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Perceiving “the harmony of the spheres.
The  Hellenes  attributed  this  faculty to
Pythagoras.   Ancient   Arabic  tradition
conferred  this honor also upon Moses.”

Alfred Sendrey, Music in Ancient Israel


The Mosaical Roots of European Musical Theory*

by Dr. Ed Metzler

Dr. Metzler-Moziani (Photo), The Metzler Formula

       §  1.  This  essay concerning musical theory is
a  by-product  of  jurisprudence.  In  July  of  1983
I  reconstructed  the  Mosaical  Tablets of the Law
from  the  well-preserved  text  of  the  Decalogue,
not  an  unlawful  project for a professional lawyer
and judge trained in ancient Roman Law as well as
Comparative  Law,  albeit  unwelcome to theology


            *Dedicated  to  my  friend  Professor  Dr.  Asher S. Kaufman,
Racah  Institute  of  Physics,  The  Hebrew University of Jerusalem,
who  shares  my  interest in the Tablets of the Law (Notearrow7 infra).


Ed  Metzler

as  a  challenge  to  its pretended monopoly on the
Law  of  Moses.  My  reconstruction proved to be
more  than  a  mere  hypothesis  because of fitting
like  a  glove  into  the  Ark of the Covenant. This
supplied the measurements of the tablets in cubits,
so  that  I  was able to calculate their volumes and
weights,  which led to my discovery of the system
of  Mosaical  metrology, prepublished in 1985 and
forming  the  first chapter of my last book entitled
the  Scientific  Study  of  the  Law  of  Moses and
Mosaical Antiquity
       §  2. The history of European musical theory
begins with Pythagoras, to whom the invention of
our  diatonic  scale  is ascribed.2) He is also said to


            1)   Cf.   Ed  Metzler,  DISCOVERING  MOSAISTICS,  Intro-
duction to the Scientific Study of the Law of Moses and Mosaical
,  Cumulative Reprint of AMMM (Archives for Mosaical
Metrology  and Mosaistics) vol. 1, nos. 15 with forewords by the
Rt.  Hon.  Lord  Hailsham  of  St.  Marylebone, KG, CH, FRS, DCL,
formerly   Lord   High   Chancellor  of  England,  and  by  Professor
Emeritus   Dr.  Johann  Knobloch,  University  of  Bonn,  Germany
(Herborn 1989) p. 24 Note 18, and pp.arrow2960.
            2)  See Hermann L. F. von Helmholtz, On the Sensations of
Tone as a Physiological Basis for the Theory of Music, translated
from   German   by   Alexander   J.  Ellis  (New  York  1954)  p.  266.


Musical  Theory

have  perceived the so-called music or harmony of
the spheres, whatever that meant,3) and beside him
only  Moses  is  credited with this ability in ancient
Arabic  tradition.4)  Indeed, this would make sense,
if  Pythagoras owed all of his theories to the Jews,
as  Hermippus  of  Smyrna claimed and I was able
to  prove.5)  The  graphical  details of the Mosaical
Tablets  of the Law, which I discovered, constitute
the  bond  between  his geometry of right triangles,
and  his  religious theories (T-orah) on the holiness
of the Tetraktys Y.H.W.H. or YaHUH (Yahuweh)
and  the  ten  spheres  (Sephirot) of the Decalogue
as well as the vegetarianism of his kosher life-style.
The  same applies to his musical theory, as I wrote
already  in  my  last  article, but this point deserves
more  extensive  treatment.6)


            3)  Cf.  Helmholtz  (N.  2)  p. 229. See below Notes 20 and 45.
            4) See  Alfred  Sendrey,  Music  in Ancient Israel (New York
1969) p. 480, literally quoted above p. 3.
            5)  Cf.  Ed  Metzler,  The  Impact of Israel on Western Philo-
sophy,  (Herborn  1993)  pp.  611,  and Note 33. On Hermippus of
Smyrna,  see  Encyclopaedia  Judaica  (German  1931)  vol. 7 at 665.
            6) See Ed Metzler, Western Philosophy (N. 5) p. 10 Note 14,
where  I  first  mentioned  the  Mosaical roots of European musical
theory,arrowwhich I discovered in spring of 1993.


Ed  Metzler

A. The Weight and Measurements of the
Mosaical Tablets of the Law
in Talmudic Tradition

       §  3.  In  the meantime, Asher S. Kaufman, to
whom this article is dedicated, has taken issue with
my discovery of the system of Mosaical metrology
on  the  weight and measurements of the Mosaical
Tablets  of  the  Law.7)  He opposes my calculated
value  of  36  kg  for  each tablet, because it seems
to  him  “to  be  far  in  excess of what Moses was
capable  of  carrying  in  the  world of reality.”8) In
his  view,  Moses  could  not possibly have carried
two  tablets  weighing  more  than 15 kg per piece,
and based on his own interpretation of the Talmud
he  proposes  13  kg  as  the weight of each tablet,


            7)  Cf.  Asher  S.  Kaufman,  The Weight of the Tablets of the
Covenant,    Where  is  Mt.  Sinai?  in  Niv Hamidrashia vol. 2223
(Tel   Aviv,  May  1990/Iyar  5750)  pp.  2536  Note  1,  where  he  is
acknowledging that his article was induced by chapter I of my book
(Notearrow1 supra), although, unfortunately, he refrains from referring
back  to  it in the further course of his argument, especially as far as
the  compatibility of thearrowtalmudic measurements of the Tablets of
the Law with mine is concerned, see below Notearrow15.
            8) See Kaufman (N. 7) p. 25.


