The Solfeggio Frequencies

The (dis)harmony of the Spheres?

Richard Dobson, April 2014

Last Edited: 30-04-14

Biblical Frequencies?

It all started with the book "Healing Codes for the Biological Apocalypse" (1999) by Dr Joseph Puleo and Dr Leonard Horowitz. Their central claim is to have discovered, within the text of the Bible, six numerologically constructed "sacred frequencies" which, moreover, they claim (by way of a particularly strange but intense conspiracy theory) to be the "lost" and even church-suppressed "original frequencies" of the musical scale associated with the names Ut-Re-Mi...(C-D-E on the modern staff) and thus the "true" basis of Gregorian plainchant. Great esoteric significance is given to the fact that the digits of each number sum numerologically to either 3, 6 or 9:
396 - 417 - 528 - 639 - 741 - 852

Some ambitious claims are made for each of these "sacred" or "healing" frequencies. This has led to sets of crystal singing bowls and tuning forks being built (more or less accurately) to the Solfeggio frequencies. As a result they are increasingly embraced (somewhat unquestioningly, it appears) not only by the complementary sound therapy community but also by musicians and composers with a focus on healing and well-being. However, the downside of this focus on precise frequencies (ostensibly to the exclusion and even avoidance of others) is that it can obscure and confuse the much more important and authentically ancient study and experience of the relationships between tones, and of the effect of those relationships on listeners. These core relationships are described mathematically as frequency ratios, and musically as intervals.

The science of pitch ratios is associated in particular with the Pythagorean tradition of Speculative Music. This studies, primarily as a philosophical, metaphysical and even spiritual exercise, the relation between music and abstract mathematics, and, by natural extension, between music and the laws of the Universe – of which the human being is naturally a part. It is central both to the idea of the "harmony of the spheres" (the macrocosm) and to the work of restoring harmony to mind, body and spirit (the microcosm), following the Hermetic principle of "as above, so below". It forms the deep background for not only much Western art music (not least including that of J.S Bach, whose interest in Pythagoreanism is well documented) but also other classical musics such as Indian Raga, which has it own version of Ut-Re-Mi, sargam, based on the Sruti scale of 22 intervals. This is in its turn closely connected to the study and practice of Nada Yoga - the "yoga of sound", an ancient musical and therapeutic tradition in its own right.

This essay is in two parts. Part 1 examines the claims of Puleo and Horowitz and demonstrates (for anyone not already persuaded by the above example) how they are variously anachronistic, impossible and impractical. By itself this would make for a predominantly negative story. Part 2 offers a more constructive complementary approach which substantially extends the numerology of the Solfeggio series and in a simple way dips a toe into the waters of speculative music. I offer below a whole "Planetary Solfeggio" comprising not six or nine but 81 frequencies – nine sets of nine frequencies forming a magic square (one set being the original Solfeggio series above), all following the same numerological principles, and even (inevitably and unsurprisingly, as it turns out) all adding numerologically to either 3, 6 or 9. Several computer-generated sound examples are provided, created (like the above example) using the open-source synthesis language Csound.

About the numbers

It is certainly plausible, at least, that such numbers (along with a myriad of others) might appear one way or another in the texts of the Hebrew Bible. The Qabbalistic practice of Gematria exploits the fact that characters in the Hebrew alphabet also represented numbers. Two words (or names, phrases) summing numerologically to the same value were deemed to have an occult or mystical connection. Within the Solfeggio we can see the tell-tale sequence of standard 9-based numerology. There is no zero digit, which is itself a relatively modern invention. Classical Arabic has a similar form of Gematria, so that, for example, a similar procedure can be applied to words and phrases of the Koran.

In the decimal sequence, 9 "wraps around" to 0, whereas in numerology it wraps around to 1. We can see this pattern clearly in the sequence of the second and third digits in the list. We can therefore add just three further three-digit numbers to the list, forming what Puleo and Horowitz agreed was the complete "biblical" Solfeggio series:

174 - 285 - 396 - 417 - 528 - 639 - 741 - 852 - 963

If we try to continue the series (by adding 1 to each digit), the 9 wraps around to 1, 963 becomes 174 and the whole sequence repeats ad infinitum. A few Solfeggists have attempted to cheat the system, so to speak, by introducing 0 and generating the new higher frequency 1074; clearly, if numerological consistency is to be respected and maintained, this is an illegitimate step. The final step is the "reduction" of each number to a single digit by progressively adding digits. For example, 174 reduces to 12 which reduces to 3.

