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A simplifi ed combinatorial analysis of Charles-
Louis Hanon’s Th e Virtuoso Pianist in 60 Exercises
By Katherine Burris and Lori Carmack, PhD
Salisbury University, Maryland
Abstract
This paper presents a mathematical application of music theory that is accessible to the undergraduate student. In particular, we
th
analyze some simple fi nger drills published in The Virtuoso Pianist in 60 Exercises by Charles-Louis Hanon (1819-1900), a 19
century French musician. Using Hanon’s exercises as a guide, we formulate a model that will generate exercises akin to Hanon’s, but
in a manner that is simpler and easier to enumerate. We then apply some basic counting methods and theory from the mathematical
fi eld of combinatorics to count the number of exercises our model can produce.
Introduction New York-based music publishing company, hands reach their initial position. The right
Enumeration is a common application G. Schirmer, Inc., fi rst printed its famous and left hands play simultaneously and in
1
of mathematics to music theory. Due to edition, making the work available to pianists unison throughout the exercise.
the complex nature of musical composi- in the United States. By the early 1900s, the When mathematically analyzing the mu-
tions, such analysis readily extends itself to use of Hanon’s drills was well established sic in Hanon’s The Virtuoso Pianist in 60
the use of graduate level mathematics, even among conservatories and music schools Exercises, a natural problem to consider is
for seemingly simple problems. For example, throughout Europe and Russia. The empha- the total possible number of exercises Ha-
counting the total possible number of differ- sis the Moscow Conservatory placed on the non could have composed in Part I, since
ent fi ve-note scales that can be created out of exercises is described by the famous com- they appear to be formulaic. As is commonly
the 12 notes of an octave can be addressed poser and performer Sergei Rachmaninoff, necessary in mathematics however, we must
using higher level group theory, a concept who graduated from the Conservatory in fi rst consider a simpler problem. That is, we
central to the fi eld of abstract algebra.2 Ideas 1891. Recounting examination requirements construct a model that will generate fi nger
from graph theory can be applied to analyze at the end of the fi fth year of the conser- exercises that are similar to Hanon’s, but are
combinatorial relationships among the 24 vatory, Rachmaninoff states, “[The student] based upon fewer notes. We then enumer-
3
major and minor chords. Because of this it knows the exercises in the book of studies ate the number of exercises our model can
is challenging to fi nd a mathematical prob- by Hanon so well that he knows each study produce. To do the counting, rather than ap-
lem in music theory that is both accessible to by number, and the examiner may ask him, plying a brute force method, we instead use
and worthy of research for an undergradu- for instance, to play study 17, or 28, or 32, a combination of basic counting principles
ate mathematics student. The basic compo- etc. The student at once sits at the keyboard and some theory from the fi eld of combina-
sitions of Charles-Louis Hanon, however, and plays.”5 Over the years, “literally thou- torics. This computation provides an inter-
provide an ideal setting for such mathemati- sands of reprints and transcriptions of the esting and challenging application of under-
cal analysis. exercises have been published in dozens of graduate mathematics to music theory. What
6
languages” . Their use continues to be a sub- follows are the details of the model and the
Background stantial and fundamental part of many pia- accompanying computation.
7
Throughout his life, Hanon produced nists’ regimen today.
various works for the organ and piano, but Hanon’s exercises are divided into three Model
he was most notably renowned for his col- parts. Etudes 1—20 make up Part I, Part In determining assumptions for our
lection of 60 pedagogical etudes, The Vir- II consists of exercises 21—43, and the re- model, we aim to produce exercises that are
tuoso Pianist in 60 Exercises, fi rst published maining 17 exercises comprise Part III. The analogous to Hanon’s, and that serve to ac-
in 1874. The purposes of the exercises are sections are purposefully designed to address complish his goals of developing the agility
to help the pianist acquire, “agility, indepen- specifi c challenges and diffi culties that virtu- and strength of the fi ngers. In addition, the
dence, strength, and perfect evenness in the oso pianists face while honing their skills. In exercises should transition easily from one
fi ngers, as well as suppleness of the wrists— this paper, we focus on the exercises in Part measure to the next. Finally, we attempt to
all indispensable qualities for fi ne execu- I. Each exercise in this section is based upon create exercises that have some type of mu-
4
tion.” The 60 drills transition easily from a simple pattern that appears in the fi rst mea- sical form and are pleasing to the ear. Two
one to another so that the musician may play sure. Subsequent measures are merely a rep- fundamental concepts in music theory are
through the entire set of exercises in a one etition of the pattern set in the previous, but the notions of consonance and dissonance.
hour sitting. At the time of fi rst publication, with the hands shifted up the scale by one These terms refer to how two or more notes
Hanon’s exercises quickly became popular note (see Figure 1). The exercise proceeds as sound in relation to each other. Generally
among pianists. By 1878, both the Conser- the hands ascend, shifting up one note at a speaking, the human ear prefers the sound
vatoire de Paris and Royal Conservatoire of time per measure for two octaves. The hands of consonant notes, and is somewhat re-
Brussels formally adopted 60 Exercises for then similarly descend down the scale, one pelled by dissonance. Both however, play an
the Virtuoso Pianist for use. In 1900, the note at a time; the exercise ends once the important role in music, as they “work to-
37 Journal of Undergraduate Research and Scholarly Excellence – Volume VI
Natural Sciences
in our model consist solely of the five mu-
sical notes C, D, E, F, and G. The notes C,
E, and G are the chord tones, and D and F
the non-chord tones. To mimic the tonality
of Hanon in our model, we insist that each
exercise begins on C and that no more than
two non-chord tones appear consecutively.
This seems to be one of the guidelines Ha-
non used to establish tonality in exercises 1
through 20 (excepting number 6).
With the above goals in mind, we formu-
late the model assumptions. It is sufficient
to describe the pattern for only the first
measure of an exercise, since this measure
determines the entire exercise. The pattern
requirements for each exercise follow:
i. The pattern consists of eight tones.
ii. We consider only “five-finger” patterns.
That is, the thumb remains on C, the first
finger on D, second finger on E, third finger
on F, and the fourth finger on G throughout
the pattern. This assumption simplifies the
counting.
iii. No note appears consecutively in the pat-
tern. Hanon’s exercises in Part I satisfy this
condition. Presumably this allows the exer-
cises to be played more fluidly.
i v. Each finger needs to play at least once.
Equivalently, each of the notes C, D, E, F,
and G must appear in the pattern at least
once. In this way, each finger is exercised.
Hanon’s Part I exercises satisfy this condi-
tion as well.
v. The pattern starts on C and ends on ei-
ther an E or an F. This assumption helps es-
tablish the tonality of the exercise and en-
sures a smooth transition between measures.
In order to develop speed, dexterity, and agil-
ity, the pianist must be able to move easily
between measures, playing as continuously as
possible to maximize effort. Ending on a C
or D would hinder the ascending motion of
the piece. Additionally, this assumption sim-
plifies the counting.
vi. At most two non-chord tones appear con-
secutively. This condition attempts to maintain
tonality.
At this point then, we can utilize assump-
tions i through vi to generate piano exercises
4
Figure 1. The first few measures of Hanon Exercises 1, 2, 10, 18 that are similar to Hanon’s. The exercises
produced from the model essentially differ
Figure 1: The first few measures of Hanon Exercises 1, 2, 10, and 184
gether to create interest, drama, and beau- Major, that is, the notes C, E, and G. There- from those of Hanon’s only in that they are
8
ty.” Dissonance brings an unsettled and ex- fore, in constructing our model we need a based upon five notes of the C-major scale,
pectant feeling, while consonance is used to condition that will guarantee a consonant whereas Hanon’s exercises are based upon
resolve that feeling. Hanon’s etudes have a sound as well as maintain the tonality of the six.
strong consonant sound. Also, the vast ma- C-Major chord.
jority of compositions, including Hanon’s 60 In the world of composition, there are Enumeration
Exercises for the Virtuoso Pianist, are based no hard and fast rules for establishing to- We next address the issue of counting
upon a specific set of pitches. As the piece nality. However, one rule of thumb is that the total possible number of exercises that
progresses, the melody may stray from this the first and last notes of a piece must be can be produced from the model. To this
foundation of tones, but never too far and of the fundamental pitches upon which end, the counting is simplified by separating
not for too long. This is what musicians refer the composition is based. Musicians refer the various main patterns into the following
9
to as tonality. Hanon’s exercises are centered to these as “chord tones,” and to all other individual cases:
7
on the consonant fundamental chord of C- notes as “non-chord tones.” The exercises A) The pattern consists of one pair of
Journal of Undergraduate Research and Scholarly Excellence – Volume VI 38
Natural Sciences
consecutive non-chord tones and six chord Principle which states that if one object can
tones. be selected in p different ways, and a second
B) The pattern consists of one pair of con- object can be selected in q different ways,
secutive non-chord tones, one single non- where p and q are positive integers, then a
chord tone, and five chord tones. choice of both the first and second object
10
C) The pattern consists of one pair of con- can be made in p×q different ways. For
secutive non-chord tones, two single non- example, an octave contains 12 notes, so
chord tones, and four chord tones. the number of two-note patterns that can
D) The pattern consists of two single non- be formed from the notes of an octave is
chord tones and six chord tones. (Note: a 12×12=144. The Multiplication Principle
pattern consisting of only one single non- extends to selecting multiple objects. Apply-
chord tone would violate assumption iv). ing the Multiplication Principle in this way
E) The pattern consists of three or four sin- to count the number of exercises our model
gle non-chord tones, and the remaining are can generate by using only assumptions i and
chord tones. ii yields 5∙5∙5∙5∙5∙5∙5∙5=5^8=390,625—an Figure 2. Examples of basic patterns generated from
F) The pattern consists of two pairs of non- impressive number. Configuration 1 of Case C that satisfy the model
chord tones and four chord tones. As an illustration of the counting proce- assumptions. The first pattern consists of the notes C-
G) The pattern consists of two pairs of non- dure, consider the following calculation that D-F-C-D-G-D-E, the second C-F-D-E-F-G-D-E, and
chord tones, one single non-chord tone, and yields the number of patterns for Configura- the third C-F-D-G-F-G-F-E. (This figure was created
1. C̅̅E using NoteWorthy Composer Software).
three chord tones. tion 1, : ̅ ̅ ̅
Because the enumeration is unique to 2. CF The remaining configurations of Case C
each case, to compute the total possible 3. C̅̅̅F are counted in a similar fashion. The results
number of exercises we must separately con- 4. C̅̅̅F total number are presented in Table 1.
total number number of number of number of number of
sider each. To illustrate how the computa- 5. C̅̅E of patterns that When counting, we can take advantage
of patterns that ( ) = (non-chord tone)×( of chord tone )
( ) = (non-chord tone)×( of chord tone )
tions take place, we detail the calculation for 6. C̅̅̅F satisfy assumptions of some obvious symmetry in the configura-
satisfy assumptions combinations combinations
combinations combinations
Case C. There are 12 different configurations ̅ ̅ ̅ tions. For example, it is easy to see that Con-
7. CF
in which one pair of non-chord tones, two total number figurations 1 and 9 of Case C are symmetric.
total numbertotal number ̅ ̅ ̅ number of number of
number of number of number of number of number of
number of number of
8. CF
number of number of number of
single non-chord tones, and four chord tones of patterns that For the sake of completion however, we will
of patterns that ( ) = ( )×( )
of patterns that
( ) = ( )×( )
̅ ̅ non-chord tone of chord tone
non-chord tone of chord tone
( ) = (non-chord tone)×( of chord tone ) = [( paired )×( single )×( single )]
9. CE
̅ ̅ =[( paired )×( single )×( single )]
occur. Let represent a chord tone (that is, present results for each possible configura-
1. CE satisfy assumptions satisfy assumptions combinations combinations
satisfy assumptions combinations combinations
̅ ̅ ̅
combinations combinations non-chord tones non-chord tones non-chord tones
10. CF
non-chord tones non-chord tones non-chord tones
a C, E, or G) and N represent a non-chord tion.
2. C̅̅̅F 11. C̅̅̅F
total number total number
tone (a D or F). The possible configurations Since the computations for the remain-
number of number of
number of number of number of number of number of
number of number of number of
number of number of number of
̅ ̅ ̅ number of number of number of
3. CF 12. C̅̅̅F
of patterns that of patterns that number of number of number of
( ) = ( )×( )
in Case C then are the following: non-chord tone of chord tone ing cases are similar, we briefly summarize
( ) = (non-chord tone)×( of chord tone ) =[( )×( )×( )]
×[( ) × ( ) −( )]
=[( )×( paired )×( single )] single
single single paired chord tones
paired single single
̅ ̅ ̅ =[( paired )×( single )×( single )]
4. CF ̅ ̅ ×[( single ) × ( single ) −(paired chord tones)]
1. CE
satisfy assumptions satisfy assumptions combinations combinations them in the following tables. Again, we com-
combinations combinations non-chord tones non-chord tones non-chord tones without a G
non-chord tones chord tones chord tones
non-chord tones non-chord tones
non-chord tones non-chord tones non-chord tones without a G
5. C̅̅E ̅ ̅ ̅ chord tones chord tones
2. CF ment that the operation of subtraction in the
̅ ̅ ̅ number of number of number of
̅ ̅ ̅ number of number of number of
number of number of number of number of non-chord tone combinations
3. CF number of number of number of
6. CF ̅ ̅ ̅ number of number of number of
=[( paired )×( single )×( single )]
=[( )×( )×( )] ×[( single ) × ( single ) −(paired chord tones)]
eliminates those in which every note does
paired single ×[(single ) × ( ) −( )]
single single paired chord tones
̅ ̅ 4. ̅ CF ×[( single ) × ( single ) −(paired chord tones)]
7. CF ̅ ̅ non-chord tones non-chord tones non-chord tones not appear, thus enforcing assumption iv. On
non-chord tones non-chord tones non-chord tones chord tones chord tones without a G
5. CE chord tones chord tones without a G
8. C̅̅̅F chord tones chord tones without a G
̅ ̅ ̅ occasion, a combination is subtracted twice
6. CF
̅ ̅ number of number of number of
number of number of number of
9. CE ̅ ̅ ̅ and needs to be added back in for accuracy,
7. CF ×[( single ) × ( single ) −(paired chord tones)]
×[( ) × ( ) −( )]
̅ ̅ ̅ single single paired chord tones as in Case G.