Musical  Theory

confident  that  “This  value is well within the realm
of  reality.”9)  However,  this would only be realistic
if  one  single  person such as Moses were expected
to  carry  the  tablets in a rucksack or two handbags
from  the  Sinai  to  the  Promised Land. Of course,
this  violates  the  biblical  rules on transportation in
the  Ark  of  the  Covenant.
       § 4. The objections raised in Niv Hamidrashia
by  my  learned  friend are entirely beside the point,
since  the issue is not at all whether Moses was able
to  carry  both  tablets, as  he does on Rembrandt’s
famous  painting or Michelangelo’s great sculpture,
which  my  worthy  opponent may have had before
his  mind’s  eye.10)  While  this is a matter of artistic
license,  the  specifications set by the Bible (Exodus
25,  16  and  21)  require the Tablets of the Law to


            9)   Sic  Kaufman  (N.  7)  pp.  26  and  34,  who  interprets  the
talmudic  passage  “and  (their thickness is) three” (Notearrow15 infra)
as  fingerbreadths  in  order  to  “preserve  the  notion  of  a Tablet”,
which  “implies  that  the  thickness  is  much less than the length or
breadth”   (Idem   pp.   2829),   a  thickness  of  three  handbreadths
beingarrowabsurd,  indeed.
            10)  As  a  design  problem  of  experimental  archaeology, the
weight  of  the  tablets had to be such as to be carried in a coffin-like
wooden  box  (Aron) by several men, but not single-handed by one.


Ed  Metzler

The  Tablets  in  the  Ark
[The printed graphics were replaced by their equivalents from this website]
Geometry of the Tablets and the Ark of the Covenant
            The rectanglearrowABCD represents the Ark of the Covenant of
YaHUH  (Yahuweh),  which  is  a  box  made  out  of  acacia wood for
transporting   the   two   stone   Tablets  of  the  Law  from  the  Sinai
to  the  Promised  Land.  The  Bible  gives  its  precise measurements:
“two  cubits  and  a  half  was  the  length of it, and a cubit and a half
the  breadth  of  it,”  (Exodus  37,  1).  In  it  is  the  circle  later  named
after  Thales  of  Miletus  touching  the  short  sides of the box in the
middle  at  points  E  and  F,  while  cutting  its  long  sides  at  points
K,  L,  M,  and  N.  The  rectanglearrowKLMN  represents the Mosaical
Tablets  of  the  Law  lying  side  by  side,  divided in two by line OP.
The  four  right  triangles  KLM,  KLN,  KMN, and LMN make use of
the  later  so-called  Pythagorean numbers 3, 4, and 5 divided by two
(3 : 4 : 5 = 1.5 : 2 : 2.5),  each  of  the  Tablets  being  1  by  1.5  cubits,
or  translated  into  handbreadths: 6 by 6 and 3 (Ammah wa-Chetzi),
as  thearrowBabylonian Talmud (Nedarim 38a) puts it, if duly corrected.
[Professor  Dr.  Asher  Kaufman’s  small  tablets  are  marked  blue.]


Musical  Theory

be  laid  into the Ark, and we know that Moses put
them  there  (1. Kings 8, 9).11) The Ark was carried
by staves (Exodus 25, 14), which may be arranged
lengthwise  at  1.5  cubits  distance  or breadthwise
2.5  cubits  apart  (Exodus  25, 10), the former like
a  stretcher  or  hand-barrow  for two carriers, and
the  latter  like  the axles of a four-wheel chariot or
sedan  with  at  least  four carriers, so that it seems
likely  that  the  tablets  had  the average weight of
an  adult  human  body.12)
       §  5.  A  well-fitting glove reveals the shape of
the  hand  for  which it was made, and in the same
way  the  measurements of the Tablets of the Law
are  implicit  in  those  of the Ark of the Covenant,
as  shown by the diagram on p. 8.13) This being so,


            11)  Cf.  1.  Kings  8,  9:  “There was nothing in the Ark save
the two stone tablets, which Moses put there at Horeb.”
            12)  See  Ed Metzler, Discovering Mosaistics (N. 1) p. 34. If
each of the Tablets of the Law weighed 36 kilograms, both of them
had  72  kilograms  or  144 metric pounds (= 158.73 English pounds
avoirdupois), which is a good average human weight to be carried
on a stretcher or in a coffin, cf. Note 10 supra.
            13) Thearrowtrouble with Kaufman’s tablets (N. 7) p. 34 is that
they  bear  no relation to the measurements of the Ark, and make it
look  like a sham package, since they are way tooarrowsmall covering
only  one  square  cubit  of  its bottom, which is 3.75 square cubits.


Ed  Metzler

the  Bible could afford to refrain from stating them
explicitly,  for  everybody  in  his right mind is able
to  infer  that  each of the tablets is one by one and
a  half  cubits.14) At some point in history, probably
during  the  Persian period with the introduction of
the  seven-palm  cubit, it was deemed necessary to
clarify  that  the  Bible means the six-palm cubit by
translating  the  measurements  of  the  tablets  into
handbreadths:  six  by  six  and  three.  These three
numbers  were  then  mistaken to refer to the three
dimensions,  reading  “and their thickness is three”
(Nedarim  38a),  which must again be emended by
deleting  the  words  “their thickness is”, that were
once  erroneously  inserted.15)


