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Ancient Music - The Mesopotamian Tunings

Introduction

In the modern, Western world, music surrounds us every day – in our cars, on radio and television, in movies, dance clubs, and on our stereo systems. And although we say we like this or that kind of music, and we just can’t stand that other kind of music, all the music we are generally talking about is exactly the same in one fundamental respect: it is played on instruments that are tuned to exactly the same pitches. (Pitch is the perceived highness or lowness of a tone or musical note.) Thus, if you are a piano player, it doesn’t matter if you’re playing rock, ragtime or Bach, all the notes you play (in fact, all the notes you can play) come from a single set of eighty-eight possible pitches (since there are eighty-eight keys on a piano). When a band or an orchestra "tunes up," all the players insure that their respective instruments will play exactly the same pitches. In other words, when an instrument plays the note called A, it should produce the same pitch as all the other instruments playing an A.

But where did these all pitches come from in the first place? Are there other pitches that could be used, and if so, what would our music sound like? How would it be different if the strings inside a piano were tuned to different pitches? How would it feel? The thesis of this paper is that since all of our modern, Western music, whether acid rock or Chopin preludes, contains pitches that are disharmonious with one another (as I will explain shortly), the effect on us, the listeners, is also disharmonious. Since we hear with our entire bodies and not just our ears, we become physically immersed in the music we listen to in a very real way: our entire bodies are forced to resonate to the music we hear.

Whether music feels good to us or not depends on many things – tempo, loudness, genre, familiarity – but beneath all these lies harmony, the feeling evoked in our bodies by musical tones or pitches played together or in sequence. The external effect of musical harmony or disharmony on us is profound, and if we don’t notice this, it is only because we have never heard truly harmonious music since it is almost never played today in Western countries. One reason is, most of our modern musical instruments cannot play exact harmonies in the first place, due to their designs. As we will see, every fixed pitch instrument – pianos, guitars, clarinets, and so forth – are tuned to pitches that are inherently disharmonious. And variable-pitched instruments – violins, trombones, fretless-basses, and the human voice – are played to duplicate these same pitches so they are not perceived to be "out of tune" with the other instruments. But what would it sound like – what would it feel like – if all of the notes were harmonious?

In our exploration of the effects of harmony, we will be looking at why Western music is tuned the way it is (and there are some good reasons), and also at other types of music with different tunings. Finally, we will concentrate on seven almost unknown tunings that are over 5000 years old that came from (as far as we know) ancient Mesopotamia –that area of the Middle East that is now Iraq, Kuwait and Iran. We will not be interested in exploring modern Arabic music at all, but instead we will apply tunings that predate all the modern cultures of this area to Western instruments and melodies so as to discover the effects of their perfect harmonies on health and altered states of consciousness. We will begin with a tuning used in early classical music up to the time of Bach called Just Intonation, then move on to the modern Equal Temperament tuning used today, and finally discuss the Mesopotamian tunings. Along the way we will touch on the blues, Indian raga music and the sitar in particular. Always our emphasis will be on feeling, how music effects us consciously and unconsciously.

It has been known from ancient times (from very ancient times, indeed) that music and mathematics are intimately related. We will discuss the mathematical aspects of the various tunings – again emphasizing feeling – but we will relegate all but the simplest fractions to another page so as not to get too sidetracked. (Although if you ever wondered why the frets on a guitar are spaced the way they are, here will be the answer. Remember logarithms?)

Octaves and Such

Okay, I need two minutes to talk about octaves and notes and things before getting to Just Intonation.

If you strike Middle C on a piano, you hear a certain pitch. If you then strike the next C key higher up the keyboard, you hear a pitch with the same "quality" only higher. These two C notes define an octave, and this is the only interval between two notes on a piano (or any other instrument) that are exactly, perfectly in tune –they are perfectly harmonious. Likewise, if you pluck the first string on a guitar, you hear an E note. If you then hold down the string exactly halfway along its length – and there is a fret in exactly in this spot for just this purpose – you will sound another E one octave higher.

