Appendix-1
Definition 2 (numeric):
The phrase consists of 2 chords, with each chord containing 3 notes. If the 2 chords being analyzed
match the computed pattern below, and are within 2 measures of each other, then the phrase is a
"musical trademark" phrase.
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The 3 notes of the 1st chord are designated as: x1, x2, and x3.
The 3 notes of the 2nd chord are designated as: y1, y2, and y3.
The external differences (chord to chord) are designated as: e1, e2, and e3.
All notes are in semi-tones and represented by numbers.
X1 represents the numeric value for a lower note of the 1st chord (n).
X1 need not be the LOWEST note, although it frequently is.
X1 can have any value (be any number) except zero.
The 1st chord has the format: x1+0, x1+4, x1+10 in semi-tones (the internal differences).
“n” (any lower note) supplies the value for x1.
The 2nd chord has the format: y1+0, y1+4, y1+12 in semi-tones (the internal differences).
Y1 = x1-1.
The 2nd chord can also be computed as: x1-1, x2-1, x3+1 (the external differences).
That is (with example numbers on the right):
n=lower note of 1st chord n=7
x1=x1+0 x1=7+0 = 7
x2=x1+4 x2=7+4 = 11
x3=x1+10 x3=7+10 = 17
y1=x1-1 y1=7-1 = 6
y2=y1+4 y2=6+4 = 10
y3=y1+12 y3=6+12 = 18
Computing differences between the 2 chords produces:
e1=y1-x1 e1=6-7 = -1
e2=y2-x2 e2=10-11 = -1
e3=y3-x3 e3=18-17 = +1
Example: 1st Chord 2nd Chord
Eb----G-----C# ----> D-----F#----D
07----11----17 ----> 06----10----18
x1+0 x1+4 x1+10 y1+0 y1+4 y1+12 (internal diffs)
-1 -1 +1 (external diffs)
Additional details for numeric definition.
Compression
Where the notes in the 1st chord or the notes in the 2nd chord are separated by more than 1 octave
from each other, pitches may be raised or lowered by octave intervals (compressed) as long as their
relative positions in terms of "higher" or "lower" remain the same. That is, x2 must be higher than
x1, and x3 must be higher than x2, y2 must be higher than y1, and y3 must be higher than y2 before
and after raising or lowering the pitches, and intervening notes removed. This is "compression", and
it applies to all notes in both chords.
g# / ab = 00
A = 01 AA = 13
A# / Bb = 02 AA# / BBb = 14
B / Cb = 03 BB / CCb = 15
C / B# = 04 CC / BB# = 16
C# / Db = 05 CC# / DDb = 17
D = 06 DD = 18
D# / Eb = 07 DD# / EEb = 19
E / Fb = 08 EE / FFb = 20
F / E# = 09 FF / EE# = 21
F# / Gb = 10 FF# / GGb = 22
G = 11 GG = 23
G# / Ab = 12 GG# / AAb = 24
Table 1
Table of Note Values (Pitches) Used In This Paper
All values are relative to the starting point and do not represent
a particular note on the scale, such as "Middle C", etc.
For example, “A” (01) can be any “A” on the scale.
Example Using Musical Trademark Example A2 – K.550 (Numeric Analysis)
1st Chord 2nd Chord
x1-----x2-----x3 ----> y1------y2------y3
x1+0 x1+4 x1+10 y1+0 y1+4 y1+12
x1-1 x2-1 x3+1
Example A2 is from K.550, Symphony Nr 40 in G Minor, 1st movement, measures 15, beat 1,
and 16, beat 1.
Eb = the lowest note in the 1st chord.
Let n = the value of Eb (7) from Table 1.
X1 = n = 7. (Y1 = x1-1 = 6).
The notes with double letters represent notes 1 octave higher than their first occurrence in the
note table.
The 2 chords are listed musically and numerically. Then, the phrase is checked for being a
trademark phrase by performing the calculations shown.
Method 1: After obtaining the note values of a phrase from Table 1, compute the trademark
values based on the first note, and compare the results to the phrase being analyzed. If the
results match (the note values are equal), the phrase is a trademark phrase.