            14)  See  Ed  Metzler, Discovering Mosaistics (N. 1) p.arrow32 on
my  discovery  of  these  measurements.
            15)  The  Babylonian  Talmud  (Nedarim  38a) reads: ha-Luchot
’Orkan Shishah we-Rochban Shishah we-(‘Oviyan) Sheloshah
, “the
Tablets,  their  length  is  six,  and  their breadth is six and (their thick-
ness is) three.” Their breadth ofarrow6 + 3 = 9 handbreadths is identical
with  that  of  the  Ark  of  the  Covenant: Ammah wa-Chetzi Rochbo,
“a  cubit  and  a  half  was  the  breadth  of  it” (Exodus 25, 10 = 37, 1),
while  the  length  of  the  Tablets  is  a cubit or 6 handbreadths each,
adding  up  to  the  length  of  the  Ark  (Ammatayim wa-Chetzi, “two
cubits  and  a  half ”),  of  which  half  a  cubit  or  3 handbreadths are
needed  for  getting  hold of the Tablets, cf. Ed Metzler, Discovering
Mosaistics  (N.  1)  p.  80  Notearrow34.


Musical  Theory

B. The Broken Tablets as Mathematical
Fractions Resulting from the
Geometry of the Tablets

       §  6.  The  thickness of the tablets is one tenth
of  a  cubit,  as  I  found out when I discovered the
three-dimensional  structure of the Ten Command-
ments.16)  Their  text  was written boustrophedon in
10  lines  of  32  letters  each,  entered into squares
like  those  of  a crossword puzzle, which disclosed
the  proportions  of the Tablets of the Law.17) Since
they  were  designed to be carried from the Sinai to
the  Promised  Land of Israel, it was only a matter
of  common  sense  to use as much of their surface
for  writing  as  possible,  both front and reverse as
the Bible says (Exodus 32, 15). Of course, the two


            16)   Cf.   Ed   Metzler,   Discovering  the  Three-Dimensional
Structure   of   the   Ten   Commandments  (Herborn  1986),  forming
chapter  II  of  my  book  Discovering  Mosaistics  (N.  1) pp. 6192;
as  well  as  Asher  S.  Kaufman  (N.  7)  p.  25,  who  writes  about it:
“His  work  reveals  some  brilliant  insights,  especially his interpre-
tation  of  the  verse  Exodus  32,  15.”
            17)  See  Ed  Metzler, Western Philosophy (N. 5) p. 8 Note 9;
and  Discovering  Mosaistics  (N. 1) pp. 71 Note 16, and 77 Note 29
on  the  arrangement  of  letters  within squares as in ancient Egypt.


Ed  Metzler

sides  had to be numbered as face A and face B by
using  the  alphabetical  letters  Aleph and Bet, and
this  is  how I got the number of 32 letters per line,
namely  by  counting  from  the initial Aleph of the
text  to the first occurrence of the letter Bet, which
is  the  first  letter of the second line on the reverse
of  the  first  Tablet  of  the  Law.18)
       §  7.  The  10 lines (Sephirot or Devarim) run
in vertical columns from front to reverse, and from
reverse  to front, passing over the top of the tablets
to  their  opposite  sides  on  each of the lines in an
uninterrupted  chain of letters, 15 up and 15 down,
one  on  top,  and  one  to  the  side connecting the
lines  and marking the space between them, as can
be  seen  from  the diagram on p. 13.19) Each tablet


            18)  Cf.  Ed  Metzler,  Western  Philosophy (N. 5) Note 21. In
counting,  double  or  triple letters had to be reduced to singles, for
the  original  text  was  written  in  an  uninterrupted chain of letters
without  spaces  between  the  words, see Ed Metzler, Discovering
Mosaistics (N. 1) pp. 69, 72, andarrow136.
            19)  The  letter  connecting  lines 5 and 6 is the Kof in Poked
Awon  Avot
  “visiting  the  iniquity  of  the  fathers”  (Exodus  20,  5
=  Deuteronomy  5,  9),  which also connects the two Tablets of the
Law,  and  being  repeated  at  the  beginning  of  the second tablet,
it  brings  the  number  of  letters  from  319  to  320,  cf. Ed Metzler,
Discovering  Mosaistics  (N.  1)  pp.  75  Note  24,  and 144 Note 34.


Musical  Theory

The   Ten   Spheres   (Sephirot)
of   the   Decalogue

The Ten Spheres (Sephirot) of the Decalogue


Ed  Metzler

has  10  vertical  and  15  horizontal  lines, yielding
150 squares of 0.1 cubit, which is also its thickness
because  of  the  letter-unit  on  its  top,  so  that  it
consists   of  150  cubic  letter-units  by  reason  of
its  geometrical properties.20) These determine both
the  number  of 320 letters on the inscription of the
Tablets  of the Law and their weight of 36 kg each,
which   is   the  heaviest  weight-unit  of  the  Bible
known  as  the  ancient Israelite talent (Kikar), and
divided   into   3000  Shekels  or  6000  drachmae,
multiples of the 150 letter-units.21)
       §  8.  In  Hebrew  the  letter-units  were called
Midot  Ketivah  or  Ketivot,  and  their  number of
32  survives  in  the  Baraita  of  32  Rules (Midot)
as  well  as  the  32  mysterious  paths (Netivot) of


            20)  Understanding  the  geometry  of  the  tablets  and  the ark
as  well  as  its  metrological and musical applications is what is meant
by  perceiving  the  harmony  of  the  spheres ascribed to Pythagoras
and Moses (above Notes 3 and 4), which like the Greek synienai used
by   Porphyrius   (Vit.   Pyth.  30)  here  stands  for  intellectual  rather
than  sensual  perception,  insight rather than hearing, shared at least
by  Moses,  the author of the 10 spheres (Sephirot) of the Decalogue,
with  Pythagoras,  cf.  Ed Metzler, Western Philosophy (N. 5) Note 7;
and  Discovering  Mosaistics  (N.  1)  p.arrow135.
            21)  On  the  number  of  3000  Shekels  per  talent,  see  Exodus
38,   2526;  and  Ed  Metzler,  Discovering  Mosaistics  (N.  1)  p.  40.


