Musical acoustics

Musical acoustics or music acoustics is the branch of acoustics concerned with researching and describing the physics of music — how sounds employed as music work. Examples of areas of study are the function of musical instruments, the human voice (the physics of speech and singing), computer analysis of melody, and determination of stylistic parameters in compositions and performances.

Methods and fields of study

 * Frequency analysis
 * Computer analysis of musical structure
 * Synthesis of musical sounds
 * Music cognition, based on physics (also known as psychoacoustics)

Sound waves
Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called "sound". Most sound that people recognize as "musical" is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease in a regular fashion, and is heard as a very "pure" tone. Pure tones can be produced by tuning forks or whistling. The rate at which the air pressure varies governs the frequency of the tone, which is also measured in oscillations per second, or hertz. Frequency is a primary determinate of the perceived pitch.



Whenever two different pitches are played at the same time, their sound waves interact with each other — the highs and lows in the air pressure reinforce each other to produce a different sound wave. As a result, any given sound wave which is more complicated than a sine wave can, nonetheless, be modelled by many different sine waves of the appropriate frequencies and amplitudes (a frequency spectrum). In humans the hearing apparatus (composed of the ears and brain) can isolate these tones and hear them distinctly. When two or more tones are played at once, a single variation of air pressure at the ear "contains" the pitches of each, and the ear and/or brain isolate and decode them into distinct tones.

When the original sound sources are perfectly periodic, the note consists of several related sine waves (which mathematically add to each other) called the fundamental and the harmonics, partials, or overtones. The sounds have harmonic frequency spectra. The lowest frequency present is the fundamental, and is the frequency that the entire wave vibrates at. The overtones vibrate faster than the fundamental, but must vibrate at integer multiples of the fundamental frequency in order for the total wave to be exactly the same each cycle. Real instruments are close to periodic, but the frequencies of the overtones are slightly imperfect, so the shape of the wave changes slightly over time.

Harmonics, partials, and overtones
The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials.

Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere near a harmonic, and are just called partials or inharmonic overtones.

The fundamental frequency is considered the first harmonic and the first partial. The numbering of the partials and harmonics is then usually the same; the second partial is the second harmonic, etc. But if there are inharmonic partials, the numbering no longer coincides. Overtones are numbered as they appear above the fundamental. So strictly speaking, the first overtone is the second partial (and usually the second harmonic). As this can result in confusion, only harmonics are usually referred to by their numbers, and overtones and partials are described by their relationships to those harmonics.

Harmonics and non-linearities






When a periodic wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), the summed wave is half-wave symmetric; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (0f, 2f, 4f, 6f, ...), it will be asymmetrical; the top half will not be a mirror image of the bottom.

The opposite is also true. A system which changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics (harmonic distortion). This is called a non-linear system. If it affects the wave symmetrically, the harmonics produced will only be odd, if asymmetrically, at least one even harmonic will be produced (and probably also odd).

Harmony
If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave will still be periodic with a short period, and the combination will sound consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave will repeat thrice and the 200 Hz wave will repeat twice. Note that the total wave repeats at 100 Hz, but there is not actually a 100 Hz sinusoidal component present.

Additionally, the two notes will have many of the same partials. For instance, a note with a fundamental frequency of 200 Hz will have harmonics at

(200,) 400, 600, 800, 1000, 1200, ...

A note with fundamental frequency of 300 Hz will have harmonics at

(300,) 600, 900, 1200, 1500, &#8230;

The two notes have the harmonics 600 and 1200 in common, and more will coincide further up the series.

The combination of composite waves with short fundamental frequencies and shared or closely related partials is what causes the sensation of harmony.

When two frequencies are near to a simple fraction, but not exact, the composite wave cycles slowly enough to hear the cancellation of the waves as a steady pulsing instead of a tone. This is called beating, and is considered to be unpleasant, or dissonant.

The frequency of beating is calculated as the difference between the frequencies of the two notes. For the example above, |200 Hz - 300 Hz| = 100 Hz. As another example, a combination of 3425 Hz and 3426 Hz would beat once per second (|3425 Hz - 3426 Hz| = 1 Hz). This follows from modulation theory.

The difference between consonance and dissonance is not clearly defined, but the higher the beat frequency, the more likely the interval to be consonant. Helmholtz proposed that maximum dissonance would arise between two pure tones when the beat rate is roughly 35 Hz.

The natural scale
Human beings distinguish sounds on the basis of their frequency. Actually what really matters is the ratio between their frequencies.

The natural scale is attributed to the Greek philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies.

After fixing the frequency of the first note — the C of the scale — the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the following octave begins and the operation can be repeated.

The following table shows the ratios between the frequencies of all the notes of the scale and the fixed frequency of the first note of the scale.

Archaeological evidence
A current viewpoint among many laypersons and scholars indicates tonal scales and tonality arise from natural overtones. Most of the archaeological evidence regarding this has been found only in the last few decades, and most, if not all, of it supports many earlier claims of the universal or "natural" evolution of the scales most widely found in human music. The evidence for this now includes the recent find of the Divje Babe Neanderthal Flute, 50,000 years old; The world's oldest known song (Assyrian cuneiform artifacts) 4,000 years old; and the recent find of many 9,000 year-old flutes in China, one of them fully still playable with 8 notes, including the octave. These finds, by independent archaeologists, reveal similarities to present day widespread musical scales.

The trio theory
Originally published in 1958, the trio theory, claims that when the most audible overtones of the three most nearly universal intervals (octave, 4th and 5th), are placed within the range of that octave, this gives rise to the most common scales: Pentatonic, major & minor (depending how many of the audible overtones are so placed).

The unequal audible strengths of the overtones determine the role and power of each note in a scale (tonic, dominant or subdominant), i.e., tonality and tonal scales.