So, is this really a scale?

The short answer is "no" – as heard above these frequences rather form a (mostly) broken or irregular harmonic series, spanning over two octaves. To the experimental musician this may make them, if anything, more rather than less interesting.

The long answer is also "no". The misunderstanding at the heart of the proposition is that the musical scale has never been and even today is not as a matter of principle defined as a series of precisely fixed frequencies. While the tuning reference of A=440 Hz is (mostly) standard today for modern instruments, it is by no means religiously adhered to. For example, the long established practice of performance of early music on "original instruments" eschews A=440 in favour of the lower value (typically) A=415, approximately one semitone lower.

The scale note names Ut-Re-Mi (or in the more modern form of solfège, Do-Re-Mi...) define not absolute frequencies but relative frequencies – ratios or intervals between two tones. They also define the written position of the note on the staff. We can describe them either sequentially – the interval between adjacent tones (the diatonic scale is based on two sizes of tone – whole-tone and half-tone), or harmonically – the interval between each tone and the bass tone or "tonic" of the scale. The Pythagorean musical scale uses seven tones, making eight when we include the final upper octave note. Thanks to the magic of polyphony we can experience both sequential and harmonic forms of the "natural major" scale together:
Sequential: tone - tone - semitone - tone - tone - tone - semitone
Harmonic: unison - M 2nd - M 3rd - 4th - 5th - M 6th - M 7th - octave
Example in Pythagorean tuning:

The reference to the upper octave gives the clue as to what really makes this set of notes a scale. The octave is, so to speak, the defined containing interval for the scale, which comprises more or less equal subdivisions of that interval, giving the discrete notes or steps of the scale. The scale then repeats over successive containing intervals. The word "octave" here describes the fundamental frequency ratio 2:1, and, indirectly, the span of the above eight-note natural scale. C is not a single note, but (in the case of the modern piano keyboard) eight notes separated by octaves. Musicologically speaking, it is not a note but a pitch class. The other notes (pitch classes) in the scale are obtained by repeated application of the generative ratio 3:2 (the perfect fifth), followed by octave reduction to normalise all pitches within a single containing octave. Together, the octave and the fifth (along with its inversion, the fourth) are at the heart of the Pythagorean scheme.

It is this vital generative aspect (each re-application of 3:2 generates a new pitch class and a new ratio, e.g. 9:4, and hence 9:8) that is absent from the Solfeggio set. No containing interval is defined, nor any procedure for deriving further notes. We are left with a rigid and inert list of, at most, nine harmonically arbitrary frequencies which, far from being subtly different from the Pythagorean scale, bear no relation to it at all. Even for the most esoteric and experimental of musicians, this is meagre material indeed. As it happens, the Solfeggio frequencies are also almost all too high for bass and tenor voices to sing.

So the claim that these frequencies are somehow the original "lost frequencies" of church planchant is beyond absurd. No unmodified adult monk could have sung them, and the great Western arts of harmony and polyphony would have been stillborn. It is time to demonstrate with a complete example.

Ut Queant Laxis...two versions

Much cited in the Solfeggio literature is the plainchant hymn Ut Queant Laxis. The initial notes of each phrase start on successive steps of the scale, which in turn take their names from the first syllable of each phrase. While the starting note is indeed Ut, the tonic note of the chant is in fact the second step, Re:
The standard (singable) version:
The Solfeggio version (Csound):
I leave it as an exercise for the listener to find the frequencies (pitches) used in the first example. Suffice it to say, the singers chose a tonic which suited them. A second exercise would be (vocal range permitting!) to try to sing along with the second example.

So, whatever the Solfeggio frequencies may be good for, as a basis for a sophisticated, evolving and increasingly polyphonic musical style and tradition predicated on singing they are terminally limited. We must be grateful indeed that the mediaeval church decided against them!

The frequency problem

To understand why there could be no such thing as a "biblical frequency", we have to remember what the unit of frequency really is. The modern notation Hz (as in "396 Hz"), named in honour of the 19th century scientist Heinrich Hertz, is here a little unfortunate. Originally the unit of frequency was simply "cycles per second". Herein lies our problem. The very concept of the unit of a second as a measure of time is really very modern indeed. Both the minute and the second existed in biblical epochs as measures of angular distance (rotation) developed by the Sumerian and Babylonian astronomers and astrologers who studied the motions of the stars and planets. That usage continues to this day.