Musical  Theory

wisdom  in  the Sepher Yetzirah.22) This basic book
of   Kabbalah,   which  hitherto  was  looked  upon
merely  as  a work of Jewish mysticism, represents
a   mystification   of   the  graphical  details  of  the
Mosaical Tablets of the Law, as I discovered when
comparing  my  results  with its detailed description
of  the  Ten  Sephirot.23)  Like  Sepher “book” and
Sopher  “scribe”,  the  term  Sephirah  has  a clear
graphical  connotation  in  Hebrew. It is a synonym
of  Davar  “word”,  which originally meant “drift”,
and  stood  for  the  10  boustrophedon lines of the
Decalogue  that were written like furrows ploughed
by  an  animal  being  driven back and forth across
the  field.  Similarly,  the  32  Netivot proved to be
the  32  letter-units  of  every  line.24)


            22)  Cf.  Ed  Metzler,  Discovering Mosaistics (N. 1) pp. 1312,
and  1378.  See  also  the Akdamut Millin by Rabbi Meir ben Itzhak
of  Worms,  Germany,  an  acrostic  poem  forming a double alphabet,
followed  by  the  author’s  name,  and ending with the 32-letter wish,
Yigdal be-Torah ube-Ma‘assim Tovim, Amen, we-Chazak we-Ematz,
which  precedes  the  Reading  of  the  Torah  on Shavu‘ot, including
the text of the Decalogue (below Note 36).
            23)  Their  identity  with the Decalogue was not recognized by
Gershom  Scholem,  cf.  Ed  Metzler,  Discovering  Mosaistics p. 150.
            24)   By   mistaking   the  initial  Kaf  for  Nun,  cf.  Ed  Metzler,
Discovering  Mosaistics  (N.  1)  pp.  132  Notearrow9,  and 137 Note 22.


Ed  Metzler

C. The Metrological Meaning of Manah
in the Omer and Tzintzenet
Laid Before YaHUH

       §  9.  Another  way  of dividing the Tablets of
the  Law is by drawing 6 vertical and 10 horizontal
lines  not  for  writing purposes, but for reasons of
weights  and  measures,  yielding  60 nearly square
rectangles  of one handbreadth or 1/6 cubit by one
tenth  of  the  length  of  the  tablets or 0.15 cubits
=   3/20  ·  1/6  =  1/40  square  cubits.  Since  the
thickness  of  the  tablets is 1/10 cubit, one sixtieth
part  or  Manah  “portion”  of them is 1/400 cubic
cubits  or  222 cm3, if the Mosaical cubit measures
44.6  cm.25)  Its weight is 6000 drachmae (Hebrew


            25) On the cubit of 44.6 cm, see Ephraim Stern, Weights and
Measures,  in  Encyclopaedia  Judaica, vol. 16 (1974) at 379. Contra,
Asher  S.  Kaufman  (N. 7) pp. 27 and 34, who uses a Mosaical cubit
of 42.8  cm,  which  he  distinguishes  from  “the normal standard of
44.65  cm,  already  in  use  at  the  time  of King Solomon”, cf. Idem,
Determining  the  Length of the Medium Cubit, in Palestine Explora-
tion  Quarterly,  vol.  116  (1984) pp. 12032. However, the Mosaical
cubit is identified by the systematic context of Mosaical metrology,
since  a  cubic cubit of 44.63 cm equals 40 Omers of 2.22 liters each,
see  Ed  Metzler, Discovering Mosaistics (N. 1) p. 24 Note 21, p. 80
Note 33, and pp. 19495 Notes 15 and 18.


Musical  Theory

Beka‘  “half-Shekels”)  divided  by  sixty, equalling
100  drachmae  or  600 grams, if the archaeological
value  of  one ancient Israelite Beka is 6.0 grams.26)
Hence  the density of the Manah weight-stone, and
of the Mosaical Tablets of the Law is calculated by
dividing  its  weight  of 600 grams by its volume of
222  cm3,  which  is  2.7  and  corresponds  to  the
specific gravity of granite.27)
       § 10. The volume of one Manah weight-stone
being  222  cm3  proved  to be exactly one tenth of
the  empirical  value, by now generally accepted in
archaeology  for  the ancient Israelite Omer, which
is  about  2.2  liters  and  one tenth of an Ephah or
Royal  Bat  of  approximately 22 liters.28) Thus, the
Manah  in  the  Omer  had a metrological meaning,


            26)  According  to  Ephraim  Stern  (N. 25) at 387, the seven
weight-stones found in Israel with Beka or B written on them have
an  average  weight  of  6.03  grams.
            27)  Cf.  Ed  Metzler,  Discovering  Mosaistics  (N.  1) pp. 25
Note  22,  34 Notearrow4, 76 Note 27, and 84 Notearrow41, confirmed by
Kaufman  (N.  7) pp.  3134. Although granite is a very hard stone,
we  know  from  the  pyramidion  of  Amenemhet III and the obelisk
of  Sesostris  I that it could be inscribed in Middle Kingdom Egypt,
regardless  of  how  it  was  done.
            28)  See  Ephraim  Stern  (N.  25)  at  380; and Exodus 16, 36:
“Now  an  Omer  is  the  tenth  part  of  an  Ephah.”