10. CF ̅ ̅ ̅
8. CF chord tones chord tones without a G
chord tones chord tones without a G
11. C̅̅̅F ̅ ̅
9. CE
̅ ̅ ̅ ̅ ̅ ̅
10. CF
12. CF ̅ ̅ ̅
11. CF 3 2 2
12. C̅̅̅F =(2∙1×2×2) × (3×3–2×2)=(2 )×(3 –2 )
= 8×5 = 40
Note that the above configurations satis- If a pair of non-chord tones appears,
fy assumption v, which states that the pattern then the pattern contains both a D and an
begins on C and ends on an E or F. F, since notes cannot occur consecutively
There are various approaches to count- (assumption iii). Subtracting the possible
ing the total number of patterns in each of number of paired chord tones that do not
these configurations. The method that we contain a G from the total number of chord
found to be most efficient is to multiply the tone combinations forces a G to appear in
possible number of non-chord tone com- the pattern. This is known in combinatorics
binations by the possible number of chord 11
tone combinations, all the while enforcing as an inclusion-exclusion principle. In this
assumptions i through vi. The most prob- way, the pattern satisfies all of the model
lematic aspect of this is ensuring that each assumptions. Figure 2 presents several first
note appears at least once in the pattern. measures of exercise that arise from Con-
This counting method uses ideas from the figuration 1 of Case C.
mathematical field of combinatorics, the
study of enumeration of various sets of ob-
jects. In particular, we use the Multiplication
39 Journal of Undergraduate Research and Scholarly Excellence – Volume VI
Natural Sciences
Case C) Case B) Case D)
(Number of Total (Number of Total (Number of Total
Non-Chord Tone Number Non-Chord Tone Number Non-Chord Tone Number
Configuration Combinations) × of Configuration Combinations) × of Configuration Combinations) × of
(Number of Chord Patterns (Number of Chord Patterns (Number of Chord Patterns
Tone Combinations) Tone Combinations) Tone Combinations)
2
̅ ̅ (23 2 2 (22 2 ̅ ̅̅̅̅̅̅ (2 – 1 – 1) × 44
) × (3 – 2 ) 40 ̅ ̅̅̅̅ ) × (3∙2 – 2) 40 3
̅ ̅̅̅̅
(3∙2 – 2)
̅ ̅ ̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅ ̅̅ ̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅ 2
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
(2 – 1 – 1) ×
̅ ̅̅̅̅
̅ ̅̅ 2 2 2 2 ̅̅̅̅ ̅ 2 2 ̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅
̅ ̅ 44
(2 ) × (3 ∙2 – 2 – 2 ) 40 ̅ ̅̅̅̅ (2 ) × (3∙2 – 2) 40
3
̅̅ ̅ ̅̅̅̅̅̅ ̅
̅ ̅̅̅̅
̅̅̅̅ ̅ (3∙2 – 2)
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅ ̅
̅ ̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅
̅̅ ̅ ̅ ̅̅̅̅ ̅̅̅̅̅̅ ̅
̅ ̅̅ ̅̅̅̅ ̅
̅̅̅̅ ̅
̅ ̅ ̅
̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅ 2
̅̅̅̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅̅̅̅̅̅ (2 – 1 – 1) ×
̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅
2 2 2 2 2
̅ ̅̅ ̅̅̅̅̅̅ ̅
̅ ̅̅̅̅ 44
̅ ̅ ̅ ̅̅̅̅ ̅
̅̅̅̅̅̅
(2 ) × (3∙2∙3 – 2 – 2 ) 40 ̅ ̅̅̅̅ (2 ) × (3∙2 – 2) 40
̅̅̅̅̅̅̅̅ 3
̅̅ ̅ ̅̅̅̅ ̅
̅̅̅̅̅̅
(3∙2 – 2)
̅ ̅ ̅̅̅̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅
̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅
̅̅̅̅ ̅
̅ ̅̅̅̅
̅̅̅̅̅̅
̅̅̅̅̅̅ ̅
̅̅̅̅̅̅
̅̅ ̅
̅̅̅̅̅̅̅̅
̅̅̅̅ ̅
̅ ̅̅̅̅
̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅
̅̅̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅̅̅̅ ̅
̅̅̅̅̅̅̅̅
2
̅ ̅̅ ̅̅̅̅̅̅
̅ ̅̅ ̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅ ̅ ̅ ̅̅̅̅ (2 – 1 – 1) ×
̅̅̅̅̅̅
̅̅ ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅ ̅̅̅̅̅̅ ̅
2 2 2 2 3
̅̅̅̅̅̅
̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ 44
̅ ̅ ̅ (2 ) × (2∙3 – 2 – 2 ) 40 (2) × (3∙2 – 2 – 2) 40 ̅ ̅̅̅̅̅̅ 3
̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅̅ ̅̅̅̅ ̅
̅ ̅ (3∙2 – 2)
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅
̅ ̅̅ ̅̅̅̅ ̅
̅̅̅̅̅̅̅̅
̅̅ ̅
̅̅̅̅̅̅̅̅
̅̅ ̅ ̅̅̅̅̅̅ ̅
̅ ̅̅̅̅̅̅
̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅
̅ ̅ ̅ ̅̅̅̅ ̅
̅̅̅̅ ̅
̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅
̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅ ̅̅̅̅ ̅̅̅̅
̅ ̅ ̅̅̅̅
̅̅ ̅
̅̅̅̅̅̅
̅ ̅̅
̅ ̅̅ ̅̅̅̅̅̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅ ̅
̅̅̅̅̅̅
̅̅ ̅
̅̅̅̅̅̅̅̅
(2 – 1) ×
̅ ̅ ̅
̅ ̅ ̅ ̅ ̅ ̅
̅̅̅̅̅̅̅̅
3 2 2 ̅̅̅̅̅̅̅̅ 2 ̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅ ̅̅̅̅̅̅
̅̅̅̅̅̅
44
̅̅̅̅̅̅̅̅̅̅
(2 ) × (3 – 2 ) 40 (2 ) × (2∙3∙2 – 2) 40
4
̅ ̅
̅̅̅̅̅̅̅̅
̅̅̅̅̅̅
̅ ̅̅ ̅ ̅ ̅ ̅ ̅ ̅̅̅̅
̅ ̅̅ ̅ ̅ ̅ ̅̅̅̅̅̅ ̅ (3∙2 – 2 - 2)
̅ ̅̅̅̅ ̅
̅ ̅ ̅
̅̅̅̅̅̅̅̅
̅̅ ̅
̅̅ ̅ ̅̅̅̅̅̅
̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅
̅ ̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅̅̅̅
̅ ̅
̅ ̅ ̅ ̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅ ̅̅
̅̅̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅
̅ ̅̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅
̅̅ ̅ ̅ ̅̅̅̅ ̅
2
̅̅ ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅̅̅
̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅ ̅ (2 – 1 – 1) ×
̅ ̅ ̅
̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅ ̅
2 2 2 2 2
̅ ̅̅
̅ ̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅̅ ̅ ̅̅̅̅
̅ ̅̅ 44
̅ ̅̅̅̅
(2 ) × (3 ∙2 – 2 – 2 ) 40 (2 ) × (2∙3∙2 – 2) 40 ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅ ̅ 2
̅̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅̅̅̅ ̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅̅̅̅ ̅
̅̅ ̅
̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅ (2∙3 ∙2 – 2)
̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅̅ ̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅̅̅̅ ̅̅̅̅
̅ ̅
̅ ̅ ̅
̅ ̅̅
̅̅̅̅̅̅ ̅
̅ ̅ ̅̅̅̅
̅ ̅̅ ̅̅̅̅ ̅ ̅
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅̅ ̅
̅̅ ̅
̅̅ ̅ ̅ ̅̅̅̅
