Formal Logic and the Structure of Music

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Divinevictory Amayo, Howard Community College, 2019, UMBC

Mentored by Dr. Mike Long, Howard Community College

 

Abstract

 

Formal logic provides a framework for formal mathematical proof in higher-level mathematics.  On the other side is music, one of the fine arts, where vocal and instrumental sounds are combined in ways to entertain audiences while challenging those performing.  For some time, music and mathematics have been connected, particularly through analysis of sound waves, which requires trigonometry.  It turns out that music and formal logic may be related as well.  This led to the question: When are musical passages equivalent in the same way that mathematical statements are logically equivalent?  Through musical analysis and use of the basic formal logic concepts of inverse, converse, and contrapositive, we explore when the equivalences exist.  It turns out that in a few cases the equivalences do exist and so passages of music are equivalent in the way that logical statements are equivalent.  But there are many instances where musical passages that might seem to be logically equivalent are not and we explore and mathematically explain why they are not.

Introduction 

Mathematics” is a word that strikes fear in the hearts of many students.  The guitar is an instrument with a rich history and an even richer sound that is loved by millions of people around the world. What do these two things have in common? In the spring of 2018, I didn’t know and I didn’t care. I was a student trying to get all of the dreadful mathematics required for his major out of the way, and a newly initiated guitarist. So, when I opened my mouth to tell my discrete mathematics professor (who is my mentor on this project), about my dislike of his field, I did not foresee that he would rope me into a challenge that would try to change my mind about mathematics. The challenge took the form of a research project that would delve into formal logic topics that were touched upon in my discrete mathematics class and combine those topics with my growing knowledge and appreciation for the music world through guitar.

Background

There are some common structures in formal logic that must be introduced to begin a discussion of this research.  The first is the conditional statement “if / then” or “if p then q” where each of p and q represent statements.  Statements in formal logic are identified as sentences that are either true or false.  Symbolically, “if p then q” is represented as  p\rightarrow q .  The second structure to consider simply reverses the order of these statements, which we would write as  q\rightarrow p .  This is known as the “converse.”  The third structure to consider involves adding negations to both statements, which we would write as   \sim p\rightarrow\  \sim q .  This is known as the “inverse,” where the squiggly line in front of the p and the q represents the negation of the statement or, more informally, “not.”  The fourth structure reverses the order of the statements and also adds negations, which we would write as   \sim q\rightarrow\  \sim p .  This is known as the “contrapositive.”  The original conditional and the contrapositive have an important characteristic in formal logic and that is that they are logically equivalent.  Similarly, the inverse and the converse are logically equivalent.  What exactly does it mean to be logically equivalent?

  • The two conditional (if / then) statements have the same truth tables, or tables used in mathematical logic to determine the validity of statements
  • The two conditional statements (if / then) can be proved from each other using techniques of formal mathematical proof

Table 1 compares the conditional statements.

Table 1:  Comparison of conditional statements
Conditional Statement Name: Symbols: Example: Logically Equivalent To:
Original Statement  p\rightarrow q If the light is red, then I will stop. Contrapositive
Converse  q\rightarrow p If I stop, then the light is red. Inverse
Inverse   \sim p\rightarrow\  \sim q If the light is not red, then I will not stop. Converse
Contrapositive   \sim q\rightarrow\  \sim p If I do not stop, then the light is not red. Original Statement

Something similar exists in the structure of music.  These structures were seen in music composed since the 1500s in Europe, but came to prominence with serial composition in the late 1800s and early 1900s, again in Europe [1].

 

