A Lacuna in Slonimsky Thesaurus

7/22/2019 A Lacuna in Slonimsky Thesaurus

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A Lacuna In Slonimsky's Thesaurus

In Thesaurus of Scales and Melodic Patterns,Nicolas Slonimsky gives a table of

twenty-one "divisions of the twelve notes into four mutually exclusive triads." [pp.!!-#

In Music Since 1900 (th ed!"$ he states% "these findings were first published in &'! ...

and have been since verified by a digital computer without finding any new solutions."

[p.''(# )his note supplies a missing classification and asserts that there are no others.

Numbering the combinations published in Thesaurus of Scales and Melodic

Patterns from one through twenty-one$ one can easily verify that *$ +$ and ' are

transpositions of , is a transposition of , $ &$ and ( are transpositions of !, *$ +$

and ' are transpositions of , $ !$ and are transpositions of , and *( and *

are transpositions of &. )hey can be subdivided into five classifications according to

the type of triad% /0 ma1or$ ma1or$ minor$ and minor /henceforth denoted by 22mm0, /*0

ma1or$ augmented$ augmented$ and minor /233m0, /+0 ma1or$ minor$ augmented$ and

diminished /2m3d0, /'0 ma1or$ diminished$ diminished$ and minor /2ddm0, /0 four

augmented /33330. )hus$ Slonimsky4s table can be reduced to six uni5ue combinations

within the five different classifications. )he extra classification comes from the two non-

isomorphic forms of two ma1or and two minor triadic subdivisions, 6xample will be

denoted by 22mm and example will be denoted by 2m2m.

Slonimsky4s examples are%

/a0 22mm 768 9:;3 :3;7; 8;


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Matriarchal Me#achords $rossmutterakkords

In Thesaurus of Scales and Melodic Patternsby Nicolas Slonimsky$ a chord using

all * note names and different intervals is called a "2utterakkord" /2other 7hord0, if

the chord is symmetrically inevitable$ then it is called a "8rossmutterakkord" /8rand

2other 7hord0. [p.*'+# In the liner notes of a recording of his music$ Slonimsky wrote

that he discovered one of the grand monsters "in a moment of plenary inspiration in the

afternoon of :ebruary +$ &+." [>rion >?S !*((# In Music Since 1900$ he further

divulges that this moment occurred during a radio broadcast of the :ourth Symphony of

)chaikovsky by the New @ork Ahilharmonic at '%+( p.m., however$ he failed to note

during which movement or who was conducting. ['th ed.$ p. '# 7omments concerning

the 8rossmutterakkord were written by 6rnst =renek$ Aaul Bindemith$ and Sergei

Arokofieff in Slonimsky4s guest book. Bindemith extended to it a long life and a good

progeny$ while Arokofieff in effect said ")o the devil with grandmothers$ let us write

musicC"

7onsidering the ordered collection of notes horiDontally rather than vertically process

a "tone row". Slonimsky incorporates his 8rossmutterakkord in ">rion"$ the first of his

"2initudes" for piano. >ther composers have also used his 8rossmutterakkord% =arl-


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inversion can be employed to define e5uivalence relations on this set. :or instance$

since the sum of the intervals is and since by transposition the first note can be

named "7"$ this reduces the number of 2utterakkords by G* to $&+*, note that the

top note will always be ":;" five and a half octaves above. )hus$ another interval of

semitones /the tritone0 will render another "7". Fith this another criterion for

e5uivalence can be defined, the "?6H>EH6" of a 2utterakkord will be the

concatenation of the se5uence following the occurrence of the tritone in the original

mutterakkord with the beginning of the se5uence$ then transposed down a tritone.

6xample% 2utterakkord% 7$ 7;$


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being a 2utterakkord to eliminate the others. )he property used was based upon the

fact that the intervalic se5uence can not contain any subse5uence which sums to a

multiple of twelve /to eliminate the possibility of the same note name occurring twice0

and no left subse5uence can sum to a multiple of twelve plus six /to eliminate :; from

occurring too early0. sing these conditions$ a computer program was written and a

systematic search was performed. )o minimiDe time$ only 2utterakkords with initial

interval smaller than a tritone /0 were sought, the others can be produced by inversion.

)he reduction to the set of non-e5uivalent /non-isomorphic0 2utterakkords was

achieved by then performing each of the e5uivalence tests$ transposing and checking

for agreement. )he search produced a list of *( 2utterakkords.


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)he interval se5uence of the '' )he interval se5uence of the +(non-e5uivalent 8rossmutterakkords non-e5uivalent self-revolves

0 -*-!-'-+--&---(- 0 -*-'-!-+---(---&*0 -*-!-'-&--+---(- *0 -+-(-----&-*-'-!

+0 -+-'-!-*--(---&- +0 -'-*-+-----(-&-!'0 -+-(-!---'--*-&- '0 -'-*-&-----(-+-!0 -'-+-*---!-(-&-- 0 -!-*-'-&----(--+0 -'-&-*---!-(-+-- 0 -!--(-+----'-*-&!0 --!-(-+--&-*--'- !0 --(-!-&---'-*--+0 --!-(-&--+-*--'- 0 -&-'-!-*---+---(&0 -&-'-!-*--(---+- &0 -&-!-*----+--(-'(0 -&-(-!---'--*-+- (0 -&-(-!----+-*--'0 -(-+----!-'-+-*- 0 -&-(-----+-*-'-!*0 -(-&----!-'-+-*- *0 -(--+----*-!-&-'+0 *--'-+-!---&---( +0 *--'-+-!--(---&-

'0 *--'-&-!---+---( '0 *----+--(--!-'-&0 *--!-+-'---&---( 0 *--&--'--(-!-+--0 *--!-&-'--!-+---( 0 *-!---&--(-'--!-+!0 *----+--&-!-'--( !0 *----&--(-'--!-+0 *----&--+-!-'--( 0 *--!--+--(-'---&&0 *-+---'----!-&-( &0 *--+-!-'--(--&--*(0 *-+---'---!--&-( *(0 +---(-'--&--!-*-*0 *--+--'----&-!-( *0 +--(--!--&--*-'-**0 *--+----'--&-!-( **0 +-*-'----&-(---!*+0 *--&--'----+-!-( *+0 +-*--!---&-(---'*'0 *--&----'--+-!-( *'0 +-'-*--!--&--(--*0 *-!--+-'---&---( *0 +---*---&-!--(-'*0 *-!--&-'---+---( *0 +-(-!--'--&-*---*!0 *-!---+--&--'--( *!0 +---(---&--'-*-!*0 *-!---&--+--'--( *0 '-+--*----&--(-!*&0 *-&---'----!-+-( *&0 '-(--&----*--+-!+(0 *-&---'---!--+-( +(0 '--*-&-----(-+-!+0 *--+----'-!-&--(+*0 *--+-!-'----&--(++0 *--&----'-!-+--(+'0 *--&-!-'----+--(+0 +--(--'--!--*--&+0 +-*--'---!---(-&+!0 +-'--*-!---(---&+0 +--*----'--(-!-&+&0 +-!-(--'----*--&'(0 +---(---!-*--'-&'0 +-(---!---'--*-&'*0 +--*-!---'--(--&'+0 '--*-+-'---&-(--''0 '--*-&---!-+-(--

p. =eyton

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A Lacuna in Slonimsky Thesaurus