"The person who does not read learns no more than the person
who cannot read." (Anon)
On the Cycle of 5ths
&"The 7-Note Solution"
Why Are So Many Music
Scales Found Only 7 Notes Long?
By Bob Fink, ivylab at shaw.ca, firstname.lastname@example.org
PRELUDE & SUMMARY
are a set of massive "coincidences" (but which are not coincidences) regarding overtones that
are (subliminally) heard from the three most nearly universal intervals:
A note, its 5th and 4th. As a result, I call my theory in the article below
the Trio theory. It's quite simple in its initial outline.
It deals with all the audible overtones not of one note, but of those three
notes, and how their overtones affect the evolution of the most widespread scales.
The article below goes into some detail of how this theory
would manifest itself in reality, increasingly supported by known and new
archaeological, anthropological, historical and physiological information.
As someone put it, the more subtle or less audible overtones
are like a family whose members' faces, male or female, have an unmistakable
family resemblance (to the "root" member), but most often, one
cannot consciously put one's finger on what exact features are most causing
the resemblance. The whole (resemblance) is recognized or
felt, but the parts that make it up are subliminal. (In music, it
would be the note's pitch or pitch-class.)
Overtones of the "Trio" lead (semi-consciously
over years of "resemblance" experience) to all these
most widely found scales and musical features:
AND the diatonic,
minor & major;
AND the major
AND these "Trio's"
overtones match the historic behavior of singers and instrument tunings
being different from each other, as is widely reported;
AND they reflect,
acoustically, the historic facts of the usually close tuning
of 4ths, 5th and octaves, but also of the wider & wilder variations
of the historic tuning of 3rds and 7ths;
AND these overtones'
relative audibility as overtones reflect or determine the most commonly
found length of the scales (5 & 7 notes) -- even when
the scales are unacoustically tuned -- or so-called "out of tune";
AND the sense
of importance -- or "tonality" or "key" -- of the notes in the scale (the tonic, 4th and
5th versus a 7th or 6th, for example) is likewise reflected in the audibility
of the overtones of the Trio;
AND the three
most used chords for harmonizing melody notes in any particular
key are also the same "Trio" -- namely, chords of the tonic,
4th and 5th, traditionally used to harmonize all 7 notes,
rather than each note in any melody getting its own
root chord -- such as this rare example: Maleguena's opening
notes & chords. [The more usual and historic harmonization is
generally like the Christmas carole Joy to the World, which actually
is the major scale, played from the top down, with its notes C and
E getting a C chord; Notes B, G and D getting the G chord, and notes A
and F getting the F chord.]
The matching of historic and archaeological facts
to the Trio viewpoint and their overtones is far too great
to be coincidence or a chance event. Some resort to "coincidence"
as an "explanation" of these parallels, but that's not scientific
in my view.
So -- Are overtones a negligible influence, or a strong
No matter how unconcerned or semiconscious the ancients
were about them, it is still a fact that overtones are heard.
It's like the frailest flower or vegetation roots whose slow persistence
can break a crack into concrete, and even lift concrete to grow through
and finally blossom. Or like the pull of gravity. We are rarely conscious
of it, but we always obey its influence upon us.
Even cross-culturally, we are forever "straightening
the picture" that is hung crookedly [to match its axis of symmetry with
our own, evolved in us by the direction of the pull of gravity].
But even then: We are never SURE if we have
straightened it exactly vertical. We feel the pull of gravity only weakly
or subliminally, and so have to step back from the picture, often many
times over, for long moments of checking & adjusting [doing it for
perhaps hours, or weeks, or even a thousand years] -- to make sure the
damn picture is hung dead straight.
So, similarly, it takes a long time to figure out what
tunings within a scale are being "suggested" by the most familiar
overtones. It takes time to focus the exact pitch tunings
of other notes that will relate best (most acoustically or
with the simplest ratios) to the most widely known and used intervals:
tone, 4th & 5th.