The natural or acoustic musical scale and its tonality (meaning a scale-form in which there are strong and weak notes, rather than all notes seeming to be equally important) arose in the most ancient times as follows, according to musicologist Bob Fink's "Trio" theory:

We hear the octave as the loudest overtone of any note, such as middle C. Next loudest [and different] note would be a tone (when we lower it by an octave) matching what is the fifth note of a scale, namely the "fifth." In the scale of C, this would be G. The note that produces middle C as its audible overtone would similarly match the 4th scale note, F.

This creates what is now called the tonic (or its octave), the fifth, and the fourth, steps ("intervals") in the scale when they are played out loud as separate notes. This "trio" of intervals come from the most noticeable of the most audibly relevant overtones to a given note.

The tonic, fourth and fifth are found in the music and scales of virtually all cultures in all periods of human music making.

When each of the intervals is sounded as separate notes, they, in turn, have their own audible overtones. The loudest of ALL these will fill in (by an evolving process) the rest of the notes found in the most widely known scales in the world and in history.

This also explains how there are strong and weak notes in the scale, why there are only 2 halftones in the scale, why notes historically entered the scale when they did etc.

Derivation of different scales
Below are shown the overtones of these three intervals. String out the three most audible (different) overtones of each, within the span of an octave, and you can get the major scale and other widespread scales (leaving out the repeated octave overtones and inaudible overtones as redundant):

TONIC C:   Overtones: C, G, E, (and B-flat; then inaudible) FIFTH G:   Overtones: G, D, B, (and F)      FOURTH F:   Overtones: F, C, A, (and E-flat;)

Using those notes and overtones, we can list these scales:

The Major scale:   C, D, E, F, G, A, B, C.

Then, substitute the three audibly weakest ones (the 3rd, 6th and 7th notes of the scale) with another three notes (which includes the even weaker next overtones listed above in parentheses), and which are flatter, and you get the minor scale. (The 6th note above is strongest of the three because it forms no halftones with adjacent notes in the major scale. Halftones in scales, as Helmholtz pointed out in Sensations of Tone, were avoided by most early musical cultures. "Many nations avoided the use of intervals of less than a tone...."):

Minor scale:       C, D, E-flat, F, G, A-flat, B-flat, C

Because those two overtones (corresponding to the E and the B) are very weak acoustically, they were the last to come into the scale. How they were tuned is a matter of historic uncertainty. Many people tuned them somewhere between minor and major (in the "cracks" on the piano), producing what are historically known as "blue" or "neutral" notes.

Or, if the 3rd and 7th notes are omitted altogether (thereby avoiding any halftones), the piano's "black notes" pentatonic 5-note scale results. This is sometimes called the "Chinese scale" but is also found in Africa, old Scottish and Irish folk music, and elsewhere.

Pentatonic scale:  C, D,    F, G, A,    C

Halftones in the Scale and Harmony
The process of tentatively adding halftones into the pentatonic scale took place in China, in Scottish music, elsewhere, and even the names given to these notes in different cultures are similar: "passing," "becoming," "leading" notes. It seems it was only this functional usefulness of the semitones which eventually allowed them into scales, as scales evolved and were recognized by various musical cultures, much as words evolve and are added sporadically into usage, and then permanently into dictionaries.

When you further consider the later advent of harmony you'll see that the first three different overtones of the notes shown (or of any note) add up to that note's major chord. There has been use of mostly the same trio -- the three chords of the tonic, dominant (5th) and subdominant (4th) -- to harmonize all the 7 scale-notes in most of the folk melodies known rather than each note in any melody being harmonized by a chord based on that note as the root of the chord. Therefore, most often, a C-major melody would have any of its notes "C" harmonied by a C major chord, but a D in that melody would be harmonized by a G chord, or a derivitive chord, and so on. This further underscores that these three near-universal trio of intervals and their overtones were fundamental semiconscious influences in the evolution of the scale's notes.

Harmony evolved as a means to enhance the inner overtone relationships between scale notes and notes in melodies. Even the names that evolved for them are perfect representations of their acoustic or tonal role, even though the names ("dominant" "sub-dominant" & "keynote / tonic") were also coined by people without acoustical knowledge.

The trio theory indicates the ear was already able to discern sounds as distinct between harmonious or dissonant because the ear could hear these acoustic properties without having to consciously know they existed or learn them solely by conditioning.

There is no doubt that acoustics alone cannot explain all musical matters, as psychology, cognition, conditioning, cultural dictums and their like are all present in the evolutionary acoustic processes outlined here.

The equal-tempered scale
In the natural scale the ratio of the frequencies of two notes which differ for one tone is not always the same. Consequently a certain melody cannot be played starting from a random note of the scale. For instance, a melody starting with the two notes C and D (ratio 9/8) cannot be transposed one tone higher, since the ratio of the frequencies of E and of D is very near ((5/4)/(9/8) = 10/9), but not equal to 9/8.

To obviate this inconveniency, we today use the so-called equal temperament, which constitutes the compromise adopted in modern western music. Earlier western music used other compromises.

Equal temperament is obtained by dividing one octave in 12 intervals, called semitones or halfsteps, so that the ratio of the frequencies of two consecutive semitones is constant and equal to $$\sqrt[12]{2}$$ — the twelfth root of two, whose numeric value is 1.059463.

This is also the value of the ratio of the widths of two consecutive frets on modern guitars. The twelfth fret divides the string in two exact halves.

The following table shows a comparison between the natural scale and the equal tempered scale:

12-TET (Tone Equal Temperament) allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch.

The following table shows the values of the intervals of 12 TET, along with one interval from just intonation that each approximates, and the percentage by which they differ: