NationMaster - Encyclopedia: Origin of scales

The natural scale is attributed to the Greek philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies. Aristoxenus of Tarentum (4th century BC) was a Greek peripatetic philosopher, and writer on music and rhythm. ...

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.

C	1
D	9/8
E	5/4
F	4/3
G	3/2
A	5/3
B	15/8
C	2

Evolution of the natural 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. To meet Wikipedias quality standards, this article or section may require cleanup. ...

The trio theory

Originally published in 1958, the trio theory (1970, in ISBN 0-912424-06-0), 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, the fifth, and the fourth, which are steps (or "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 related 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 influences from the loudest of all these overtones suggests (by an evolving process) what notes can fill in the rest of the notes found in the most widely known scales in the world and in history.

This also explains why there are strong and weak notes in the scale, why there are only 2 halftones historically accepted in the scale, and 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.

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

Halftones in the scale and the evolution of 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 further considering the later advent of harmony it can be seen 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 "C" notes harmonied by a C major chord, but a D in that melody would be harmonized by a G chord (or a derivative chord); an "E" would be harmonized by a C chord; an "F" (or an "A") in the melody would be harmonized by an F-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. An equal temperament is a musical temperament, or system of tuning, in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

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. The Twelfth root of two is a quantity representing the frequency ratio between any two consecutive notes of a modern chromatic scale in equal temperament. ...

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 neck of a steel-string acoustic guitar showing the first four frets. ... Different kinds of guitars The guitar is a fretted and stringed musical instrument, used in a wide variety of musical styles, and is also widely known as a solo classical instrument. ...

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

**Cent values of equal temperament**
Tone	C¹	C♯	D	E♭	E	F	F♯	G	G♯	A	B♭	B	C²
Cents	0	100	200	300	400	500	600	700	800	900	1000	1100	1200

12-TET (Tone Equal Temperament) allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch. An equal temperament is a musical temperament, or system of tuning, in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... Music notation is a system of writing for music. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

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: In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ...

Name	Exact value in 12-TET	Decimal value	Just intonation interval	Percent difference
Unison	1	1.000000	1 = 1.000000	0.00%
Minor second	$sqrt[12]{2^1} = sqrt[12]{2}$	1.059463	16/15 = 1.066667	−0.68%
Major second	$sqrt[12]{2^2} = sqrt[6]{2}$	1.122462	9/8 = 1.125000	−0.23%
Minor third	$sqrt[12]{2^3} = sqrt[4]{2}$	1.189207	6/5 = 1.200000	−0.91%
Major third	$sqrt[12]{2^4} = sqrt[3]{2}$	1.259921	5/4 = 1.250000	+0.79%
Perfect fourth	$sqrt[12]{2^5} = sqrt[12]{32}$	1.334840	4/3 = 1.333333	+0.11%
Diminished fifth	$sqrt[12]{2^6} = sqrt{2}$	1.414214	7/5 = 1.400000	+1.02%
Perfect fifth	$sqrt[12]{2^7} = sqrt[12]{128}$	1.498307	3/2 = 1.500000	−0.11%
Minor sixth	$sqrt[12]{2^8} = sqrt[3]{4}$	1.587401	8/5 = 1.600000	−0.79%
Major sixth	$sqrt[12]{2^9} = sqrt[4]{8}$	1.681793	5/3 = 1.666667	+0.90%
Minor seventh	$sqrt[12]{2^{10}} = sqrt[6]{32}$	1.781797	16/9 = 1.777778	+0.23%
Major seventh	$sqrt[12]{2^{11}} = sqrt[12]{2048}$	1.887749	15/8 = 1.875000	+0.68%
Octave	$sqrt[12]{2^{12}} = {2}$	2.000000	2/1 = 2.000000	0.00%

References

More details on the evolution of scales and of music can be found in the few full-length books on this subject, such as

On the Origin of Music, 2004, ISBN 0-912424-14-1, which includes a bibliography & reviews.
See also: The Origin of Music, ISBN 0-912424-06-0; [1] and
Wallin, Nils, Bjorn Merker, and Steven Brown, eds., The Origins of Music, MIT press, 2000, ISBN 0-262-23206-5 — a collections of essays relating to music archaeology, music psychology, music and language.

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