The tunings of the fourth, & fifth themselves are
quite easier to get acoustically perfect. It's all those subliminal "overtone
flies" buzzing 'round our ancestors' heads while playing 4ths, fifths
and a root tone [which are making those overtones happen], that are hard
to nail down in pitch. The sum of these 9 or so different
overtones is what contains the subtle suggestions for what eventually get
settled as "fill notes" within the skeletal tone-4th-fifth-octave
"scale," for lack of a better word. [Perhaps a 3-note+octave
This process of on-going persistent overtone influence
-- over the long term -- trumps the other, perhaps more "conscious"
influences in the formation of scale pitches -- such as influences from
overblowing, or voice-breaking, especially in non-literate or very early
pre-history, perhaps even before any musical instruments were ever made. If we don't consider the trio explanation, then we are left with miraculous coincidence.
E S S A Y
I would like readers to learn why I abandoned use of
the Cycle of 5ths in reckoning how the pentatonic and diatonic
scales came into being. Instead I developed what I call my "Trio"
theory (1958) that the overtones of the three most widely found
intervals [a tone, its fourth and fifth], led to the evolution of these
The chart of these overtones looks like this:
TONIC C: Overtones:
C, G, E, (& Bb, then inaudible)
FIFTH G: Overtones:
G, D, B, (and F)
FOURTH F: Overtones:
F, C, A, (and Eb)
These will be referred to as the essay proceeds.
These views were written in response to a reader who
originally wrote in defense of the cycle of 5ths in the formation of scales.
His view and writing style was difficult to follow, and
is not published here in full. However, a few of the ideas he raised are
briefly referred to or quoted in my comments anonymously.
The best I could glean was that he is correct that early
musicians [those who had the cycle of 5ths available to them] can and did
tune the scales mentioned, after simple math and/or specialized tools for
producing a cycle came into existence, but in my view, that is not
how still-earlier prehistoric or preliterate humankind originally
arrived at musical scales.
Scales evolved from other processes and existed prior
to the cycle of 5ths. Later, as the ability to make a cycle of 5ths
was discovered, the cycle came to be a tool by which one could quickly
and accurately tune the pre-existing and long traditional scale(s).
My reader could not agree, and argued [from what little
I could follow of the syntax of his arguments], that the cycle of 5ths
"could have been known" by prehistoric humankind and was used
to discover and tune the pentatonic and the diatonic scales.
The overtones of my "trio" of intervals, he
asserted, had nothing to do with it, and were usually not pitched in nature
close enough to acoustic intervals to tune the scale as true to the consonant
ratios that the scales could attain if tuned through use of the cycle of
Through the heat and prodding of that debate, there emerged
a full reproduction of the views I want known against using the cycle of
5ths in the origin of music, for what that's worth for the more specialized
My "pro-cycle" reader asked: "If a prehistoric
man can find the first fifth, why can he not also find the 5th of the 2nd
note, then the 5th of the 3rd, 4th and 5th?
A prehistoric person (not just a prehistoric "man")
doesn't "find" the fifth but hears it as a "pitch class"
in the overtones of a flute, bamboo, wind whistle, all through nature.
Whether a perfect 5th or not, the barely heard 5th -- as an overtone --
can suggest (when a fifth is actually produced in the neighborhood of another
note, by singing or another instrument) a vague feeling of it having been
heard before or somehow belonging to or having some affinity to a just
previous sounded-out note or notes.
Most things are measured by the senses within a range,
and the ear and mind can modify their perception of things like a 5th,
to suit a need, such as, for example, hearing a consonance even when it's
slightly mistuned, or getting a feeling of recognition or recall.
Most ears are very forgiving. Millions of people have
adapted to the tempered "dissonance" of piano tunings -- They
hear it with the same self-delusional mental "adjustments" that
we must make in a drive-in theatre when we first hang up the speaker in
a back window of the car. An analogy will clarify this:
At first we think "it's gonna be a long irritating
night" listening to Clark Gable's voice coming from the back
seat speaker, while his lips are moving silently from the screen out-side
the front window. What an annoying drag.