So what’s going on with these octaves? Is it just a coincidence that a string half as long produces a pitch exactly one octave higher than the original string? No. To keep matters simple (and to stay under my two minutes), I’ll just say that an octave is defined as the pitch difference between a plucked whole string and a half string with the same tension. In a piano, the string lengths, tensions and thickness’ are all balanced to produce octaves. When a piano tuner goes to work on your out-of-tune piano, the first thing he/she does is get the octaves right.

One way this is done (nowadays) is with a little electronic gadget that measures the frequency of a note. Whatever frequency Middle C is tuned to, the next C above it is tuned to a frequency exactly twice as large. This holds for any two notes that are one octave apart. But this is rather amazing: How can two notes that sound "the same" have numbers that are related by a factor of two? Why not 6 or 3.14 or something else? This is the first clue that music and mathematics are related. As we shall see, mathematics is indispensable for good musical vibes.

Just Intonation

Now, what about all the other notes between octaves? Why are there twelve notes in a piano octave? Why was that half-length guitar fret the twelfth fret?

We’ll find it convenient to skip over most of musical history for now, and begin in the seventeenth century, picking up what we need from the Greeks, some blues singers, and the likes of John Cage as we go along, eventually ending up back before the dawn of recorded history.

First, we need a simple way to write down the ratio of two notes. We’ll use the standard notation of writing a fraction, where the numerator is the relative frequency of the higher note, and the denominator that of the lower note. So the ratio of an octave will be written as "2/1." We use relative frequencies because we want only to capture how two notes are related to each other, not, as yet, any absolute pitch or its frequency. In a similar way, a unison, two notes sounding the same pitch, will be written as "1/1."

Now we want to assemble a sequence of fractions that represent the pitches of notes in an octave. Starting on the left, we will have 1/1, and on the right, 2/1. In between, we’ll need several other fractions for the intervening notes. In Europe in the early 1600’s, clavichords and harpsichords were tuned to a scale called Just Intonation, which had intervals with the following ratios:

1/1

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

9/5

15/8

2/1

C

C#

D

D#

E

F

F#

G

G#

A

A#

B

C

Figure 1. Just Intonation

The letters in the second row correspond to the notes on a piano (or guitar or flute). The pitch of these notes depends on only two things: one, the starting frequency of the first C (because all the other pitches are derived from this one), and two, the ratios above the letters. Mathematically, if the first C were tuned to, say, 500 Hz (Hertz or cycles per second), the E above it would have a frequency of 5/4 x 500 or 625 Hz. The last C, which actually starts the next octave, would be 1000 Hz. There are twelve pitches from C to the B above it, each pitch being given a note name. It is important to remember that the relative pitch comes first, and a note name is then assigned to it. There is nothing magical or mystical about an F; it doesn’t correspond to a color, a chakra, or a dimension any more than it corresponds to a fruit. It is the tone or pitch to which the note name refers that may or may not resonate with or correspond to some other feature of the universe.

Speaking of the universe, the twelve tone scale was originally a seven tone scale, attributed (in the historical record) to Pythagoras and later to Plato. Pythagoras felt that whole number ratios were universally important, and that musical scales reflected aspects of the universe in important ways. We won’t get into those ideas here, other than to say Plato’s Republic can be interpreted in a primarily musical way, expressing similar ideas.

Why were these whole number ratios used? Why couldn’t any old intervals be used? Because they sounded good; they felt good. Musically, such intervals are called consonant. A perfect fifth – the interval from C to G in Figure 1, for example – evokes a very pleasant sensation in our bodies. Other intervals, especially those that cannot be represented by the ratios of two integers, sound dissonant, unpleasing or at least unfinished. Composers use more or less dissonant harmonies in music today to heighten suspense or discomfort. (This is especially noticeable in movie soundtracks; If you don’t get the sense of danger or suspense from the visuals, you certainly will get it from the music.)

Remnants of the seven tone scale still exist today –they are the white keys on a piano, and are indicate by the non-sharped note names in Figure 1. The other five tones were added later, and in fact there was never common agreement about their tuning. There were several "Just" scales, which varied primarily in the intervals of the black keys. The exact ratios were selected by the composer to fit the needs or the mood of the piece being performed.

This brings us to the major problem of all Just scales: the intervals between successive notes were not all the same size. This meant, that if I were a composer performing an evening of my music, I would have my harpsichord or pianoforte tuned to the exact intervals needed for my first piece, say in the key of G. My harmonies would be exact for the first piece, giving the audience the experience I desired. However, if my next piece were written in the key of D, say, the instrument would have to be retuned to sound right in that key. This meant that the audience would take a break – the origin of the intermission – while the strings were all retuned. Well, this was a drag for everybody. And as music became more complex, composers wanted to write music in different keys during different portions of their compositions, and of course, this was impossible with Just tunings. What to do? Enter …

Equal Temperament and Dissonance

Skipping a lot of musical theory that we don’t need for our purposes, equal temperament was a way to open up performing and composing to whole new worlds of musical possibilities. First invented in the early sixteenth century or perhaps earlier, equally tempered tuning meant forcing the successive intervals between notes to be the same size, regardless of the loss of harmonic purity between the notes that this caused. Equal temperament did not become popular until J. S. Bach’s time (the first half of the seventeenth century) on the Continent, and was not officially adopted in England until the 1840’s.

In ET, the interval between each of the twelve notes in an octave is exactly one-twelfth of the octave. So, for example, the interval between C and C# has a ratio denoted by the twelfth root of 2. This number is approximately 1.05946…, where the … indicates there are indefinitely more digits. This is called an irrational number, and cannot be expressed as a fraction of two integers. In effect, each of the ratios in the Just tunings were changed to a greater or lesser extent away from their harmonically perfect values. Whereas the fifth in Just tuning was 1.5 (3/2), in ET the fifth has the value 1.498…, which is very slightly flatter than 1.5. The ET major third (the E in Figure 1, or 5/4 = 1.25) is 1.2599…, which is noticeably sharper than the Just major third. So instead of perfect harmonies, we have just the opposite: disharmonies between all notes except octaves.

These changes in tuning are generally regarded as being too subtle to be readily noticed in today’s music by any but trained performers (or listeners). But I contend that we all perceive the dissonances on some level, where the word "perceive" is to be taken in the widest possible sense. The reason for this is a phenomena called resonance. The cilia (tiny, fine hairs) in our inner ears each vibrate with and only with certain sound frequencies. But in the same way, other structures in our bodies (both physical and non-physical) resonate with sound frequencies, thereby making us very high-fidelity microphones, receiving all kinds of sound frequencies, not all of which are directly audible to our ears. (And it is because of resonance that we should not place microwave transmitters near our heads – certain molecules in our bodies resonate unhealthily with the frequencies cell phones put out. Like water molecules. But that’s another story.)

In order to represent ET in a table like Figure 1, we need a representation that avoids all those irrational numbers. Fortunately, one was invented in 1885 by musicologist Alexander Ellis. The interval between two successive notes in ET tuning is called a semitone, and Ellis proposed dividing each semitone into 100 cents. Thus, since an octave contains twelve semitones, there are 1200 cents in an octave. Now, ET can be represented as in Figure 2.

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

C

C#

D

D#

E

F

F#

G

G#

A

A#

B

C

Figure 2. The Equal Temperament Tuning

There are mathematical equations to convert between integer ratios and cents; see the Mesopotamian Temperament page for the details.

Other Experiments in Tunings

Not all music written and performed today is in equal temperament tuning. Most of what is called world or ethnic music – from Bali, China, Japan, Africa and India for example – are written in tunings unique to each culture. Here, we will concentrate on Indian music, and we will see how different tunings affect the audience (and the performers).

Indian ragas are based on selecting (usually) seven, eight or nine tones from a total of 22 possibilities within one octave. Figure 3 is one such tuning. One tone, usually the lowest, is played constantly and is called the drone. The other tones are played around the drone, giving the music its mood and rhythm. But what is most important is how a particular tuning is chosen.

1/1

9/8

32/27

4/3

3/2

8/5

16/9

2/1

Figure 3. The Indian Asavari Tuning

There are said to be over 5600 different Indian raga tunings. For a particular concert, one tuning is chosen in the following way: Consideration is given to the purpose of the concert; perhaps it is a religious celebration for a certain Indian saint. Also taken into consideration are the time of year, time of day, the weather conditions, and the mood the musicians wish to convey to the audience. These and other factors determine the tuning.

Next, and most important for us, the concert begins with the tuning up, which can last for 45 minutes or more. Its purpose is to attune the players and the audience to the "mood" of the concert. In the West we might say a certain quality of resonance – both physically and in more subtle ways – is set up in the environment of the concert, so that when the actual performance begins, the music can be fully appreciated from the beginning. Part of this "appreciation" manifests in a subtle, or even dramatic, alteration in consciousness of all in attendance. Not only does this get the musicians "in the groove" so to speak, it allows the audience to follow where the music leads them. Physically, the Indian sitar has moveable frets, unlike a guitar. The frets are moved to produce the exact tones desired; the frets are then fixed in position for the duration of the concert.

Here in the West, the most famous performance of Indian music was Ravi Shankar’s opening music for the Concert for Bangla Desh in 1971 in Madison Square Garden. Before beginning, he asked the audience to sit quietly, not to smoke, and to be respectful of the sacred music they were about to play. The reason for this is now easy to see: sacred means being able to change conscious awareness in deeply profound ways that we are generally unaware of in the West. Music – any music – can create an inner holodeck for us, where we can experience other realities, other levels of awareness. India, with many thousands of years of musical development, has this skill down pat. For the rest of us, since most of the music we listen to only has the intent to entertain (at best), we remain in the dark about the hidden possibilities music possesses. There are rare exceptions – notably the music of Mozart, where recent research has indicated that his music has the power to relax, calm, even heal.

Even here in the West, there is some music that just isn’t suited to straight equal temperament. In blues music, there are certain tones that involve ratios containing the number seven (derived from African scales). One of these tones is the blue third, with an interval of 7/6, or 1.1667. This is equivalent to 267 cents, or between the note names D and D# in Figure 2. The only way to reproduce an approximation of this tone on a piano is to hit two or three keys together, hoping that the right tone will be "in there" somewhere.

There is also no rule (nowadays, at any rate) that a musical scale must have exactly twelve tones. Some early keyboards in the seventeenth century had multiple black keys between the white keys, each tuned to a slightly different pitch; the musician would use the key that sounded "best" for the piece at hand, meaning "most in tune." More recently, John Cage started a movement in the 1940’s by putting literally dozens of pitches in the space of one octave. Many experiments, and many bizarre instruments, have been created in order to plumb the depths of musical tunings.

The Mesopotamian Tunings

Several years ago, I ran across seven tunings (or modes) in a computer program called The World Music Menu written by Steven Nachmanovitch. They were grouped together under the heading "Mesopotamian," and were found by Lou Harrison on a Old Babylonian cuneiform tablet in the British Museum. This program, when connected to an electronic synthesizer, would automatically retune it to any of several dozen tunings Steven had collected from around the world. In fact, without designing custom instruments that can be tuned to play different scales, only an electronic synth can play all these scales.

(I recently found a recording of Lou Harrison talking about the Mesopotamian tunings in a 27 minute radio interview broadcast on KPFA in Berkeley, California in 1971. It is on the radiOM.org site. You'll have to register on the site, but it's free and they don't bother you. Lou plays the tunings on a recreation of a Sumerian harp, and demonstrates one way of retuning it from one mode to the next. Hats off to the radiOM folks for preserving old bits of arcana like this!)

I had taken my keyboard to Sedona to visit some friends. Before leaving home, I had programmed the keyboard to reproduce these seven tunings. Once there, I began improvising, a few minutes on each tuning in no particular order. We were doing an informal therapy session on a friend, when suddenly she began to experience a sharp pain in her right leg. I stopped playing after a few minutes, the pain was dealt with, and we began to discuss what had happened.

It seemed that the music had evoked an emotional childhood memory, constellated in that person’s leg. Everyone was surprised that simple music, played only for a short time could be so evocative. I played through the seven scales later that day and the next, and we were careful to notice what mental, physical and emotional reactions everyone felt. Each tuning seemed to evoke something, but that something was different for each person.

When I returned home a few days later, I again looked at the mathematical ratios of the tunings. I discovered that there was a natural order of the scales that was different than that printed in the computer manual. I was able to perform for several more people and then several groups of people during the next few weeks. In most of the cases, I asked those present to be aware of what they were experiencing while the music was playing. There were similarities and differences. No two people had the same experience, or even the same type of experience. However, without exception, their experiences came to an end precisely the same time I finished the last piece.

I had decided that the tunings should be played in spiral fashion. That is, I played an improvised composition in each of the seven tunings in the order I had discovered, then played through the tunings in the reverse order, ending where I had begun with the first tunings. This made thirteen pieces, played in the order 1-2-3-4-5-6-7-6-5-4-3-2-1. Each time I played, the entire cycle took about an hour.

Once I had recorded the music, I began doing journey work with people individually which I called Sonic Repatterning Therapy or SRT. For the most part, this consisted of some initial energy balancing as the recorded music got underway, then a period of silence to let the music open an internal journey for the client, and finally a series of questions to the client, such as, "What are you experiencing now?" and "What do you see now?"

My conclusions at that point were that the particular harmonies in these tunings, played in a certain order, relax the body, the mind, and indeed the whole self, in such a way that the inner Self can be heard, or at least sensed. These sensations are variously visual, auditory, even kinesthetic. It is also relatively easy for the therapist to be an escort for the client; in other words, the therapist can share and even anticipate the experiences the client is having, and thereby act as a guide and assistant to help the client understand the experience.

There is one other common thread I have witnessed: in nearly every case, whatever specific experience was evoked in the listeners, it related to some past or present event in the person’s life, or illuminated some choice the person was in the process of making for the future. Several people described the experience as similar to being shown an inner movie or a series of scenes that helped them understand and often release a past experience so they could get on with their lives.

The Mesopotamian Tuning Ratios

The seven Mesopotamian octave tunings as I found them are shown in Table 6. As you can see, each of the seven notes in the scale are represented by a fraction – a ratio of two integers that result in ascending numbers between one and two. Each tuning has a different set of ratios, which give it a characteristic sound and "feel".

For several years I just composed music in these tunings, without being overly concerned with where these particular fractions came from. I also just took it for granted that there were only seven notes in each tuning, and restricted myself essentially to playing only the white notes on the keyboard. This restricted the harmonic complexity of the music – which was my goal, although the music was still rich and varied.

As an aside, the process of retuning my electronic keyboard (a Kurzweil 2500S) turned out to be fairly simple, after I developed some basic mathematical formulas for doing so. (These formulas are apparently well-known, but I didn’t have access to them at the time.) The problem was that the Kurzweil only knew about Equal Temperament semitones, which could be adjusted by cents, whereas I was starting with a sequence of fractional ratios. How to convert ratios to cents, that I could then program onto the keyboard?

Skipping over the derivation, given any ratio, the corresponding value in cents is given by (cover your eyes if you don't dig logarithms)

So, for example, using a ratio of 3/2 or 1.500 yields 701.9 cents. From Figure 2, the fifth, which is G, corresponds to 700 cents in the ET tuning. To produce an exact fifth, the G needs to be raised by 1.9 rounded to 2 cents – an easy adjustment on the Kurzweil. In this way, I programmed each of the seven tunings and stored them for easy recall.

It is also possible to adjust the entire keyboard either sharper or flatter. The zero adjustment corresponds to the concert pitch (at least here in the U.S.) of A below middle C of 440 Hertz, meaning that all instruments in the orchestra are tuned to this one pitch, so that the violin in the third chair is in tune with the piano and the clarinets. This seemed quite arbitrary to me. In fact, in France, the tuning is set to A 435, and has varied in different places over recent decades. As I was attempting to recreate the original effects of these tunings (without, of course, having any idea of what kind of instruments the ancient Sumerians used), I wanted to at least get the "concert" pitch right.

But what in the world, literally, could I use as a reference pitch? The Sumerian civilization is the oldest anywhere on Earth we have any detailed record of; it was in full flower by at least 3500 BC. What could possibly be contemporary with them? At first I settled on the following, possibly apocryphal story: I had heard that many years ago, the musician and composer Paul Winter had gone into the Great Pyramid in Egypt and had struck the stone "sarcophagus" in the King’s Chamber and measured the frequency at which it resonated. The result was 440 Hz, the same as the modern-day concert tuning. This was as good as I was going to get, I thought, since depending on whose dating you believe, the Pyramid was nearly as old or even older than the Sumerians. So I left the master tuning page on the Kurzweil as it was originally set, and played the music based at A 440.

It was not until much later that I discovered that the motions of the Earth have certain frequencies associated with them. For example, the 24 hour day, the sidereal year and the precessional year all have associated frequencies. (There are also frequencies associated with the moon, the other planets and the sun.) I will discuss this more fully in a later section. Suffice it to say here that this discovery caused me to adjust the keyboard to produce these frequencies in my music.

Early on in my exploration of these tunings, I asked myself how the seven tunings are related to each other. First of all, the different fractions appeared, usually, in several of the tunings, so there seemed to be some underlying pattern. One thing I noticed was that each fractional numerator and denominator was a power of 2 or 3. For example, in Qablitum, 256 is 2^8 (2 raised to the 8th power) and 243 is 3^5. Further on, 1024 is 2^10 and 729 is 3^6. This discovery was rather amazing. I had wondered why the Sumerian tunings were just these fractions; why not, say, the Just Intonation fractions, or some others? Here was a hint, although I didn’t understand it. Two and three are prime numbers; all the fractions in all seven tunings are powers of these two prime numbers. Was this important at all? Not being a believer in coincidences, especially numeric ones, I thought it was important in some way.

When I got to Nis gabri, I found the number 721 which is neither a power of 2 or 3. Since it was the only such number in all seven tunings, I assumed it was a misprint, and subsequently used 729 instead. This adjustment was corroborated in what I next found.

Since all the tunings were comprised of a common set of fractions, I wanted to find out if and how they might be related. I had assigned each tuning a musical key. When the keyboard was tuned to Nis gabri, for example, I played in the key of G; for Ishartum I played in A. I had made these assignments based on how the music sounded and felt. But what would happen, I asked, if I transposed one of the tunings into a different key? I started with Ishartum. I could derive a new tuning in B by taking the second note of the scale in A, which is B, and making its ratio 1/1 instead of 256/243. By doing this eight times, I would have a new scale that ran from B to the next B one octave above it. To convert 256/243 to 1/1 you multiply it by its reciprocal, 243/256. So I multiplied each fraction in the Ishartum tuning by 243/256, and arbitrarily tacked 2/1 to the end to complete the octave.

To my surprise, the result was the exact ratios of the Nis gabri tuning! What was going on here? I next took Embulum and multiplied its ratios by 8/9. I got Ishartum. I found that each of the tunings could be turned into one of the other tunings. There was some king of reciprocality among the tunings. It would be two years before I discovered why this was so.

The Circle of Fifths

After the octave, the perfect fifth is the most common interval in modern music. (If you sing the first two notes of "Amazing Grace", this interval is a perfect fifth. The first and third notes of the spiritual "Kumbaya" are also a fifth.) Nearly every pop, folk and rock song ever written features notes or chords that have the relation of a fifth.

The exact frequency relation between the key or tonic note and its fifth is 3/2 or 1.5. If one starts with C, its fifth is the G above it (on a piano). But one can then take the G and find its fifth, which is the next higher D with a frequency 1.5 times higher than the G. If one continues in this manner 12 times, each note name on the piano will be encountered once. Taking one more fifth returns one to C where we started. (Well, almost. There is the matter of the so-called "comma", because this raising by fifths doesn’t come out quite even. I’ll get back to the comma below.)

In modern music theory, the repeated raising to the next fifth is called the Circle of Fifths, and is used to show the written sharps and flats in the key of the tonic or root note. For example, music written in the key of C has no sharps or flats. One fifth up, the key of G has one sharp, the key of D has two sharps and so on. The circle ends with Bb with two flats (after five sharps, we switch to flats), F with one flat, then back to C with none.

While working on another musical project, I wanted to know what the different frequencies would be starting arbitrarily with F and going once around the circle of fifths. A fifth above F is C, and a fifth above C is G. I had to stop here, because suddenly I was in the octave above the F where I had started, and I desired frequencies in the same octave. This was a simple matter, though, because to get back to the original octave, I just had to divide the frequency of the G by 2. (Remember the octave relation was 2/1 – the octave note is exactly twice the frequency of the starting note.) So I continued along, reducing octaves a total of seven times before I got back to C, one full circuit around the fifth circle.

There were two things I noted here. One was common musical knowledge – the comma. The other was completely astounding to me, and I was right back in Mesopotamia – ancient Sumer. Let’s take the comma first.

I had started with an F frequency of 172.06 Hz. After multiplying the fifth ratio of 1.5 thirteen times, and halving the results as necessary to stay in the same octave, I got back to an F, but with a frequency of 174.40. I was sharp by over 2 Hertz from the original 172.06.

Essentially what I had done was to multiply and arbitrary frequency F by 1.5 twelve times, and divide by 2 seven times along the way. If you do this on a pocket calculator, you get

This number is called the Pythagorean Comma. Starting with any harmonic interval such as a fifth always results in a comma. Various ways of getting around the comma have been devised; one of them was the Equal Temperament tuning. But this was exactly what I was trying to get away from: My goal was to produce music that was as harmonically pure as I could make it, and I was determined to stick to these ancient ratios. Fortunately, the Kurzweil handles alternate tunings octave by octave; that is, the octave relationship always remains exactly the same; the keys within each octave are retuned separately, eliminating any sensible effects of the comma. (There is much more that could be said about the comma – for example, does its existence imply that an octave should in some sense be slightly less than 2 /1? – but I feel this would take us too far afield here.)

The second outcome of this circle of fifths exercise was a complete surprise. After I had written all the frequencies down within a single octave, and rearranged them in ascending order – A, A#, B, C, C#, D, D#, E, F, F#, G, G# – I decided to calculate the ratios between the lowest A and each of the others. The first ratio was 1.0678. This was close to 1.0667, which is 16/15, a ratio in the Just Intonation tuning. Maybe, I thought, this tuning is what the circle of fifths reduces to. Although close, it was not exact. So I continued.

The second ratio was 1.125, which is 9/8. This is indeed the interval of a major second exactly in the Just tuning. The third interval was 1.2013, again missing the expected Just ratio of 1.200, which is 6/5. Two hits and two misses. When I finished, all the ratios missed the expected Just ratios by a small amount except the fourth, 1.333, and the fifth, 1.5. It looked like a mish-mash. To simplify things, I looked only at the six un-sharped notes, looking for some kind of pattern.

Suddenly, I saw many – no, all – of the ratios of Nid murub! I had found out where the Mesopotamian tuning ratios had come from! Nid murub was derived from the circle of fifths, reduced down to one octave, just as I had done. And the ratios weren’t just approximate, as the Just Intonation ratios had been; they were exact to four decimal places. But I had found something else. The original tunings I had found in a book were only seven-note scales. What I had just derived were twelve-note scales; one relative pitch for each of the white piano keys and for each of the black keys, too.

How had the Sumerians, the earliest and most advanced of the early civilizations in Mesopotamia, come by these ratios? The exactness of the ratios, 1024/729 for example, show a mathematical sophistication that we have not had ourselves for more than a hundred years. This knowledge is as amazing as their astronomical knowledge. For it was the Sumerians (or their antecedant civilization) who gave us the 24 hour day, divided into 60 minutes per hour and 60 seconds per minute. They also knew that one year was 365.25 days, and astoundingly, that the precessional year was 25,920 years. No one knows the answers to these questions. Some, like Zacheria Sitchen, have made guesses. The question is still open in my mind, but there is much we don’t know about those times, 5500 years and more in the past, and we know virtually nothing about earlier times.

Now a few words must be said about the Just Intonation tuning that was so close to the frequencies calculated from the circle of fifths. Pythagoras is generally credited with discovering the principles of note ratios that are small, whole-numbered fractions. In fact, there is a tuning called Pythagoras’ Lydian that is the seven-note equivalent of the twelve-note "Basic" Just Intonation. The ratios are: 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1. Now the Sumerian civilization predates the Greek one by almost 2000 years. It is my belief that the earlier knowledge was mostly lost; and the mathematical facility of the Sumerians was certainly lost. What Pythagoras discovered, or rediscovered, was an approximation of what the Sumerians knew two millennia earlier. The ratio 5/3 was substituted for 128/81 because it was easier to represent and deal with the former number than the latter. (There were other reasons: 5 was a crucial and magical number for Pythagoras, but it would take us too far afield to pursue them further here.) So Pythagoras’ tuning was the best approximation he could make to the earlier, and I believe, more accurate tunings of the Sumerians.

The "Greek" modes

There are seven Mesopotamian tunings. By 500 BC, there were seven "Greek" modes, another word for a tuning. I put the word "Greek" in quotes because they are approximations of the earlier Sumerian tunings. The correspondences are shown in Table 3.

Ionian

Nid Murub

Dorian

Embulum

Phygrian

Ishartum

Lydian

Nis Gabri

Lesbian (Mixolydian)

Mitum

Aeolian

Kitmun

Locrian

Qablitum

Table 3.

This table was derived by noting similar interval sizes: major and minor seconds and thirds. The two sets of tunings correspond exactly, including the awkward tritone (an F# in the key of C) in the Lydian and Nis gabri tunings. In fact, the Greek modes are EXACTLY the same as the Mesopotamian ones if the differing intervals are shifted by the Pythagorean Comma! The reason for this is easy to see. The Greek modes used the interval of a third, 5/4, which the Mesopotamian tunings do not have. Much scholarly arguing exists about the supposed advantages thirds give to the richness of music composed in the Greek modes; Nearly every music theory textbook tackles this problem. In reality, though, the two sets of modes have nearly identical intervals. I have not done experiments, but I'll wager the effects each has on us subjectively will also be nearly identical. I'll still side with the Sumerians and Plato, and stick with the "purer" intervals generated by the Mesopotamian Circle of Fifths.

For more on the Mesopotamian tunings, continue on to the Mesopotamian Temperament page.

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