produce music with modern Synthesizers, software with micro tuning capabilities
or design / invent a new instrument based on this Temperament.

SONIC GEOMETRY: THE LANGUAGE OF FREQUENCY AND FORM

IMPORTANT FOOTNOTES ABOUT THE MOVIE "SONIC

GEOMETRY:There are some major mistakes
in the mathematical grid of the movie.
Some numbers listed are simple miscalculations,
but a more crucial mistake is that one note is missing in the scale!!!

Miscalculations:

B in column 2 has to be 468 instead of 456,
643 in column 3 has to be 648 and 3755 in column 5 has to be 3744.

Missing freqencies:

• 135.9 (between C# and D in column 1)
• 243 (between 234 in column 1 and 525 in column 2)
• 486 (between 456 in column 1 and 504 - missing - column 3)
• 504 & 522 (between 486 - missing - and 540 in column 3)
• 972 (between 936 in column 3 and 1008 in column 4)
• 1044 (between 1008 in column 4 and 1080 in column 4)
• 1116 (between 1080 in column 4 and 1152 in column 4)
• 1944 (between 1872 in column 4 and 2016 in column 5)
• 2088 (between 2016 in column 5 and 2160 in column 5)
• 2232 (between 2160 in column 5 and 2304 in column 5)
  
  Movie screenshot:
  

Another footnote to make is related to what is being said
in the movie about Concert Pitch and instruments.

In the movie Eric Rankin mentions that most modern
musical instruments have been tuned to 432 for decades

(until A4=440Hz became the International Standard).

This is not correct, 432Hz has never been a standard,
and only some old instruments seem to / might have been
build for (or close to) 432Hz as Concert Pitch.

There are many old instruments in museums,
as well as old Pitchpipe (Church) organs
with various pitches ranging between

A4=360Hz up to A4=460Hz.

Instruments for Baroque music (1600-1750) for example,
were designed for a Concert Pitch 415Hz.

Below the movie "Sonic Geometry" (by Eric Rankin).

Publicado por Jose Alfonso Hernando en 9:20 AM

http://noticiasseleccionvaldeandemagico.blogspot.ca/2014/11/432.html

 

are the Songline harmonies
-- from the 265 Giga-Terra Hz superluminal frequency
of Mother Proton's inner ring, to the 7 Giga-Terra Hz
of her outer ring, to the lesser luminal tonic of 'Child' Electron
-- are they 'Octave' based, of the seven/twelve (artificial?)
notes of our Western Music Scale?

or do they sound a five-note Pentatonic Hierarchy,
or other, true, kinematic structure-based scale of resonant harmonies ...
wherein the nodal resonances above/below
and around/within/without each other note
are determined not only by the differing wavelengths,
but also by the differering (subluminal, luminal, superluminal) Velocities
of each electromagnetic light/song note/being!

Millennium Twain


each note a colour, a structure, a personality, an individual!

 

Wide Music scale in 6 directions over the heart.
Soon to add the inner lines - starting from scratch.
Abyssal Dionysus

 

The polyhedral Tree of Life is correlated
with the eight Church musical modes.

These musical scales, which start and finish with the D scale,
comprise 48 notes between the tonic and octave
— a set of 50 notes that is represented
by the 50 corners of the first (6+6) regular polygons
of the inner Tree of Life.

They comprise 14 different notes
(eight with tone ratios of the Pythagorean scale,
six with non-Pythagorean tone ratios).

The eight scales have 168 rising intervals
and 168 falling intervals between notes above the tonic.

These intervals are symbolised by the (168+168) yods
in the first (6+6) polygons other than their corners.

These are the musical and Tree of Life counterparts
of the 168 automorphisms and 168 antiautomorphisms
of the Klein Quartic.

Its symmetry group PSL(2,7) is isomorphic
to the symmetry group SL(3,2)
of the Fano Plane representing the multiplication of octonions.

Their group of automorphisms is the exceptional group G2.

Its seven pairs of roots correlate with the seven pairs
of notes in the seven musical scales,
explaining why the correspondence exists,
viz. Pythagorean music and physics based upon octonions
share the same principles.

The 48 vertices of the 144 Polyhedron that stem
from the 48 faces of the underlying disdyakis dodecahedron
express the 48 basic degrees of freedom
manifested by a holistic system.

In the context of the musical modes,
these are notes, which can be grouped into eight sets
of six notes between the tonic and octave.

The icosahedron with 12 B vertices in the disdyakis triacontahedron
represents the 12 different notes between the tonic and octave of these scales.

Eight of them consist of fours pairs of notes
and their complements,

one of which is Pythagorean and the other non-Pythagorean.

The 216 edges of the 144 Polyhedron represent the 216 intervals
other than octaves between the notes of the eight Church musical modes.