But within minutes, the mind will "hear" what
it re-interprets is the correct or "real" direction of the voice,
and will "place" the sound in front of us, as if coming from
Gable's lips. Believing is seeing -- or should I say "believing
is hearing"? All our sense perceptions can be "tempered"
or altered by the beliefs of the mind prior to it telling us what we sensed.
[Although, people with perfect pitch rather than relative pitch have more
difficulty adapting to off-key tunings.]
Similarly, the ancients would take the "suggestions"
offered by frequently heard overtones of the trio [of tone, 4th and 5th
intervals], and fit them into their long-term efforts at tuning them as
consonantly as possible to create a scale that they found pleasing to them.
It's like adjusting binoculars to make an image come into focus. The exact
place of maximum focus is a trial & error process, often taking some
fiddling with the turnscrew on the lens. Similarly with the more subliminal
and subtle overtone "suggestions": It probably took many hundreds
or thousands of years to fine-tune them to conform with the simplest consonant
ratios on a more or less societal "standardized" level.
Along the way, many scales that finally were arrived
at often were are still not acoustically "natural," due to workmanship
limits, existing instruments dimensions, different opinions and/or other
influences -- but these tunings averaged close to acoustic tones, and obviously
were tolerable or even revered by conditioning and associations. I cannot
conceive of any other way early people would have evolved music scales,
or interval usage that matches the approximate pitch classes of overtones,
without being stimulated to do that by the overtones themselves.
The prevalence of even such "out-of-tune" or
non-acoustic scales being just 5, or limited to 7 notes long (or 6 and
8 if you include the octave), which are found widely and repeatedly in
many cultures and times, indicates the pressure [to find scale notes to
add within the basic three intervals], comes from overtones.
Here's how that happens: The three most
widely or universally used intervals are a note, its 5th and 4th. The audible
overtones of these three intervals will produce notes or pitch-classes
of notes numbering from the loudest 5 different overtones up to only 7
or 9 different overtone pitches. After about 9 of them, the overtones become
inaudible to the human ear. [See chart of the three intervals and their
combined overtones.] That they may be tuned by ancients to acoustically
perfect intervals, or not, only indicates that the influence
of the overtones is more or less subliminal, not a deliberate, obvious
or conscious process. Tuning until acoustic intervals eventually prevail
is a long-term process. But when 5 & 7 scale notes are caused, that's
an immediate reflection matching how many differently pitched overtones
are audible. So therefore, the overtones are the likely cause
limiting the lengths of scales to 7, while avoiding adding more unnecessary
semitones to the scale. [More on this later.]
The inconsistency of the tuning of these "fill notes"
to form a scale shows there was no serious nor obvious concern to meet
some already existing "standard" of tuning. This is also shown
when, as found in many non-literate cultures, singers, for ex., sing perfect
acoustic intervals while their accompanying traditional instruments (with
fixed tones, like a flute) do not always play such perfect intervals. The
reasons for this are explained more fully elsewhere regarding the evolution
and use of temperament.
In any event, ancient people, lacking the power of math,
could not have discovered the diatonic through math. Thus the math of the
cycle of 5ths is irrelevant to the issue of the origin of music. And if
Darwin is right, prehistoric music and scales possibly happened even before
speech, and so certainly before math was developed.
There is no evidence to believe an early person would
even know that a 5th came from some specific original note
until repeated experience indicated that. By then, music, instruments and
rudimentary scales likely would already have been formed by
Certainly, the ability to make a long cycle of fifths,
beyond singing the first two of them within ordinary voice ranges, was
non-existent in prehistory. The whole Newtonian idea of "experimentation"
like that has no evidence of existing then. The empirical method wasn't
fully developed even with Galileo. Nor the desire to achieve a formal "scale"
could hardly have existed as an ancient abstract goal or concern, as both
the evidence -- and lack of it -- suggests. It would be like seeking to
invent the fender or brakes before knowing even that a wheel could
The first such evidence of any device for producing a
cycle of fifths comes from the Ancient Greeks -- people like Pythagorus.
Thus the cycle of fifths could not have played any role
in the creation of acoustic-like pentatonic or diatonic scales among
the recently found prehistoric examples of these scales -- and more
and more of these are being found in archaeological activity as time goes
Therefore, nothing in my theory uses the cycle of fifths
to explain the origin and evolution of scales and music. That's as absurd
to me as assuming that the ancient pyramids were erected with power cranes
and other heavy lifting gas-engine equipment, or that the ancients used
tuning forks to tune their flutes.
Another time my detractor wrote: "If a person can
find (hear) the fifth of one note I see no reason whatever why he can't
find (hear) the fifth of the second note and so on. When he has five notes
he has a pentatonic scale. Two more makes the major scale."
But there are problems with assuming this method of discovering
My reader's cycle would look like this (example implies
a scales made in the key of C): F-C-G-D-A-E-B- and so on, one fifth after
another. Go these five and you get a pentatonic: F-C-G-D-A. Putting it
in the key of C, it would be c,d,f,g,a and octave c'.
(There are no semitones in this scale.)
Go two more (E and B) and it can make the diatonic: c,d,e,f,g,a,b,
and octave c'. (Now there are two semitones or half-tones in the
Helmholtz, the "father of acoustics," wrote
that many nations have avoided the half-tone in their scales. At least
as noted in the more widely used or "near-universal" scales.
So, as Helmholtz wrote in his founding of the science
of acoustics (Sensations of Tone, p. 280), "The old scale of
five tones [pentatonic] appears to have avoided Semitones as being too
close. But when two such intervals already appear in the [diatonic] scale,
why not introduce more?"
Helmholtz saw no answer. It was this which, even if a
long series of 5ths or cycle could have existed in prehistoric times,
led me to abandon the cycle of 5ths as an historical means by which notes
were added in a scale.
There is no reason not to keep adding more,
if we rely only on the cycle of 5ths. This cycle is equal -- each 5th before
or after the other is just as loud as its adjacent 5th. There is nothing
to stop the cycle there, nor after 7 fifths, nor after 20, for that matter.
But something caused the process to make
stops. This must be explained.
My critic wrote, to answer the Helmholtz quote: "All
12 are on your keyboard, so more semitones have been introduced."
That makes no sense:
Helmholtz knew the 12-tone division existed
all around him in his time and had existed for centuries. So WHY would
he ask "when two such intervals already appear in the [diatonic] scale,
why not introduce more," when he would have known the
12 semitones were already existing? Either Helmholtz was a
moron, or the answer offered by my challenger incorrectly confuses and
equates the terms "division of the octave" with a "scale."
Maybe: Is it because Helmholtz wasn't talking
about "divisions of the octave," but was talking about introducing
semitones only into scales instead?
Scales are a series of tones that can be made from the
divisions of the octave chosen by society. Indian musical systems have
several divisions of the octave, including one that has 22 notes. Another
has 17 tones if I recall correctly. These divisions are not scales. I.e.,
music was never composed using all these notes as if they were all part
of the same scale or "key" or "mode." Rather, scales
were made from these divisions. Some had 7 tones, others 5,
6 and so on. Music could then be composed using these fewer scale or modal
tones. (Some of these divisions were so theoretical, that they were rarely
used in practice.)
In ancient Greece, there was no known division of the
octave formally established, that I know of. In our modern Western society
there are, of course, 12 notes in the division of the octave. Our major
and minor scales can be made from that division.
Only atonalists have developed the 12 tones into a "scale,"
and they compose music that initially requires use of all 12 before repeating
any of them. On the other hand, in tonal music, only 7 of the 12 semitones
are used, and sometimes, if a "black key" is used, it is aptly
called an "accidental" indicating it doesn't normally belong
to the scale in use.
I developed the view that overtones emerging from the
"Trio" of intervals -- a tonic, fourth and fifth -- contained
built-in "stops" in the scalenote-adding process:
The first [pentatonic] stop avoids halftones. The second stop [after adding
half-tones to make the diatonic] allows & tolerates these halftones
in the scale historically in many pentatonic cultures, but only as "passing
notes." The names assigned show this. By the Scottish:
"crossing over" notes -- or, by the Chinese: "Pien"
tones, or "becoming" notes. Each name
[independently generated] is parallel to what we call "leading
tones" -- to the most important 5th and the octave.
And further: Here we now see emerging a concept of "importance"
or dominance of notes in the scale: Thus we have the "Tonic, Dominant
& Sub-dominant" -- words coined by people who knew virtually
nothing of acoustics & ancient history.
The cycle of 5ths implies equality of each added note.
But the "Trio theory" I use recognizes that notes, as overtones
of the Trio of tonic, 4th and 5ths (which are heard most frequently from
the earliest music made), have different strengths of being audible
to the human ear. Not all are equally as loud. The loudest overtones became
the most important notes [and also have simpler ratios, to which the ear
responds regarding their being most consonant].
The Cycle of 5ths cannot explain any of this sense
of inequality among scale notes (known now as tonality),
but my "Trio theory" can explain it.
Further: The cycle of 5ths cannot explain the minor scale:
The next notes in the cycle of 5ths after the major diatonic are F#
and C# (F-C-G-D-A-E-B-F#-C#) -- but no widely found scales
in any time nor place contain these notes -- and they also do not make
the minor scale. You need to go higher and longer into the cycle of 5ths
before you can reach the notes needed for a minor scale.
But the Trio's list indicates its next audible overtones
[among the trio of intervals] are Bb and Eb (or the "pitch-class"
tones known as Bb and Eb). These are two of the notes that would, if they
replaced the major E and B, make the scale minor. Thus, no additional
halftones are added, in conformity with the general historic avoidance
of them (except their use as leading tones); and again, the audibility
of these Bb and Eb overtones are next within the list of lessening audibility
of overtones of the trio. [Again See chart of the 3 intervals and their
How "coincidental" is all this? Or more likely,
better to think: How "fitting" -- in terms of the semi-conscious
efforts made regarding how to tune notes in the large pentatonic scale's
"gaps" (between D and F , and between A and C'). Some ancient
tunings of the 3rd and 7th scale notes are major, some minor. And many
are "blue note" tunings, or as in Africa and the orient,
called "neutral" notes, tuned as a compromise between
the major and minor.
Being weak or less audible overtones, their tuning in
most cultures has been widely noted as hesitant and varied. The 5th is
often tuned perfectly, but the 3rd and 7th tunings are all over the place:
A. M. Jones is quoted by Merriam [Continuity and Change In African
Cultures, pp. 71-72]: "I have lived in Central Africa for over
twenty years, [and] never heard an African sing the 3rd and 7th degrees
of a major scale in tune.";
Merriam notes: "There has been some discussion of an African
scale in which the third and seventh degrees are flatted or, more specifically,
neutral between a major and minor interval. ...in jazz usage, these two
degrees of the scale -- called ‘blue' notes -- are commonly flatted and...the
third...especially, is ...given a variety of pitches in any single jazz
Helmholtz op.cit., p.255) notes that the "history of
musical systems shows that there was much and long hesitation as to the
tuning of the Thirds."
This discrimination regarding different notes in the
scale (as to tuning consistency) must be explained.
The cycle of 5ths explains nothing of this. The Trio
theory, by having a built-in stop after 7 (of only 9 audible overtones
available from the chart of the most universal intervals), explains the
tuning hesitation among these two last, and least audible, overtones. [In
the key of C, these are Eb and Bb]. These stops, as noted earlier, also
explain the persistence of 5 and 7 notes scales so widely found throughout
the world and times, no matter how "true-tuned" acoustically
or not they are.