1st Chord 2nd Chord
x1 x2 x3 y1 y2 y3
Eb-----G------C# ---> D-----F#------D <== Phrase to analyze
Eb-----G-----CC# ---> D-----F#-----DD <== Notes expressed uniquely
from Table 1
07 11 17 ---> 06 10 18 <== Numeric values from Table 1
To compute the phrase’s qualification for Trademark eligibility given only the lowest note
(x1), the following algorithm can be used:
x1 x2 x3 y1 y2 y3
x1=7 x1+4 x1+10 <====================== Calculate 1st chord
. . . x1-1 y1+4 y1+12 <== Calculate 2nd chord
. . . (x1-1 x2-1 x3+1) <== The "x2-1" and "x3+1"
. . . . . . can be used instead of
. . . . . . "y1+4" and "y1+12". The
. . . . . . results are the same.
07 11 17 ---> 06 10 18 <== Computed values using
trademark algorithm
The computed values match the actual values in the 2 chords. Therefore, the phrase "07 11 17,
06 10 18" is a trademark phrase. Note that the algorithm needed only 1 note as a base for the
calculations: E-flat, or "07" from the table.
Method 2: Calculate the internal and external differences between the notes for a phrase.
If the results are “+0 +4 +10, +0 +4 +12, -1 -1 +1”, the phrase is a trademark phrase.
07 11 17 06 10 18
-07 -07 -07 -06 -06 -06
--- --- --- --- --- ---
+0 +4 +10 +0 +4 +12 <==Internal Diff Calculation
06 10 18 <==2nd Chord
-07 -11 -17 <==1st Chord
--------------
-1 -1 +1 <==External Diff Calculation
All results match the required values. Therefore, this phrase matches the definition of the musical
trademark phrase.
Usefulness of a Numeric Definition
Defining the trademark phrase numerically makes the definition potentially utilitarian. Since Mozart
used this phrase in many different keys, using a single numerical formula or model simplifies the
search process.
Additionally, if large quantities of music by Mozart and other composers were suitably digitized, a
computerized search for the phrase could be undertaken. (Any such computer program would also
need to be able to handle the interpolation of notes, delayed resolutions, allowing or disallowing one
note from a phrase to qualify as part of the trademark phrase, and probably other factors).
The numeric definition and translation of notes into numbers is not intended to be used for playing
the phrase (although it can be done). They can be used for identifying the phrase and listing the
notes from the phrase. The sheet music containing the phrase should corroborate the identification.
Numeric proof for example A1:
Chord locs: 36-4, 37-3.
Interpolation: Yes
As Written: Eb-Eb----------C#-G ---> D-D---------D--F# K397-aw.mid
Interpolated: Eb-Eb---[G]----C# ---> D-D—--[F#]--D K397-int.mid
Compressed: Eb---[G]----C# ---> D—--[F#]--D
Musical TM: Eb---[G]----C# ---> D—--[F#]—-D K397-tm.mid
From Table 1: Eb---[G]---CC# ---> D—--[F#]-—DD.
From Table 1: 07 11 17 ---> 06 10 18
Proof: +0 +4 +10 +0 +4 +12
Proof: -1 -1 +1
Numeric proof for example A2:
Chord locs: 15-1, 16-1.
Interpolation: No.
As Written: Eb-Eb---G-Bb-C#-Bb-C# ---> D-D----F#-A--D-F#-D
Compressed: Eb---G----C#-Bb ---> D----F#----D-F#
Musical TM: Eb---G----C# ---> D----F#----D
Note values and calculations of differences:
Eb----G-----C# ---> D----F#-----D <==Musical Trademark
Eb----G----CC# ---> D----F#----DD <==From Table 1
07 11 17 ---> 06 10 18 <==From Table 1
07 11 17 06 10 18
-07 -07 -07 -06 -06 -06
--- --- --- --- --- ---
.+0 +4 +10 +0 +4 +12 <==Internal Diff Calculation
.06 10 18 <==2nd Chord
-07 -11 -17 <==1st Chord
--------------
-1 -1 +1 <==External Diff Calculation