Ed  Metzler

The  Broken  Tablets  in  the  Ark,  and
the  Omer  full  of  Manah
laid  before  YaHUH

[The printed graphics were replaced by their equivalents from this website]
The Broken
Tablets in the Ark

            The   legendary   “broken   tablets”  of  Moses  were  mathematical
fractions   of   the   Mosaical   Tablets  of  the  Law,  which  according  to
talmudic   tradition   bore   no   inscription,   and  filled  the  empty  space
of   half   a  cubit  or  three  handbreadths  in  the  Ark  of  the  Covenant,
one   handbreadth   between   the  tablets,  and  one  between  them  and
the   wood   on   either   side,   being   sideways  wrapped  in  cushioning
fleece   (Tzintzenet,   diminutive   of   Tzinah,   from   Tzon   “sheep”),   as
indicated   on   the  above  diagram  by  the  six  vertical  blue  lines.  And
Moses   (Exodus   16,   33)   said   about   the   Ark   (El   ha-Aron,  correct
for   El   Aharon):   Take   a   little   fleece   (Tzintzenet),  and  put  therein
an  Omer  full  of  Man,  i.  e.  10  Manah  weight-stones, and lay it before
YaHUH,    i.    e.    the    first    tablet    beginning   with   “I   am   YaHUH”
(Exodus  20,  2).  The  Omer  is  a  vertical  slice  of  one  sixth  of  a  tablet
or   one  handbreadth,  which  is  horizontally  cut  up  into  ten  portions
(Hebrew   Manah)   of   0.15  cubits,  and  0.1  cubit  thick  as  determined
by  the  one  letter-unit  on  top  of  the  tablets,  divided  for  writing into
150   cubic  letter-units  of  0.1  cubit  (pp.  8  and  13).  Since  the  Manah
is  one  handbreadth  wide  and  wrapped  in  fleece, the other spaces are
filled   with  stones  of  0.1  cubit,  leaving  6  ·  1/45  cubit  for  the  fleece.


Musical  Theory

rather  than  a  biological  one, constituting the daily
ration   of  grain  for  one  person  distributed  from
Omer  pots  by  judges  over  tens  in  conformance
with  the  decimal  system  of  judicial structure and
the  biblical  measures  of  volume.29) The Omer full
of Manah laid before YaHUH (Exodus 16, 33) was
neither  “before  the  Ark”,  as  Rashi  thought, nor
a  “golden  pot”  within  it, as Rabban Gamaliel the
elder  taught  Saint  Paul,  but  a row of ten Manah
weight-stones  adding  up  to  one  Omer,  and  laid
in  front  of  the  opening words “I am YaHUH” of
the  first  Tablet  of  the  Law.30)
       §  11. When carrying the Ark of the Covenant,
the  broken  tablets  filled  its  empty  space  of half
a   cubit   (Luchot   we-Shivrey  Luchot  Munachim
,  as  the  diagram  on p. 18 demonstrates,


            29)  Even  today a quorum of 10 Jews is called a Minyan after
the  10  portions (Manah) in the Omer, see Ed Metzler, Discovering
Mosaistics  (N.  1)  pp.  445,  118  Note 46, and 178 Note 38. Contra,
the entomologist Frederick Simon Bodenheimer, who believed that
it was the excretion of scale-insects on tamarisks.
            30)  Cf.  Paul’s Epistle to the Hebrews 9, 4: the Ark, “wherein
was  the  golden  pot that had manna, and Aaron’s rod that budded,
and  the  tables of the covenant.” The rod may be a Torah “pointer”
(Moreh  “teacher”  or  “road sign”), whence “herm”, below Note 45.


Ed  Metzler

in  order  to  prevent  the Tablets of the Law from
sliding  sideways  and  damaging each other.31) The
broken  tablets  had  to  be  wrapped  in  fleece for
cushioning them off against the Tablets of the Law
so  that the Manah stones could only be used once
in  front of YaHUH, while the spaces between and
behind  the  tablets  were  filled  with  twice fifteen
of  the  smaller cubic letter-units, making up 1/5 of
one  tablet  and  weighing  600  Shekels  or  7.2 kg
called  a  Tzinah “golden fleece” (1. Kings 10, 16),
in  contrast  to  Tzintzenet  “little  fleece”  (Exodus
16,  33),  being  but  1/6  tablet  of  500 Shekels or
6  kg.32)  After  the  geometry  of the Tablets of the
Law  had  fallen  into  oblivion, they were believed
to  have  been  broken  in  anger.33)


            31) See Ed Metzler, Discovering Mosaistics (N. 1) pp. 4243
and 58. The length of the Ark (above p.arrow18) adds up to 2.5 cubits
from  right  to  left:  1/45  +  1/6  +  1/45  +  6/6  +  1/45  + 1/10  + 1/45
+  10/10  +  1/45  +  1/10  +  1/45  =  225/90  =  2.5  cubits.
            32)  Cf.  Ed  Metzler,  Discovering  Mosaistics  (N.  1)  pp. 43
Note   30,   51  Note  48,  and  195  Note  16.  The  Golden  Fleece  of
Greek  mythology  (below  Note 46) preserves the original idea that
Tzinah  and  Tzintzenet  derive  from  Tzon  “sheep”.
            33)  See  Exodus  32, 19. The Talmud reports that the Broken
Tablets  bore no inscription, and lay in the Ark of the Covenant, cf.
Ed  Metzler,  Discovering Mosaistics (N. 1) pp. 4243 Notes 2728.


