Fundamentals of Music
Music has been around perhaps as long as man’s consciousness. However, it wasn't until Pythagoras (570–495) quantified the rules of music that the system enjoyed in the Western world was developed. Building on the foundation laid by Pythagoras, we have uncovered not only its intrinsic beauty but the mathematics that lies beneath it.
The Development of Scales
There are many legends told about the Greek philosopher, mathematician and polymath Pythagoras best known for his Pythagorean Theorem in Mathematics. Although we attribute several things to him, many scholars believe his discoveries are the product of the Pythagorean School or perhaps were known in other parts of the world. He studied in Egypt for 22 years and therefore some of the revelations attributed to him may have already been discovered. For the sake of argument and simplicity, we will tell Pythagoras’ story, be it a myth or fact, because it does shed insight on the relationship of notes within a scale.
As legend has it, Pythagoras heard hammers pounding and noticed that some of the sounds were harmonious and others were not. Being a man of science and philosophy, Pythagoras set out to find the source of this phenomenon. It had long been known that some sounds produced by hammering could be consonant. It was also well known that this phenomenon could be produced by strings of various lengths and thickness and with wind instruments. What Pythagoras did was to quantify the relationships that caused such phenomena.
Instead of using hammers, Pythagoras started by plucking a string of a given length, tension and thickness; then he listened to its sound as he continued to pluck. Of course, it was harmonious with itself. Next, he cut the string in half and plucked it along with the first string, the two were also harmonious. He continued, cutting the string in thirds, and the sound was also harmonious with the other strings. Next, he cut the string into fourths and not to his surprise; they all were in harmony with each other. What Pythagoras discovered were basic notes or vibrations of frequencies that were most harmonious with each other. Moreover, they were associated with each other by whole numbers.
The distance between notes is referred to as intervals, and for that matter, when we speak to each other we speak in intervals. Sounds that share the same interval relationship are harmonious to each other and that is why some words and phrases rhyme. The plucking of the first string created what is called unison and it is also referred to as the tonic, the second string the octave, the third string the dominant.
By cutting strings into these various whole number proportions, what Pythagoras did was to increase the vibrations of the strings in corresponding proportions. That is to say, the string with half the length vibrates twice as fast, a third the length three times as fast and so forth.
The formula for strings could be expressed (given a constant length) as: T =MV^2 where T is the tension, M the mass of the string and V the velocity of the sound traveling through the air.
Although he had his eureka moment, Pythagoras was by no means finished. After all, you could make some music with these basic notes, and to be honest, people around the world had been making music with them for centuries if not millenniums before Pythagoras. However, making music wasn’t the goal, identifying musical relationships was. In music, it is ultimately the relationships of notes (frequencies) to each other that counts and what Pythagoras had discovered were those notes that have the strongest relationship with each other.
What Pythagoras did next is perhaps the most ingenious part. He made a string with a thickness that gave the sound of the previous string that was divided into one third. He had noticed that the string that was divided in half was essentially the same sound only higher in pitch. However, the string that was divided into one-third was unique. This is perhaps the most essential aspect of the dominant tone; it is the simplest way to generate new tones.
He continued by dividing this new string into one third. He plucked it and a new sound was heard that was also harmonious with the other sounds. This process continued until he found a relationship that was not harmonious, and he stopped. What he had was a scale, a collection of eight notes all harmonious with each other. This became the C major scale which corresponds to the white keys on the piano. In effect, he divided an octave by using harmonious fifths as the yardstick of music.
The following is the scale: C D E F G A B C
The ratios are as follows:
C 1:1, D 9:8, E 5:4, F 4:3, G 3:2, A 5:3, B 15:8, C 2:1
It is also known as the diatonic scale. The last C is an octave above the first one. Now it is important to note, that plucking a string with one-half the length of the original string was also significant because it defined the range of notes that could be contained without repeating a fundamental tone.
With this formula in place, additional octaves were later developed with the keys on the piano essentially representing the range of notes of most instruments from about 30 Hz to 4100 Hz, about eight octaves. Scientists would refer to this as bandwidth. However, notes have additional harmonics associated with them, and since our hearing range extends to 20,000 Hz, we can hear some of these harmonics.
