"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, green@webster.sk.ca
 
PRELUDE & SUMMARY
There 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.
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:
 
The pentatonic,
AND  the diatonic, minor & major;
AND  the major chord;
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  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 one?
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 "proto-scale"?]
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 scales.
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 5ths.
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 reader.
 
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.
(See the URL: http://www.greenwych.ca/natbasis.htm Search for keywords "temperament" and also for the "Tracey" letter.)
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 other methods.
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 exist.
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 by.
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 scale notes:
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 semitone 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 scale.)
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." [My emph.]
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 combined overtones.]
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:
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 infinitum ).
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 already occurred.
 
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 :
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.
 
 Links
* 4o,ooo Year-Old Neanderthal Flute Matches Notes in Do, Re, Mi Scale --by Bob Fink
* Oldest Song in the World
* Harmony in Ancient Music
See also The Drone
* Analysis of Atonality
* Origin of Music [new book]
* Origin of Music -- an essay [book]
 
1/15/05