It is apparent to me that what is taking place over time
is a seeking of the proper tuning to match the subliminal suggestiveness
among the 9 most familiar overtones. The desire that half-tones or dissonance
be avoided (and the decreasing audibility of the overtones) explains the
general refusal or failure to create scales with 8, 9, 10, 11 or more notes,
keeping usually to 5 or 7 note limits, despite the fact that among the
widest developed number systems preferred through history are TEN
based systems (10 toes or 10 fingers). And along with this, the unequal
spacing of tones defies the usual penchant of minds to divide things equally
in most other areas of life and design (inches, feet, meters, sidewalk
slabs, windows on buildings, telephone poles, thermometer degrees, ad
My antagonist claims I "cannot know" that making
a cycle of 5ths was not possible for the prehistoric people who made the
first music, saying that I "wasn't there."
But yes, we CAN know or infer many likely
limits of ancient peoples from their artifacts, tools, bones, DNA and skeletal
structures. What has been found are tools and musical instruments whose
workmanship defies possibility of producing a full cycle of fifths. Plus
the motive for producing a cycle would be incomprehensible until a certain
technological and/or linguistic level is reached -- but by that time, the
making of music, rudimentary scales, finished scales and such, would have
Several different theories may explain facts and events,
or at least some of them. But the best theory is the one that makes
the fewest assumptions, nor needs them, to explain the same
or more of the facts and events.
But my critic assumes: "I expect they wailed
and howled in twos and threes and fours making all manner of discords like
cats until the natural intervals were discovered."
That, on its face, is absurd and impossible. Even if
people for unknown and mysterious motives sought out harmony by such trial
and error, not even knowing consonance could be
found, they could just as easily have ended up conditioning themselves
to their noise and ended up preferring it to any eventual acoustic harmonies.
Kids make that noisy activity for years, and discover nothing of making
harmony -- even when it's done in societies where there are radios playing
Mozart or "My Fair Lady" tunes & harmony everywhere.
Besides, there is no reason in prehistory for harmony to exist when individual
notes are far more consonant than even what we usually call consonant chords,
as I explain at: "Stages in the Evolution
of Scales..." etc :
"When any harmony is added to a given note...each tone added will
produce its own additional audible overtones...conflicting with
those of other notes, producing ...dissonant properties....
"So why did humanity bother with development of harmony at all??
"The answer is that there are relationships of overtones between
the notes of a melody, even in a rudimentary pentatonic melody. That
is the heart of any ‘scale....'
"The role of harmony in a melodic musical culture is this:
Harmony tends to overtly reveal between the notes of a melody what is hidden
(in overtone relationships) by playing some of these overtone relationships
"The dissonances of harmony now become tolerable once a musician
is concerned with the connections between notes of a melody. [Only with
the advent of melody, can harmony and chords take on any meaning].
...a musician in an ‘earlier' stage, prior to such melodic practices, is
concerned more with the aesthetic effects of single or pure tones, and
so would avoid the dissonances of deliberate harmonies [with the possible
exception of using octaves and fifths, which are assigned to different
voice ranges as a form of ‘unison']. Like organum.
"Therefore, we can so far explain from this why, in the limited
data existing of the most ancient music, we likely will never find any
deliberate harmonic practices existing unless and until the diatonic scale
(or perhaps a pentatonic that includes occasional leading tones) has been
Lastly is this comment from my reader: "I believe
my view on scales is actually the majority one and it's your group of mathematically
challenged overtone enthusiasts who are the ones with the funny ideas."
No further reply came from Bob Fink. After having
completed a study on the statistical probability of the spacing of the
holes on the Neanderthal Flute (see that essay's Appendix, as well
as his math analysis article in the book: Studies in Music Archaeology
III, from the 2000 conference of international music archaeologists),
Mr. Fink was recently found wandering the streets and lanes of his home
town searching and looking for something. Asked what he was looking for,
he claimed he was looking for his mathematical ability. He was afraid it
fell out of his pocket or something, and that he could not continue to
reply on this webpage because he felt he would remain "mathematically
challenged" until he relocated his lost math, or if stolen, until
it was returned.