̅ ̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅ 2
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅
̅ ̅̅̅̅
̅ ̅ ̅ ̅ ̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅ (2 – 1 – 1) ×
̅̅ ̅ ̅ ̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
2 2 2 2 ̅̅̅̅ ̅ ̅
̅ ̅̅ ̅̅̅̅̅̅ ̅
̅ ̅̅
̅̅ ̅ ̅ ̅̅̅̅ 44
̅̅ ̅
̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅̅̅̅ ̅
̅ ̅ ̅ (2 ) × (3∙2∙3 – 2 – 2 ) 40 ̅ ̅̅̅̅̅̅ (2) × (2∙3∙2 – 2 – 2) 40 ̅̅̅̅ ̅̅̅̅
̅ ̅ ̅ 2
̅ ̅ ̅
̅̅̅̅ ̅
̅ ̅ ̅
̅ ̅̅̅̅
̅̅̅̅ ̅
̅ ̅̅̅̅̅̅ (2∙3∙2 – 2)
̅ ̅ ̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅̅̅̅
̅̅̅̅ ̅̅̅̅̅
̅ ̅
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅ ̅̅ ̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅
̅̅̅̅ ̅
̅ ̅̅̅̅
̅ ̅̅
̅̅̅̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅ ̅
̅̅ ̅ ̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅
̅̅ ̅ ̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅ ̅
̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅̅
̅ ̅
̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅̅̅̅ ̅ 2
̅ ̅̅ ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅ ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅ ̅̅̅̅ ̅
̅ ̅̅ ̅̅̅̅̅̅ ̅ (2 – 1 – 1) ×
̅ ̅̅̅̅̅̅
̅̅̅̅ ̅ ̅
̅ ̅̅̅̅
̅̅ ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅
̅̅ ̅ 2 2 2 2 ̅ ̅̅̅̅ 2 2
̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅ 44
̅ ̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅ ̅ (2 ) × (2∙3 – 2 – 2 ) 40 (2 ) × (2 ∙3 – 2) 40
̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅̅
̅̅̅̅̅̅̅̅̅̅ 2
̅ ̅
̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅ ̅̅̅̅̅̅̅̅
̅̅̅̅̅̅ ̅
̅ ̅̅ (2∙3∙2 – 2)
̅ ̅̅ ̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅ ̅
̅̅ ̅ ̅̅̅̅ ̅ ̅
̅ ̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅ ̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅ ̅̅
̅ ̅ ̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅ ̅
̅̅̅̅ ̅̅̅̅̅
̅̅̅̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅̅̅̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅̅ ̅
̅ ̅̅ ̅ ̅ ̅̅̅̅
̅ ̅̅
̅̅̅̅ ̅ ̅ ̅̅̅̅ ̅
̅̅ ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅ ̅
̅̅̅̅̅̅ ̅̅
̅̅ ̅ ̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅̅̅ ̅
(2 –1) ×
̅ ̅ ̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅
̅ ̅ ̅ 3 2 2 2 ̅ ̅̅̅̅̅̅̅̅
̅̅̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅̅ 44
̅ ̅ ̅ ̅ ̅̅̅̅
(2 ) × (3 – 2 ) 40 ̅̅̅̅ ̅̅̅̅ (2) × (2 ∙3∙2 – 2 – 2) 40
̅̅̅̅̅̅ ̅ 3
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅ ̅̅ ̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅ ̅
(2∙3∙2 – 2 – 2)
̅̅ ̅ ̅̅̅̅̅̅ ̅̅
̅̅̅̅̅̅ ̅ ̅ ̅̅̅̅ ̅
̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅ ̅̅̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅ ̅̅̅̅̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅ ̅ ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅ ̅