Musical  Theory

D. The Harma (Chariot) of Harmony as
Greek for Hebrew Merkavah
of Jewish Mysticism

       §  12.  The  deification of the Mosaical Tablets
of  the  Law  resulted  from their being deposited in
the  Holy  of  Holies  of  the  First Temple, built by
King  Solomon  in Jerusalem almost 500 years after
Moses  had  made  them  in the Sinai, thus idolizing
the  very  same  stone  that  prohibits  idol worship,
and  naming  Israel’s god YaHUH (Yahuweh) after
its  first  two  words.34)  Consequently,  the wooden
box (ha-Aron), in which they were carried, became
the  divine  chariot  (Hebrew  Merkavah) of Jewish
mysticism,  whence  the  Pythagorean  harmony of
the  spheres  (from  Greek harma “chariot”).35) The


            34)  Cf.  Ed  Metzler,  Western  Philosophy (N. 5) Notes 5 and
29;  Discovering  Mosaistics  (N. 1) pp. 86 Note 46, and 142 Note 31;
as  well  as  p.  18  supra.
            35)  Anthropomorphism  changed  the box in which the stone
tablets  lay  into  the  dwelling  (Mishkan), in which YaHUH resided,
the  throne, on which he sat, or the chariot (Merkavah) that he rode
(above  Note 12), likewise personified as Harmonia and Hermes, see
Ed Metzler, Discovering Mosaistics (N. 1) p. 146 text accompanying
Note  40;  and  Western  Philosophy (N. 5) Note 32; cf. Note 45 infra.


Ed  Metzler

biblical  basis  of  Merkavah  mysticism is the first
chapter  of  the prophet Ezekiel, which has always
been  understood  as a description of the graphical
details  of  the Tablets of the Law, and this is why
it  has  been selected as the Haftarah for Shavu‘ot
“Pentecost”,  which  is  a holiday commemorating
the  legislation  of  the  Decalogue.36)
       §  13.  After discovering the Jewish blueprints
of  Pythagorean  philosophy  in geometry, religion,
and  vegetarian diet reminiscent of Daniel 1, 816
I  investigated  whether  or  not the musical theory
of Pythagoras likewise had Mosaical roots, and so
I  found  the congruence of our diatonic scale with
the 10 or 15 horizontals of the Tablets of the Law,
as  superimposed  in  the diagram on p. 23.37) First


            36)  The  Sefer  Yetzirah  (chapter 1, no. 8) quotes correctly
Ezekiel  1,  14  on  the  boustrophedon Decalogue, see Ed Metzler,
Discovering Mosaistics (N. 1) p. 137 Notearrow21.
            37) Our diatonic scale is not a natural necessity, as pointed
out  by  Helmholtz  (N. 2) p. 249, but a man-made geometrical con-
struction resulting from mathematical convenience and feasibility,
mainly  from  the  fact  that  7/10  is  a  good  approximation  of the
square  root  of 1/2 = 0.5, which divides the octave into two equal
intervals,  while  the  further  subdivision  of each half-octave into
three  tones  or  six  semitones  is  simply a matter of mathematical
interpolation, cf. Notes 43, 49, and 50 infra.


Musical  Theory

The  Congruence  of  the  Diatonic  Scale
with  the  10  or  15  Horizontals  of
the  Tablets  of  the  Law

[The printed graphics were replaced by their equivalents from this website]
Congruence of the Diatonic Scale







256.0 =
273.1 =
274.3 =
284.4 =
288.0 =
295.4 =
301.2 =
320.0 =
320.0 =
320.0 =
341.3 =
349.1 =
365.7 =
384.0 =
384.0 =
426.7 =
426.7 =
480.0 =
512.0 =
               The   first  adjoining  column
represents  an even-tempered scale
beginning    with    the   arithmetical
pitch  of  c256  =  28  vibrations  per
second, each subsequent semitone
being   calculated   by   multiplying
with  the  12th  root  of  2  =  1.0594,
and  each  subsequent quartertone
by  multiplying  with  the  24th root
of  2  =  1.0293. The second column
is  the  physical  scale  with  middle
c   of   256   vibrations  per  second,
while  the  third  column  gives  the
respective   frequency  ratios.  The
tempered   quartertones  are  given
only  between  c  and f# of the first
column   in   order   to   enable   the
reader   to  compare  the  values  of
the  physical  and  tempered scales.


Ed  Metzler

I  noticed  that  the  major Seventh (c : b = 8/15) is
defined  in  fifteenth,  the common denominator of
the  frequency  ratios  in the upper tetrachord from
the  Fifth  (c : g = 2/3 = 10/15)  to  the major Sixth
(c : a = 3/5 = 9/15),  and  ending  with  the  Octave
(c : c’ = 1/2 = 7.5/15).38)  In  the  lower  tetrachord
only  the  major  Third (c : e = 4/5 = 12/15) can be
turned   into  fifteenth,  the  Fourth  (c : f = 11/15),
the  major  Second  (c : d = 13/15),  and the Semi-
tone  (c : c# = 14/15)  being less than a quartertone
off the even-tempered standard.39)
       §  14.  Next I observed that the 10 horizontals
of  the  Tablets  of  the Law are equally relevant to
our  diatonic  scale,  supplying better values for the
lower  tetrachord,  which proved to be constructed


            38)   On  the  definition  of  the  major  Seventh  (c : b = 8/15)
as  well  as  the  other frequency ratios see Helmholtz (N. 2) pp. 17,
337, and the Table of Intervals by Ellis pp. 45356.
            39)  According  to  Ellis  in  Helmholtz  (N. 2) p. 526, even if
they  were  a  quartertone  off, European ears do not appreciate the
neutral  interval  of  a  quartertone  between  two  semitones of our
chromatic  scale,  and  hence persist in mistaking it “sometimes for
one and sometimes for the other.” Therefore, the above deviations
from the normal frequency ratios are negligible quantities, because
the  differences  c#  15/16    14/15  =  1/240,  d  8/9    13/15 = 1/45,
and f 3/4 11/15 = 1/60 arearrowvery slight.


Musical  Theory

in  analogy  to the upper tetrachord, since what the
fifteenths  are  for  the  one,  the tenths are for the
other.40)   At   the   top   of   the  lower  tetrachord,
there   is   the   Fourth   (c   :   f  =  3/4  =  7.5/10)
corresponding to the Octave (c : c’ = 1/2 = 7.5/15)
on  top  of  the  upper  tetrachord, followed by the
major  Third turned into tenths (c : e = 4/5 = 8/10)
corresponding  to the major Seventh (c : b = 8/15),
by  the  major Second (c : d = 9/10) corresponding
to  the  major Sixth (c : a = 3/5 = 9/15), and finally
by  the  fundamental  Prime (c : c  =  1/1 = 10/10)
corresponding  to  the Fifth (c : g = 2/3 = 10/15) in
the  upper  half-octave.41) The interval between the
two  tetrachords  is  neither  tenths  nor  fifteenths,
but  f    g  =  3/4 2/3 = 1/12 = (1/10 + 1/15) : 2,
the  arithmetic  mean  of  both.42)


            40)  What  is so special about tenths and fifteenths that the
common  denominator  of  the  frequency  ratios  in the lower tetra-
chord  is  10,  and in the upper one is 15, has mathematical reasons
connected  with  the  central  importance of the even-tempered Tri-
tone  f# 7/10,  of  which Helmholtz-Ellis were not aware, who have
but  f# 99/140,  whereasarrow7/10 = 98/140 is lacking (below Note 49).
            41)  For  the major Second (c : d = 9/10) see Helmholtz-Ellis
(N.  2)  pp.  17,  332,  and  454,  from  which 8/9 differs by only 1/90.
            42)  Cf.  Notearrow51  infra.


Ed  Metzler

E. Moses and the Muses: the Music
of the Ten Spheres (Sephirot)
or Ten Commandments

       §  15.  To Greek or Roman ears Mosheh, the
Hebrew  form of Moses, transcribed as Mousai or
Latin  Musae,  does  not  sound  like  a  masculine
singular,  but  like  a  feminine  plural,  and  hence
the  personification and deification of the Mosaical
arts  and  sciences  adopted by the Graeco-Roman
world  led  to  the  nine  goddesses  known  as the
Muses.43)  In  keeping  with  the  Torah  of Moses,
the Muses are concerned with music and literature
rather  than  sculpture  and  painting prohibited by
the  Decalogue  as  objects  of  idol worship.44) On
Greek  soil  the  Ark  of  the  Covenant  or  divine


            43)  The  Mosaical  arts of alphabetical music and writing pre-
vailed  in antiquity, as the Muses did over the Sirens, and as Apollo
did  over  Marsyas, because they were superior to their competitors,
and so did modern European music because of the superiority of its
man-made scale, cf. Notearrow54 infra.
            44)  Cf.  Exodus  20,  4  =  Deuteronomy  5, 8: “Thou shalt not
make  thee  any  sculpture (Pessel), or any picture (kol-Temunah)”,
because  “Thou  shalt  not  bow  down thyself unto them, nor serve
them!”  (Exodus  20,  5  =  Deuteronomy  5,  9).


Musical  Theory

chariot (Harma) was also personified as Harmonia,
who  became  the  wife  of Kadmos, the brother of
Europa   and   legendary  founder  of  Thebes,  the
Boeotian  city of David, and deified as Hermes, the
Latin  Mercurius  (by syncope from Mercavarius),
who  like  the Harma (Merkavah) whence his name
travels between heaven and earth.45)
       §  16.  In  Greek  mythology Apollo’s lyre was
given  to  him  by Hermes, as the Golden Fleece of
the  Argonauts  was  also  his gift, or translated into
plain  demythologized  language,  both  came  from
the  Ark,  the  Golden Fleece being an ingot of gold
equivalent   in   weight  to  the  twice  fifteen  cubic
letter-units  between  and  behind the Tablets of the
Law,  named  after  the  fleece  (Tzinah),  in which
they  were  wrapped,46)  and  the  classical 10-string


            45)  The  Greek  god  of  travelling  salesmen  Hermes derives
from  harma  “chariot”,  the  Ark  of  the  Covenant  of YaHUH, and
similarly  the  Roman  god  of merchants Mercurius “Mercury” from
Merc(a)v(a)rius,  whence  also  mercari  “to  trade” and merx “mer-
chandise”;  cf.  Notearrow35 supra. The brother of Moses, Aharon (by
metathesis  from  ha-Aron  “the  Ark”)  is likewise a personification,
see Sendrey (N. 4) p. 61; and above p. 18. As Jews wear hats before
the  Ark,  so  does  Hermes  holding  his  caduceus  or Aaron’s rod.
            46) See 1. Kings 10, 16; and above Notearrow32.


Ed  Metzler

The  10  Vertical  Lines  on  the  Tablets
as  Blueprints  of  the  Kinnor

[The printed graphics were replaced by their equivalents from this website]

The Tablets of the Law as Blueprints of the Kinnor

            If the 10 strings of an ancient Hebrew lyre (Kinnor or Assor)
are  tuned  to  produce  the musical note c = 15/15, then c’ = 7.5/15;
b = 8/15;   a = 9/15;  g = 10/15;  f = 7.5/10;  e = 8/10;  d = 9/10;  c = 1.