The instruments that we are all familiar with today have a specific range and instruments are distinguished from each other by their range and their harmonic content. By harmonic content, I mean, although two instruments my play the exact note, they vary in how that note propagates through the air, thus a Middle C, 261 Hz, on the piano and one on the guitar will vibrate at the same fundamental frequency, but each will have different combinations(harmonics) of other frequencies that gives it its timbre. This is why one instrument sounds different from another.
Experimentation with the eight notes in the diatonic scale gave birth to modes each assigned a Greek name. Instead of starting on the tonic, it started on another note within the key. For instance, C Dorian would start on D and end on D, etc. The modes for the key of C are as follows. Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian as shown below. The W designates a whole step, and the H a half step.
During the Middle Ages, Church music was heavily dependent on modes. However, in the late Middle Ages and early Renaissance, musicians, composers, and instrument makers discovered that although interesting, Pythagoras did not go far enough in his exploration of scales. Even though the note F# that was produced by Pythagoras’ successive adding of perfect fifths was not harmonious with the unison or tonic, it was harmonious with its fifth note G. As a result, of their experimentation, another scale based on G was born which substituted F# for F.
G A B C D E F# G
By repeating the process of adding fifths, other notes and scales were developed leading to our current system known as the chromatic scale. The additional notes and scales were designated as sharps, # and flats, b. A sharp raises the tone a half step and a flat lowers a tone a half step, thus Db and C# are the same note. The chromatic scale is as follows:
C D♭/C♯ D E♭/D♯ E F G♭/F♯ G A♭/G♯ A B♭/A♯ B
Instead of having one diatonic scale, there were now twelve. This was the second great epiphany in music theory although, like Pythagoras, no one knew the source of the phenomena.
The additional five notes correspond to the black keys on the piano. It is most ironic that the fifth turned out to be the perfect slice of the musical pie because, unlike the octave, it produced a harmonic frequency that was not generated by C. The chromatic scale is divided into semi-tones and the relationship to each note is equal to the twelfth root of two or approximately 1.06. Hence C# is 1.06 times C.
The following is the chromatic scale as illustrated on the piano:
Note that the black keys on the piano, the sharps and the flats, form a pentatonic scale (scale of five notes). Therefore, it could be said that the chromatic scale is a combination of two scales, the diatonic (white keys) and the pentatonic (black keys). Only eight notes are diatonic to a key. For C, it would be the white keys on the piano.
The Pentatonic scale is any scale consisting of five notes an appears in several cultures around the world. One of the most popular pentatonic scales is the Blues Scale consisting of I, IIIb, IV, V, and VIIb notes. This scale is a powerful improvisation tool and is often used in Rock, R& B and Jazz.
Chords
In addition to modes and key signatures, chords, or the use of several notes played at the same time, were developed and could be played together on polyphonic instruments such as the piano or the guitar. These chords were generally built on triads and were diatonic to a key. For example, take the C diatonic scale
C D E F G A B C
A C Major triad would consist of notes: 1, 3, 5 or CEG. Since these three notes are also diatonic to the G and F keys, the chord is also diatonic to all three keys.
Each chord could be extended to include 7th, 9th, etc. Jazz musicians normally play chords in their higher extensions (at least a 7th chord). This allows them to substitute chords while maintaining the same or similar tonality. The Jazz saxophonist John Coltrane developed Coltrane Changes (substitutions) which, in its simple form, a chord a third above or a third below could be substituted for the tonic, e.g., Em7 (iii)or Am7 (vi)for Cmaj7 (I) and keep the tonality because three out of the four notes are common.
Chords that are diatonic to the keys are normally given the Roman Numeral designation for example: In the major scale, the I chord is called the tonic, the ii chord is minor and is called the subtonic, the iii chord is also minor and is called the mediant, the IV is major and is called the subdominant, the V chord is also Major and is called the dominant the vi chord is minor and is called the submediant and the vii chord is diminished and is called the subtonic. Below are the chords for C Major and C Minor keys. Although keys may share common Minor and Major chords, the V7 is unique to a key and in this regard defines the key.