The 180 edges of the disdyakis triacontahedron
represent the 12 basic notes and the 168 intervals
other than octaves between notes above the tonic.

The 13 rising intervals and the 13 falling intervals
between the tonic and the other types of notes in the Church modes
are the musical counterparts of the 13 Catalan solids and their duals.

Just as the disdyakis triacontahedron has 26 sheets of vertices
that are perpendicular to either B-B or C-C axes
and 33 sheets perpendicular to A-A, B-B & C-C axes,
so it is the 26th member of the family of Archimedean
and Catalan solids and the 33rd stage in the development of polyhedra.

The disdyakis triacontahedron contains 1680 vertices,
edges & triangles, where 1680 is the number of yods
in the lowest 33 Trees of Life constructed from tetractyses.

The 28 polyhedra defined by its 62 vertices have 3360 hexagonal yods
in their faces.

This is the number of yods needed to construct the inner Tree of Life
from 2nd-order tetractyses.

It is also the number of helical turns in one revolution
of the ten whorls of the basic unit of matter described
over a century ago by Annie Besant & C.W. Leadbeater,
showing how the disdyakis triacontahedron
embodies geometrically the structural parameter of this object,
identified in earlier work by the author
as the E8×E8 heterotic superstring constituent of up and down quarks.

http://www.smphillips.8m.com/article-31.html

from: https://www.facebook.com/ArtoftheInitiate/photos/a.213895748801200/370392349818205/?type=3&theater

 

The seven notes D–C' above the tonic of the C scale
therefore comprise three pairs of notes (D, E), (F, G) & (A, B)
and the octave C' as well as a (3:3:1) pattern.

The two triplets (D, F, A) and (E, G, B) correspond
in the Tree of Life (Fig. 2) to the two triads of Sephiroth of Construction:

Chesed-Geburah-Tiphareth and Netzach-Hod-Yesod,
whilst the last note of the C scale, the octave C',
corresponds to Malkuth, the last Sephirah of Construction,
which completes the emanation of the Tree of Life.
The three pairs of notes (D, E), (F, G) & (A, B)
separated by a tone interval correspond to the pairs
of Sephiroth of Construction on the three pillars of the Tree of Life.

This parallelism

2

suggests that the musical scales, both collectively
and in their mathematically perfect version
— the Pythagorean scale, conform to the Tree of Life,
the Kabbalistic representation of Adam Kadmon, or ‘Heavenly Man.’

http://www.smphillips.8m.com/article-31.html

from: https://www.facebook.com/ArtoftheInitiate/photos/a.213895748801200/370393009818139/?type=3&theater

 

1. The ancient musical scales/Church modes
Historically speaking, musical scales were always divided
into eight notes because the ancient Greeks
regarded them as composed of two tetrachords (sets of four notes).

If the pitch, or ‘tone ratio,’ of the starting note (‘tonic’)
of a scale is given the value of 1, the eighth note of the scale (‘octave’)
has a tone ratio of 2, that is, it has twice the frequency of the tonic
and is the tonic of the next higher set of eight notes.

The arithmetic mean of these two frequencies is (1+2)/2 = 3/2.

This is the tone ratio of the ‘perfect fifth,’
so-called because it is the fifth note in the ascending scale,
counting from the tonic.

The musical scale based entirely on octaves and fifths
is called the ‘diatonic scale.’

The tone ratios of the eight notes making up an octave of this scale are:

1 9/8 (9/8)2 4/3 3/2 27/16 243/128 2

This diatonic scale is also called the ‘Pythagorean scale'
because Pythagoras is generally thought to have discovered
its mathematical basis.

It comprises five tone intervals (T) of 9/8 and two intervals (L)
of 256/243, called in Greek the leimma, or ‘left over,’
which corresponds to the modern semi-tone,
although slightly less than it.

The interval pattern of the Pythagorean scale is:

tone–tone–leimma–tone–tone–tone–leimma.
T T L T T T L

In the C scale, the tonic is labelled ‘C’ and subsequent notes
in the scale are labelled D, E, F, G, A & B, the octave being
written as C'.

Their tone ratios are:

1
9/8
(9/8)2
4/3
3/2
27/16
243/128
2
C
D
E
F
G
A
B
C
As proved in Article 14,1 the six notes above the tonic
of the C scale can form only two triplets of notes
with the same proportions of their tone ratios.

They are (D, F, A) and (E, G, B),
corresponding members of which are separated by a tone interval (Fig. 1).

http://www.smphillips.8m.com/article-31.html

from:
https://www.facebook.com/ArtoftheInitiate/photos/a.213895748801200/370392803151493/?type=3&theater

 

A number of non-snark hypohamiltonian graphs,
including a number previously discussed,
are shown above, ordered by vertex count.

https://www.facebook.com/ArtoftheInitiate/photos/a.213895748801200/370391319818308/?type=3&theater