The second as a measure of time only began to appear with the invention of the first mechanical clocks in the 15th century - where it corresponded, in effect, to a single swing of a pendulum or a single tooth on a gear wheel. Until that point, there was no means by which anyone could measure time even that precisely – which in any case was, by modern standards, not very precise at all. To measure a frequency as being 396 Hz rather than 397 Hz requires the ability to measure time with at least a precision better than their reciprocal - to, say, 0.0025 seconds or better. Even today, that is a fairly demanding requirement beyond the capacity, for example, of the simple electronic tuners widely used by guitarists.

Any measurement depends on the availability of some universally agreed reference. So the questions then become: what is that reference, and how is it measured and defined? Not only has the unit of the second not somehow existed for ever, neither have the metre or the kilo. Those imaginary biblical musicians and scribes had no technology which could even begin to measure such a small unit of time. Even in the 20th century, the second has been redefined more than once, as tools such as atomic clocks have been built to greater and greater accuracies.

One instrument that conceivably was available to those (moderately) ancient musicians and scribes was the Pythagorean monochord, a single stretched string (of any convenient length) over a moveable bridge. With this it is possible to find and measure the ratios of all musical intervals (consonant and otherwise), not by reference to some absolute time, but by a simple comparison of string lengths. There is considerable evidence, including from surviving cuneiform tablets, that the seven-tone diatonic scale first appeared in ancient Mesopotamia, and that Pythagoras learned them from Sumerian and Babylonian musicians during his time as a prisoner in Babylon.

The precision problem

A number such as 396, written down, is perfectly precise. So is 397. However, according to the Solfeggio claims, only the first is sacred – 396 Hz is a frequency with great healing and mystical power, but 397 Hz is not. That is a difference of a mere 0.26%. 396.1 Hz is also, one presumes, not quite sacred. This is a doctrine of division where none is needed or warranted – a suggestion that sacredness is a matter of precision of measurement. It is like claiming spiritual superiority by standing one foot closer to the sun. It is also seriously impractical, given that, as we are to understand from Puleo and Horowitz, human beings sang these notes. Even the minority of humans with perfect pitch (who have, so to speak, a reliable internal reference) are not accurate to that degree. We will find that only modern digital technology can possibly approach the sort of precision demanded here.

Yet even computer-based audio systems are not perfectly accurate - there is always a margin of error. Clocks drift over time. It is why, for example, professional sound studios run all their digital audio hardware off a single "master" clock, to ensure they are perfectly synchronised together over long periods of time. Such margins of error are inescapable where any physical system is involved; they will mean that even the carefully tuned frequencies of crystal singing bowls are unlikely to be perfectly exact to any of the Solfeggio frequencies. The margin of error needs therefore to be at least as sacred as the frequency itself.

Herein lies the difference between musica speculativa and musica practica – between the philosophical contemplation of idealised pure number and pure form on the one hand, and the exigencies of practical music making with approximate physical tools and indeterminate human bodies on the other. Plato famously expressed this dichotomy quite uncompromisingly when he argued "music is far too important to be entrusted to musicians!".

Musica Practica – tuning in practice

Working musicians, needless to say, took and continue to take the opposite view. It is possible (indeed, a matter almost of routine for instrument tuners) to tune the intervals of the natural scale to high precision by listening to beats (a form of vibrato). The beats indicate the difference between the two frequencies. For example, 396 and 397 sounded together will produce a beat of 1 Hz. Eliminate the beat and the interval (in this case a unison) is then tuned exactly.

Beats can also arise between wider intervals. If the frequencies are high enough (and the difference is large enough), the beat is heard (through a psycho-acoustic trick of the ear) as a new audible pitch, technically called a "resultant tone". In effect, one frequency serves as the reference for the other; the trick is to tune the two primary tones so that the resultant tone is in tune with them. One could say that the whole purpose of the system of just intonation is to ensure that as many resultant tones as possible are in tune in this way.

Interestingly enough, in the Solfeggio series there is one perfect fourth (396-528, G-C, ratio 4:3); the other intervals are variously "a bit off" and "way off". This does not make them wrong or uninteresting, it just makes them difficult to sing and more generally something of an acquired taste. If however we base our investigation on seeing the frequencies not as a scale but as a more or less disjunct harmonic series (in effect, a chord), a host of new sonic possibilities opens up.