A Chord Progression is a series of chords played in a sequence which usually establishes the harmonic content of a musical piece. Melodies are normally played atop of a harmony which gives the music a sense of completeness. Below is a popular chord progression called a turnaround consisting of a ii, V7, I7.
By using Roman numerals, musicians can outline chord progressions to be played in any key. Furthermore, there is a relationship between the modes of a key and the chords that are diatonic to a key. For example, in a Major key, a Dorian scale could be improvised over the Dm7 (ii) chord and likewise a Mixolydian scale over G7 or the V cord and so forth.
Modulation
A Modulation is a change in key. Modulations are used to create movement. Some argue that the first rule of music is to maintain the key, and the second rule is to break the key, To this end, there are four types of modulations: Direct, Prepared, Pivot Chord and Transitional.
In Direct Modulation, the key changes from chord to chord without preparation. In Prepared Modulation, however, normally a series of cords (cadence) is played between keys. In Pivot Chord Modulation, a series of chords that are diatonic to both keys are played as part of the transition. And finally, a Transitional Modulation occurs when following a cycle of chromatic ii-V’s or a sequence of dominants, with the music winding up in another key.
The Cycle of Fifths
With the final musical structure in place, relationships between the eight notes in a diatonic scale and the twelve notes in the chromatic scale were developed with each note having a musical key associated with it. Each one of the notes in the diatonic scale is representative of partials or overtones of the key signature.
As mentioned previously, the major tones in any diatonic collection are the tonic, the octave, the dominant and the subdominant. This is also true of the keys. That is to say, that the key a fifth above (dominant) and a fifth below (subdominant) are harmonically closer to the tonic and their respective scales differ by only one note.
For example, the musical keys closest to the key of C, are the key of F (subdominant) and the key of G(dominant). These relationships are the most aesthetically pleasing to the ear, and it is as if they are ingrained into the human psychic. There are countless songs written around this basic relationship and it is common in the music of all cultures. They are as common to us as the primary color, which by the way are frequencies as well with each color assigned to a specific frequency (wavelength). The V7 chord is the second power chord in the key and tends to pull toward the I chord.
The Cycle of Fifths forms the roadmap for the harmonic structure of music. It is like a clock with note C located at twelve o’clock with its most dissonant note or scale opposite it at six o’clock. This is true of any key. The following chart illustrates the Cycle of Fifths.
The outer circle denotes the number of sharps or flats in each key, the next outer circle denotes the Major key and the inner circle represents its Relative Minor key which has the same key signature but its starting point is shifted relative to its Major key.
For instance, for the Major key of C, instead of starting on C, you would start on A which would yield the A Minor scale. Therefore, by adding five additional notes to the original scale, we progressed from one Major key and one Minor key to twelve Major keys and twelve Minor keys.
The Major and Minor keys are like night and day. Typically, the Major keys have a happier sound and the Minor keys a sadder sound. Moreover, as Nature would have it, the musical cycle was finally complete. As seen from the chart, keys and scales that are harmonically close to each other are adjacent in the clockwise and counterclockwise directions
The beauty of the Cycle of Fifths lies in its simplicity and its symmetry. It is important to note, that harmony is maintained not by the individual notes or scales, but by the relationships of notes and scales to each other. This dictates which notes or keys are valid and the migration paths (modulation) from one key to the other. Each note added to form a successive key is also a fifth apart from each other, e.g., C to G, G to D, etc. The standard ii-V7-I chord progression is so powerful because it moves counterclockwise in the Cycle of Fifths.
The subdominant of any key and the dominant are in many ways the mirror image of each other, one is a fifth above the tonic and the other is a fifth below the tonic. Moreover, the tonic is the fifth or dominant of the subdominant. In terms of vibrations or frequencies, the ratio is 3/2 in the clockwise or 4/3 in the counterclockwise direction of the Cycle of Fifths.