We first consider a passage or line of music as our baseline structure, which we might view as similar to the original conditional statement in formal logic (Figure 1 and Table 2).  The second musical structure to consider is the “retrograde.”  In music, the retrograde of the original line or passage is simply the original line or passage played in reverse starting with the last note or rest of the original passage and ending with the first note or rest of the original passage. This would be similar to the converse in formal logic where the order is reversed. The third musical structure to consider is the “inversion.”  In music, the inversion of the original line or passage is created by literally flipping the notes upside down on the musical staff, changing the contour of the music.  In this structure, the first note is the same as that of the original line or passage of music (some composers have modified this definition a bit and used a different starting note, but in this research the starting note will not be changed).  This structure would be similar to the inverse in formal logic, which is the negation or opposite of the original statements.  The final structure to consider is the “retrograde inversion.” In the retrograde inversion, the original line or passage is flipped on the musical staff, like the inversion, and then the order reversed, like the retrograde (the retrograde inversion is most often the retrograde of the inversion, but some composers convert in the opposite order, which has different results).  In this structure, the first note is the same as the last note of the inversion while the last note is the same as the first note of the original passage.  This structure would be similar to the contrapositive in formal logic, which is the negation or opposite of the original statements with the order changed.

Hypothesis 

We hypothesized that the original musical statement and the retrograde inversion would be the same in rhythm and in tonality, an identical match in the same way that the original statement and the contrapositive in formal logic are logically equivalent.  This idea emerged because the original line or passage of music and the retrograde inversion were seemingly equivalent in a similar way that the original statement and contrapositive in formal logic were equivalent. 

Initial Results

We have already shown the parallels between the original statement and the original line or passage of music and the retrograde inversion and the contrapositive.  However, when examining our first pieces of music, we quickly realized that our hypothesis was not correct.  The rhythm and tonality of the original line or passage of music and the retrograde inversion were different.  As a result, we changed the idea of this research work to come up with a working definition of “musical equivalence” to parallel “logical equivalence.” 

Identifying Musical Equivalence 

A simple passage of music with three quarter notes and a quarter rest was created (Figure 1) to better describe the musical structures:  the original musical line or passage, followed by the retrograde, the inversion, and finally the retrograde inversion.  The circles with vertical lines attached are the notes that tell the musician which pitch to sound and for how long.  The French bracket type symbols are the rests that tell the musician not to play for a period of time.  This simple passage contains four measures.  The vertical lines indicate where each new measure begins.

 

Figure 1: Musical passage consisting of four measures

 

Table 2 breaks the simple passage of music up into four parts, by measure.  The table also adds comparisons to formal logic.  This table also makes it easier to understand why the first hypothesis was made.

Table 2:  Comparison of conditional statements and music structures. Note that symbols are added in the inversion and retrograde inversions.  This is to keep the steps between the notes the same. The concept of “step” is discussed in the text.
Music    
Name Original Line or Musical Statement Retrograde Inversion Retrograde Inversion
Parallel in formal logic Conditional Statement Converse Inverse Contrapositive
Symbols  p\rightarrow q  q\rightarrow p   \sim p\rightarrow\  \sim q   \sim q\rightarrow\  \sim p
Equivalence Contrapositive Inverse Converse Conditional Statement

 

To understand the discussion of “musical equivalence, it is useful to understand how “steps” are defined in music. The easiest way to understand steps in music is to look at the piano keyboard.  Two white keys that have no black keys in between are separated by a half step.  Two white keys that have a black key in between are separated by a whole step.  To break this down more, movement from a white key to an adjacent black key is a half step and then from the black key to the next white key is also be a half step.  Similarly, two black keys that have a white key in between are separated by a whole step.  Breaking this down, movement from a black key to the next white key is a half step and then from the white key to the next black key is also a half step.

 

In the original line or passage of music, notice how the notes go up a step each time on the staff and the notes are then followed by the quarter rest.  In the retrograde, where the order is changed, the quarter rest comes first and then is followed by the three notes in the opposite order.  Also, since each of the notes in the original statement goes up a step each time, in the retrograde, each of the notes goes down a step each time.  In the inversion, the quarter rest follows the three quarter notes, which is identical to the original statement.  However, each of the notes goes down a step each time, which is the opposite of the original line or passage of music where the three notes each go up by a step.  In the retrograde inversion, the order of the inversion is changed and so the quarter rest is followed by the three quarter notes.  However, the three notes each go up by one step as in the original statement, but the pitches are different.