The Planetary Solfeggio

Back to the numbers

Whether or not we accept all the pseudo-historical, pseudo-musicological and pseudo-conspiratorial baggage attached to the Solfeggio frequencies, we can achieve much by focussing on the numbers themselves, and observing something more of the patterns involved. We have already seen the cyclical nature of the digits in Pythagorean numerology, no different in principle from that of any mechanical counter such as a vehicle odometer except for the exclusion of the zero digit. This is therefore a crude sort of "offset base-9" counting system, where the usual base-9 counting digits 0,1,2,3,4,5,6,7,8,0... are offset by adding one to each. The nececssary qualification to this is that numbers in numerology are not used for counting or mathematics, as such. We recall that these numbers are regarded in symbolic and divinatory terms as indicators of inner conditions, and in gematria as substitutes, anagrams, for words, names and phrases. There is no mechanism within numerology whereby three digits can become four.

The novelty in the case of the Solfeggio numbers lies in the one additive step where each number is formed by adding the three-digit number 111 to the previous one, and wrapping the digit 9 back to 1 as required; a form of modular arithmetic. The somewhat banal observation to make here is that the digits of 111 sum to 3. Thus, given that the "root" number 174 itself reduces to 3, it is inevitable and obvious that all subsequent numbers in the set will numerologically sum to a multiple of 3. The question then arises – is this property of the number 174 coincidental, or is there a patttern to discern here?
Indeed there is. The digits are separated by a common difference, in this case by six. This is therefore a small example of a simple arithmetic progression, but where values above nine are numerologically reduced. In this case, the number 13 reduces to 4. From this simple observation we can define a complete set of nine root numbers, with the common difference ranging from 0 to 8. In turn, each root number generates a new nine-number set by repeatedly adding 111.

The Complete Set

The Solfeggio Magic Square
111 123 135 147 159 162 174 186 198
222 234 246 258 261 273 285 297 219
333 345 357 369 372 384 396 318 321
444 456 468 471 483 495 417 429 432
555 567 579 582 594 516 528 531 543
666 678 681 693 615 627 639 642 654
777 789 792 714 726 738 741 753 765
888 891 813 825 837 849 852 864 876
999 912 924 936 948 951 963 975 987
The first set (111-999) in this table is significant in that it is the only one forming a true harmonic series (with 111 Hz as the fundamental pitch). There is no point within the set at which 9 wraps around to 1. This means that all the frequencies are exactly harmonically related (the basis of just intonation). The result is a harmonic spectrum (as is common to most string and wind instruments, and also as heard in overtone chanting) – when sounded together using pure sine waves the ear hears them as making a single bright note. Funnily enough, the series stops just at the point where the intervals begin to form a recognisable scale.
Conversely, each of the other columns features two such wraparound points, indicated in grey. This is why I describe them as forming a "broken" harmonic series. The spectrum becomes inharmonic – more characteristic of metallophones such as bells and singing bowls. In the sound examples below, the frequencies (all sine waves) are presented firstly in sequential overlapping tones, and secondly in groups defined by the wraparound points.

Following the tradition in Speculative Music of defining cosmic correspondences I have associated each set of tones with the planets of the Solar System, from Mercury to Pluto - highest root frequency to fastest planet. It happens to lead to a possibly meaningful synchronicity with respect to the original Solfeggio set. I have also indicated the arithmetic factors for each root frequency. None of them is itself a prime number but a few have a relatively high prime factor, something which some readers may find interesting.
Sounds of the Planetary Solfeggio
planet root freq factors sound
Mercury 198 22×9,33×6
Venus 186 31 × 6
Earth 174 29 × 6
Mars 162 3 × 6 × 9
Jupiter 159 53 × 3
Saturn 147 7 × 7 × 3
Uranus 135 3 × 5 × 9
Neptune 123 41 × 3
Pluto 111 37 × 3
This example represents a compact outward journey through the solar system using a single chord for each planet, the harmony resolving, so to speak, with Pluto.