̅ ̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅ ̅̅̅̅ ̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅̅̅ ̅̅
̅̅ ̅ ̅ ̅̅̅̅
̅̅̅̅ ̅̅̅̅ 2
̅ ̅ ̅ ̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
̅̅̅̅̅̅̅̅ ̅
̅ ̅ ̅ ̅̅̅̅̅̅ ̅ ̅ ̅̅̅̅̅̅
̅ ̅
(2 – 1 – 1) ×
̅ ̅̅̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅ ̅ ̅ ̅̅̅̅
̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅̅
2 2 2 2 ̅ ̅̅̅̅ 3
̅̅̅̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅ ̅̅ ̅̅̅̅̅̅ ̅ 44
̅̅̅̅ ̅ ̅
(2 ) × (3 ∙2 – 2 – 2 ) 40 ̅ ̅̅̅̅ (2) × (2 ∙3 – 2 – 2) 40 ̅ ̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅ 2
̅̅ ̅ ̅̅̅̅̅̅ ̅
̅̅̅̅̅̅ ̅̅
̅̅̅̅ ̅̅̅̅ ̅ ̅̅̅̅̅̅ (2 ∙3∙2 – 2)
̅ ̅ ̅
̅̅̅̅̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅
̅ ̅ ̅̅̅̅ ̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅ ̅̅ ̅ ̅̅̅̅
̅̅̅̅ ̅ ̅
̅ ̅̅̅̅ ̅
̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅̅̅̅̅̅ ̅̅
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅
̅ ̅ ̅
̅̅̅̅̅̅̅̅ ̅
̅̅̅̅̅̅ ̅ ̅ ̅̅̅̅̅̅̅̅ 2
̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅ ̅̅ (2 – 1 – 1) ×
̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅̅ ̅ ̅̅̅̅ ̅ ̅
2 2 2 ̅̅̅̅ ̅ 2 2
̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅̅̅̅ ̅̅ 44
(2 ) × (3∙2∙3 – 2 – 2 ) 40 ̅̅̅̅̅̅ ̅ (2 ) × (3∙2 – 2) 40
2
̅ ̅ ̅
̅ ̅̅̅̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅̅̅̅̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅ (2 ∙3∙2 – 2)
̅̅̅̅ ̅ ̅̅̅̅̅̅ ̅
̅ ̅̅̅̅ ̅̅̅̅ ̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅ ̅ ̅̅̅̅̅̅
̅̅̅̅̅̅ ̅̅
̅ ̅̅̅̅
̅ ̅ ̅ ̅̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅ ̅̅̅̅̅
̅ ̅̅̅̅̅̅̅̅
̅̅̅̅̅̅̅̅ ̅
̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅
̅ ̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅
̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅
̅̅̅̅ ̅ (2 –1) ×
̅̅̅̅ ̅̅̅̅
̅ ̅̅̅̅
̅ ̅ ̅ 2 2 2 2 2
̅̅̅̅ ̅
̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅ 44
(2 ) × (2∙3 – 2 – 2 ) 40 ̅̅̅̅̅̅ (2 ) × (3∙2∙2 – 2) 40 ̅̅̅̅ ̅̅̅̅̅
3
̅̅̅̅̅̅
̅̅̅̅̅̅ ̅
̅̅̅̅ ̅ ̅
̅̅̅̅ ̅ (2 ∙3∙2 – 2 – 2)
̅ ̅̅̅̅
̅̅̅̅̅̅
̅ ̅̅̅̅
̅̅̅̅̅̅̅
̅̅̅̅ ̅ ̅̅̅̅ ̅̅̅̅
̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅
̅̅̅̅̅̅̅̅ ̅
̅̅̅̅̅̅
̅̅̅̅ ̅̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅
̅̅̅̅̅̅̅
̅̅̅̅ ̅
̅̅̅̅̅̅
̅̅̅̅̅̅ ̅
̅̅̅̅̅̅̅
̅̅̅̅̅̅ ̅̅
̅̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅ 2
̅̅̅̅ ̅
(2 – 1 – 1) ×
̅̅̅̅̅̅
̅̅̅̅ ̅ ̅̅̅̅̅̅̅̅ ̅
̅̅̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅
2 2
Table 1. Pattern counts for Case C configurations (one ̅̅̅̅̅̅̅
̅̅̅̅̅̅ ̅ 44
̅̅̅̅̅̅ (2 ) × (3∙2 – 2) 40
̅̅̅̅ ̅ 3
̅̅̅̅̅̅̅
̅̅̅̅ ̅
̅ ̅ ̅
̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅
̅ ̅ ̅ (2 ∙3 – 2)
̅̅̅̅̅̅̅
̅̅̅̅̅̅̅̅ ̅
pair of consecutive non-chord tones, two single non- ̅̅̅̅ ̅̅̅̅̅
̅̅̅̅̅̅
̅̅̅̅̅̅̅
̅ ̅ ̅
