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

Discussion in '~THE NEW EXCHANGE~' started by CULCULCAN, Sep 28, 2014.


    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    "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)

    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    -1 + 0 + 1 = 0
    ~susan lynne schwenger

    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    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.


    A tuning system is Concert Pitch + Musical Interval System
    (+ Temperament).

    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.

    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 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.


    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


    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.


    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


    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.


    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    Monday, November 17, 2014


    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.


    "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)."


    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 21/15, 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:

    Factor 9
    Factor 9
    Factor 9
    0D288288 0 1/11/1
    1Eb (D#)301.630680104.95540950040728 22/2117/16
    2E315.9324160 203.9100017307748734/319/8
    3F330.8342240 297.513016132302685/7419/16
    4F#↓ (Gb)346.5360320386.3137138648348 77/645/4
    5Gb (F#)362.9378400470.78090733451234 63/5021/16
    6G380.0396480551.3179423647566 95/7211/8
    7Ab (G#)400414560628.2743472684155 25/1823/16
    8A416.8432640701.9550008653874 55/383/2
    9A436.5450720772.6274277296696 97/6425/16
    10Bb (A#)57.2468 800840.5276617693107 27/1713/8
    11B478.8486880905.8650025961624 133/8027/16
    12B501.4504960968.8259064691249 47/277/4
    13C525.252210401029.5771941530866 31/1729/16
    14Db (C#)55054011201088.2687147302222 21/1115/8
    15D57657612001200 2/12/1
    *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.

    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.



    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!!!


    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

    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    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!

    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

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

    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    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.


    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    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


    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.’


    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member

    11027997_370392803151493_8175453548716617731_n. 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:

    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:

    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).


    CULCULCAN The Final Synthesis - isbn 978-0-9939480-0-8 Staff Member


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