 

As mentioned, the original idea for this research was that since the original statement in formal logic and the contrapositive of the original statement were logically equivalent, the original line or passage of music and the retrograde inversion would be the same passage of music in some way.  However, this simple passage shows this is not the case.  It was quite obvious that a truth table could not be used to show that the original line or passage of music and the retrograde inversion were equivalent in the same way, similar to the relationship shared by the original statement and the contrapositive in formal logic.  Also, a truth table could not be used to show that the retrograde and inversion were equivalent in the same way, similar to the converse and inverse.  Therefore, a new direction for the research was taken to identify another equivalence, a “musical equivalence,” that can be explored and defined, which does not require a truth table or formal mathematical proof to show the equivalence.

Identifying “Musical Equivalence”

There are some interesting characteristics that show up in the simple piece of music above that perhaps lead to a definition of “musical equivalence.”  In this simple example, the original musical line or passage and the retrograde inversion are not exactly the same.  Other than the first note of the original passage and the last note of the retrograde inversion, the notes are not the same.  The rhythmic structure of the original line or passage of music is not the same as the retrograde inversion since in the original musical passage the three quarter notes were followed by the quarter rest, while the retrograde inversion has a quarter rest followed by three quarter notes.  The same can be said for the retrograde and the inversion and their “musical equivalence.” They are not the same.  Other than the first note of the inversion and the last note of the retrograde, the notes are not the same.  Also, the rhythmic structure of the inversion is not the same as the retrograde since in the inversion, the three quarter notes were followed by the quarter rest, while the retrograde has a quarter rest followed by three quarter notes.  From this, it appears there is only one characteristic that might contribute to the notion of “musical equivalence,” and that is a guarantee they have one note in common, which may not be in the same position.  In fact the same notes will be the first note in one structure and the last in the second structure when the two structures are “musically equivalent,” and that only works in one direction.  For example, the first note of the original line or passage of music and the last note of the retrograde inversion are the same, but vice versa cannot be added here since this idea only works in one direction.

 

A deeper dive is needed to develop or dispel the idea of  “musical equivalence,” and that dive is in the actual musical structure or patterns in the pitches of the notes.  Let’s consider the following scenarios from a more complex original line of music.

  • The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the retrograde will go up a half step, and then down by a whole step, and then down another whole step:

Original Musical Line                     Retrograde

  • The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the inversion will drop down by a whole step, and then down another whole step, and then back up another half step:

 

      Original Musical Line                         Inversion

 

  • The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the retrograde inversion will go down a half step, then up a whole step, and then up another whole step:

 

         Original Musical Line                     Retrograde Inversion

 

The original music line here is a little more complex than the original simple passage that was created, as the movement goes up and down and there are whole and half steps.  With the original passage, it was thought that patterns in the pitches of the notes would be critical in defining the notion of “musical equivalence.” However, that was not the case.  We started by comparing the original musical line or passage and the retrograde inversion since they are similar to the original statement and the contrapositive in formal logic, which are logically equivalent.  In the original simple musical passage, the original musical line or statement and the retrograde inversion both had movement up by two steps, but the notes or pitches were not the same.  Here, the idea of the movement between the notes was still promising.  However, in the later more complex original music line, the original musical line or passage and the retrograde inversion had very different movement.  In the original musical line or passage, the movement was two whole steps up and a half step down, while in the retrograde inversion, the movement of the notes was a half step down followed by two whole steps up.  This example alone dispels the idea that the patterns in the pitches of the notes would be critical in defining “musical equivalence.”

Summary and Conclusions

While there were ideas about a way to define “musical equivalence,” each time counterexamples proved that the ideas were not true.  The only result was that the first note of one of the structures is the same as the last note of the other structure and that this only happens in one direction.  Further research should be done toward defining characteristics for “musical equivalence.”  Another opportunity for research would be to develop a structure for determining “musical equivalence,” similar to the truth table in formal logic, which cannot be used here.  Additionally, research might examine when the notes or pitches of the “musically equivalent” structures might be the same or when the rhythmic structures are the same.

 

For me personally, I have realized that mathematics is not a word to be feared, but is a discipline that should be appreciated more.  I have learned that mathematics is all around in places I never thought.  I conclude also that math is ……

 

Contacts: divinevictoryamaya@howardcc.edu, mlong@howardcc.edu

 


References

[1] Smith, R. B.  (1966).  Serial Composition.  Oxford University Press.  1966.

 

 

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