Musical  Theory

Hebrew  lyre  being  designed  after the 10 vertical
lines  on  the  Tablets  of the Law, as illustrated by
the  diagram  on  p.  28.47)  Since they look like the
warp (Hebrew Manor) of a loom, the lyre is called
Kinnor in Hebrew (by assimilation from Ki-mnor),
while  the  10  or  15  horizontals of the tablets are
like  its  weft,  and  yield  the most ancient method
of  tuning  the  lyre  down  to the time of Orpheus
(by  metathesis  from Hebrew Rophe’-us “healer”)
into  the  tetrachords  c貿  and  g膨’.48)
       §  17.  The middle of the Octave (c : c’ = 1/2)
dividing  it  into  two  equal  intervals  is  the even-
tempered  Tritone  (c : f# = 7/10), which is a good
approximation  of  the  square root of 1/2 available
already  at the time of Moses, for (7/10)2 = 49/100
and  50/100  =  1/2.49)  From  this  fixed  point, the


            47)  Cf.  Josephus  Flavius, Jewish Antiquities, VII 12, 3; and
Sendrey  (N.  4)  pp.  27475, who mentions (p. 61) that Cinyras, the
Greek  mythical  hero,  is a personification of Hebrew Kinnor “lyre”.
            48)  See  Helmholtz  (N.  2) p. 255. The Kinnor is also like the
furrow (Hebrew ke-Nir) or boustrophedon lines of a ploughed field
(Note  24  supra). On Orpheus/Ropheus cf. Italian Orlando/Roland.
            49)  Cf.  Richard  Gillings,  Mathematics  in  the  Time of the
Pharaohs  (MIT  Press  1972) pp. 21417; and Sendrey (N. 4) p. 125.
Ellis  in  Helmholtz  (N. 2) p. 455 fails to givearrow7/10 for the Tritone.


Ed  Metzler

lower  half-octave  10/10 7/10 = 3/10  is  further
subdivided   by   mathematical   interpolation   into
three  tones  or  six  semitones, and so is the upper
half-octave   7/10 5/10 = 2/10 = 1/5 = 3/15  with
an  aberration  of  plus/minus  1 % or less from the
even-tempered   standard,   as   elaborated   in  the
charts   on   p.   31.50)   This  explains  the  musical
significance  of  the  10  or  15  horizontals  on the
Tablets  of  the  Law,  which  are  not  an arbitrary
whim,  but  a  mathematical  necessity. By moving
each half-octave a semitone away from the middle,
the  two  tetrachords become disjunct, creating the
seventh tone of the heptatonic scale.51)
       §   18.   The  hebdomadal  periodicity  of  the
Sabbath  is a sign of Jewish identity, shared by the
heptatonic  scale  and  the  seven-day week.52) Yet,


            50)  Aarrowquartertone  is  + 2.9302 %  or  2.8468 %.
            51)  A  semitone  up  f# g  =  7/10 2/3  =  1/30 = (1/15) : 2,
and  a  semitone  down  f f# = 3/4 7/10 = 1/20 = (1/10) : 2  yields
the  whole  tone  between  the  tetrachords  f g  = 3/4 2/3 = 1/12
=  (1/10) : 2 + (1/15) : 2  =  (1/10 + 1/15) : 2,  see  text  accompanying
Notearrow42  supra.
            52)  Cf.  Exodus  31,  13 and 17: “It is a sign between me and
the children of Israel for ever”, just like circumcision performed on
the  octave  of  birth  (Genesis  17,  11  and  12).


Musical  Theory

The  two  Conjunct  Heptachords  of
the  Twelve-Semitone  Scale


The  two  Conjunct  Tetrachords  of
the  Hexatonic  Scale


The  two  Disjunct  Tetrachords  of
the  Heptatonic  Scale



Ed  Metzler

music  was  only  part  of the cultural package deal
transmitted  from  Solomonic  Israel  to  Greece in
the  golden  age  of  the  Israelite (18th) dynasty of
Egypt  under the House of David-Thutmosis I, and
during   the   centuries   that   followed,  when  the
Hebrew-Phoenician  alphabet gradually superseded
Linear  B  without  the  intermission  of  the Greek
Dark  Ages,  as  the  alleged  relapse  into  illiteracy
is  called,  invented to deny the impact of Israel on
Egypt  and  Greece.53) Like the system of Mosaical
metrology,  which  I  discovered ten years ago, the
European  tonal  system springs from the geometry
of  the Ten Commandments or Sephirot “spheres”
on the Tablets of the Law conceived by Moses and
perceived  by Pythagoras, being the music of the
geometrical  mind.54)


            53)  Cf.  Ed  Metzler,  Western Philosophy (N. 5) Notes 4, 24,
andarrow26; Discovering Mosaistics (N. 1) p. 50 Notearrow47.
            54)  Similarly,  our  law  is  called  “the  law  of  the  geometric
mind”,  cf.  Bernhard  Grossfeld,  Zahlen und Geometrie als Rechts-
symbole,  in  Festschrift  Rüdiger  Schott  (Berlin  1993)  pp. 34560
Note  4.  See  also Ed Metzler, Discovering Mosaistics (N. 1) p. 120
Notearrow52  on  the  alphabet and the geometrical mind. The musical
alphabet  of  our  diatonic  scale  is a geometrical construction, and
as such an expression of universal mathematical truth.


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