Each chord can be described as creating a harmony - just not one understood within classical tonal musical languages. These days such things are far from being unfamiliar territory. As noted above, Tibetan singing bowls typically feature inharmonic intervals between their partial tones, and also small microtonal differences generating beats. A single bowl may be used as a point of focus for meditation, but it is also common to play a whole collection of bowls together creating a sustained abstract pan-harmonic texture remote from any tonal musical framework. This suggests that one effective way of using the Planetary Solfeggio frequencies would be, rather than play one in isolation, to play them together as a group process in the style of spontaneous improvised vocal toning. This extended example explores that approach using the frequencies of the original Solfeggio set.

Coda: a few numerical musings

Other Patterns. In any magic square, significant patterns are found in the numbers on the diagonals, as well as in the columns and rows. This is clearly the case for the Planetary Solfeggio. A further nine frequency sets can be constructed, e.g. 111,219,318... up to 198,297,396,495 etc. The latter is interesting in that the numbers are multiples of 99, which becomes in effect a missing (secret!) fundamental. This set of numbers therefore defines a harmonic series starting from the second harmonic. It is however not quite perfect – the final frequency should be 990; of course as explained above the zero is not available to us here, if the numerology is to be consistent. It is clear that such microtonal differences are central to the character of the Planetary Solfeggio, even though (or perhaps because) they take the ear away from the familiar territory of conventional tonal harmony.

Tyranny of the nines. There are 729 possible three-digit numbers not containing a zero digit. 729 = 9³. Or, equivalently, 9 × 81. This gives a clue to the distribution of numbers whose digits sum numerologically to 3, 6 or 9: there are 81 of each, so there are 243 in all. That's a lot of numbers to get excited about. Likewise there are 81 numbers for each of the sums 1, 2,4,5,7 or 8. Truly, once we start down the dark path of the nines, forever will it dominate our destiny. It is the price we must pay for excluding zero.

A Solfeggio of Light. Another remarkable aspect of the Solfeggio story is that these frequencies, as well as being biblical, are also said to be "electromagnetic". Why is far from clear, but it is consistent with the broadly anachronistic approach. Sound waves in air, needless to say, are nothing of the sort, and "frequency" is simply a term signifying a speed, and applies to anything which periodically repeats. The term electromagnetic properly applies to such phenomena as radio waves, and, most significantly here, light. This suggests an alternative interpretation of the numbers as wavelengths in nanometres (nm). The range from 174 nm to 963 nm nicely covers the range of visible light (the rainbow), extending from ultraviolet (wavelengths shorter than 400 nm) to infrared (wavelengths longer than 700nm).

Retuning Middle C. One idea which to some extent unites Solfeggists, speculative musicians, and even computer scientists, is that it would be really rather nice if the frequency of "Middle C" were a round 256 Hz, rather than the (approximately) 261.6 Hz of modern equal temperament based on A=440. 256 is a round number (at least to computer scientists) because it is one of the "powers of 2" (2 multiplied by itself 8 times). In binary (base-2) arithmetic, it would be written as 100000000. Readers paying attention with their ears as well as their eyes will have noticed that the Pythagorean natural scale example above is based on C=256.

440 v 432? One consequence of C=256 is that, using Pythagorean tuning, concert A has the frequency 432 (see it also in the magic square above). This has inspired an even more strange conspiracy theory, that the modern tuning standard of A=440 is a malefic influence, subversively introduced as a way of suppressing the more "sacred" frequency 432 and thereby somehow "detuning" humanity (how those directing this avoid being affected as well is not documented). The call has therefore gone out to all musicians to retune to A=432. There is the usual caveat here regarding precision; even A=440 is honoured more in the breach than in the observance, most of the time. I will simply note that, as it happens, the frequency C=528 of the original Solfeggio set is a perfect minor third (ratio 6:5 or 1.2) above 440. Sound them together and the difference (88 Hz) makes a low F resultant tone to form a perfect virtual F Major chord (example created using Audacity):

You will need to play this suficiently loudly (but not painfully!) to hear the resultant tone. The phenomenon depends only on the availability of a means to play those tones strongly and steadily enough (such as a pipe organ, or a skilled wind/brass player). As it happens, these tones also cohabit very happily and harmoniously with Solfeggio G=396 (perfect minor third with perfect fourth). 432 with 528 is by comparison "a bit off". So 440 Hz (despite the number having a zero in it) is perhaps not so bad after all. We can all sleep easy in the knowledge that all conspiracy theories about musical frequencies (including the electromagnetic ones) can be safely ignored, and that the variably and approximately tuned music we listen to will do us much more good than harm!