It is also important to note that the overall ratio between the diatonic scale and the chromatic scale in terms of the number of notes in each scale is 12/8 or 3/2. Thus, the ratio of 3/2 seems to be integral to the rules of harmony, and as noted previously was the first ratio that Pythagoras observed as truly being different.
The rules of harmony must adhere to the relationships delineated by the Cycle of Fifths and from such relationships, a musical hierarchy is developed — from notes to chords, to chord progressions to songs and scores. As the history of music attests, the number of musical permutations from these basic building blocks seems to be inexhaustible.
Music and Mathematics
Gottfried Leibniz, the co-inventor of Calculus once said: Music is the pleasure the human mind experiences from counting without being aware that it is counting. The relationship between music and mathematics has long been noted. Some have claimed that listening to music helps with cognition, a phenomenon known as the Mozart Effect. Notwithstanding, there appears to be a relationship between the cognitive skills required for music and mathematics. Physicist Dr. Stephon Alexander outlines such relationships in his book entitled: The Jazz of Physics: The Secret Link Between Music and the Structure of the Universe.
In addition to cognition, music is directly related to mathematics as illustrated by the following series. The Fibonacci series is a series of numbers starting with consecutive ones and adding them together to get 2, then add 2 to 1 (the previous number) to get 3, then 2 to 3 to get 5, then 3 to 5 to get 8 and so on. The first 25 numbers of the Fibonacci series are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4184, 6765, with each new term being the sum of the previous two terms. From the series, Fibonacci boxes can be drawn, e.g., 1x2, 2x3, 3x 5, 5x8, 8x13.
A careful examination of the piano keyboard reveals a relationship with the Fibonacci series. First, a black key is always between two white keys, ratio 1 to 2 Fibonacci ratios. The first group has two black keys and three white keys, 2 to 3, which are Fibonacci ratios, the second group has 3 black keys and 5, 3x5, white keys, also Fibonacci ratios, in the third group, there are 5 black keys and 8 white keys 5x8 which are Fibonacci ratios and finally in the total set; there are 8 white keys and 13, total keys 8x13, which are also Fibonacci ratios.
In addition to the Fibonacci series, the Fourier series explains why instruments sound differently when playing the same fundamental note. The theory states that any periodic waveform could be deconstructed into a series of sinewaves. In this regard, each instrument has its signature, differing in amplitude and relevant harmonic frequencies (overtones).
Within the vibration of the bass guitar string, there are multiple vibrations at different frequencies as shown below. Pythagoras accounted for the initial or fundamental vibration of the string, but he did not account for the harmonics that were produced by the fundamental frequency. The harmonic series is defined as the sum of: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 … and has been proven by several mathematical proofs to continue to infinity.
These frequencies vary with amplitude in accordance with the following formula.
Some sounds have odd harmonics, some have even and some have both (that is why we have sine and cosine terms in the equation). A true sinewave, however, does not have any harmonics. What the Fourier series does is explain how the ear processes sound and by doing so connects all sounds, including music to the mathematics that produced it. It breaks down sounds into their sinusoidal components. This discovery gave birth to synthesized instruments that could emulate the sounds of various instruments or create new sounds hitherto unheard by varying the harmonic content of the fundamental frequency.
Throughout nature, we repeatedly find that these two series. Fibonacci named after the Italian mathematician, Leonardo Pisano Fibonacci (1170–1240 or 1250), who brought Arabic Numbers (0–9)to Europe and the French scientist Joseph Fourier (1768–1830), tend to define the rhythm of Nature’s song. Both men developed their series without music in mind. In this regard, music is telling us more than meets the ear. It is singing the song of Mathematics that our brains readily comprehend and makes us want to vibrate or dance with it.
Pythagoras proposed that there is a rhythm or harmony to the universe which he denoted as the harmony of the spheres. Perhaps, in this regard, Pythagoras was right. We now know that baryonic matter (the stuff that stars and life) is composed of, can be deconstructed into a series of vibrating waves or strings at the subatomic level. Maybe we do live in a clockwork universe where everything is composed of notes played in a symphony conducted by the mathematical principles of the cosmos.