̅̅̅̅̅̅
̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅
̅ ̅ ̅
̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅
chord tones, and four chord tones)
̅̅̅̅̅̅
̅̅̅̅̅̅̅
̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅
̅ ̅ ̅
̅ ̅ ̅
̅̅̅̅̅̅ ̅ (2– 1) ×
̅̅̅̅̅̅̅
̅̅̅̅̅̅ 3
̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅
̅̅̅̅̅̅ ̅̅ 44
̅̅̅̅̅̅̅
̅ ̅̅̅̅ (2) × (3∙2 – 2 – 2) 40
2
̅ ̅ ̅
̅̅̅̅̅̅̅̅ ̅
̅̅̅̅̅̅̅ (2 ∙3∙2 – 2 – 2)
̅ ̅ ̅
̅ ̅̅̅̅
̅̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅
̅̅̅̅̅̅ ̅̅
̅ ̅ ̅
̅̅̅̅̅̅̅
̅ ̅ ̅
̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅
̅ ̅̅̅̅
̅ ̅̅̅̅
̅ ̅ ̅
̅̅̅̅̅̅̅
̅̅̅̅̅̅̅
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅ (2– 1) ×
̅ ̅ ̅
̅ ̅̅̅̅̅̅
2
̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅
Case A) 44
̅ ̅ ̅ (2 ) × (2∙3∙2 – 2) 40 4
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅
(2 ∙3 – 2 – 2)
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅ ̅ ̅
̅ ̅̅̅̅̅̅
̅ ̅ ̅
̅̅ ̅
̅ ̅̅̅̅
(Number of ̅̅ ̅
̅ ̅̅̅̅̅̅
Total ̅ ̅̅̅̅ 2 Table 4. Pattern counts for Case D configurations (two
̅ ̅̅̅̅̅̅
̅̅ ̅
̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
(2 ) × (2∙3∙2 – 2) 40
̅̅ ̅
Non-Chord Tone
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
Number
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅ single non-chord tones and six chord tones)
̅̅ ̅
̅̅ ̅
Configuration Combinations) ×
̅ ̅̅̅̅̅̅
̅ ̅̅̅̅
̅̅ ̅
̅ ̅̅̅̅
̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
of ̅̅ ̅̅̅̅
̅̅ ̅
(Number of Chord ̅ ̅̅̅̅̅̅
̅̅ ̅
̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅ ̅
2
̅̅ ̅̅̅̅
Patterns
̅̅ ̅
(2) × (2∙3∙2 – 2 – 2) 40
̅ ̅̅̅̅̅̅
Tone Combinations)
̅̅ ̅
̅ ̅̅̅̅̅̅
̅̅ ̅̅̅̅
̅̅ ̅̅̅̅
̅̅ ̅
̅ ̅̅̅̅̅̅
̅̅ ̅̅̅̅
̅ ̅̅̅̅̅̅
̅̅̅ ̅
̅̅ ̅
̅̅̅ ̅
̅̅ ̅̅̅̅
̅̅ ̅
̅̅ ̅̅̅̅
̅̅̅ ̅
̅̅ ̅
̅̅ ̅̅̅̅
4 ̅̅̅ ̅
̅̅̅̅ ̅̅ ̅̅̅̅
(2) × (2 – 1) 30 2 2
̅̅ ̅
̅̅ ̅̅̅̅
̅̅̅̅ ̅̅ ̅ (2 ) × (2 ∙3 – 2) 40
̅̅̅ ̅
̅̅̅̅
̅̅̅ ̅
̅̅̅̅ ̅̅ ̅̅̅̅
̅̅ ̅
̅̅̅ ̅
̅̅ ̅
̅̅ ̅̅̅̅
̅̅̅̅ ̅̅̅ ̅
̅̅̅̅
̅̅ ̅̅̅̅
̅̅̅ ̅
̅̅ ̅̅̅̅
̅̅̅ ̅
̅ ̅̅̅ ̅̅̅ ̅
̅̅ ̅̅̅̅
4
̅̅̅ ̅
̅ ̅̅̅ (2) × (2 – 1) 30 ̅̅ ̅̅̅̅ 2
̅ ̅̅̅
̅̅̅ ̅ (2) × (2 ∙3∙2 – 2 – 2) 40
̅̅ ̅̅̅̅
̅ ̅̅̅ ̅̅ ̅̅̅̅
̅̅̅ ̅
̅ ̅̅̅
̅ ̅̅̅ ̅̅̅ ̅
̅̅̅ ̅
̅̅ ̅̅ ̅̅̅ ̅
̅̅ ̅̅ 4 ̅̅̅ ̅
̅̅ ̅̅
(2) × (2 – 1) 30 ̅̅̅ ̅ 3
̅̅̅ ̅
̅̅ ̅̅ (2) × (2 ∙3 – 2 – 2) 40
̅̅ ̅̅
̅̅ ̅̅
̅̅̅ ̅
̅̅̅ ̅
̅̅̅ ̅
4
̅̅̅ ̅ (2) × (2 – 1) 30
Table 3. Pattern counts for Case B configurations (one
̅̅̅ ̅
̅̅̅ ̅
pair of consecutive non-chord tones, one single non-
̅̅̅̅
̅̅̅̅
̅̅̅̅ chord tone, and five chord tones)
̅̅̅̅ 4
(2) × (2 – 1) 30
̅̅̅̅
̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
5
̅̅̅̅̅
̅̅̅̅̅ (1) × (2 – 1 – 1) 30
Table 2. Pattern counts for Case A configurations (one
pair of consecutive non-chord tones and six chord
tones)
Journal of Undergraduate Research and Scholarly Excellence – Volume VI 40
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