Where did the specific ratios in the Mesopotamian tunings come from? Why these numbers and no others? After playing with these numbers for a while, I realized that all the ratios in all seven of the tunings are based on just one initial ratio: three over two, or 1.5. This is called a fifth interval, and is the entire basis for the seven Mesopotamian tunings.
How could this be? Can the simple fifth account for the ratios 729/512 and 1024/729? It turns out that it can, and that's what we'll discuss here. (How the Sumerians, a supposedly primitive people, came up with this scheme is discussed elsewhere.)
We will start arbitrarily with a frequency of 400 Hertz. If this is done in a spreadsheet, any starting frequency can be used. We use a nice round number to illustrate how quickly this roundness disappears. And we will also arbitrarily call this note C, since on a modern piano, the major scale in C can be played on the white keys, thus starting as simply as possible. But again, this is an arbitrary selection; any note name could be used.
The idea is to multiply each frequency by 3/2 or 1.500, thus determining another note of a scale. Since there are seven white notes in an octave (before then next C is reached), we will repeat this multiplication seven times.
So 400 x 1.5 is 600.00 Hertz. (We will print two decimal places, but will calculate with at least six internally to avoid round-off errors.) What note does 450.00 represent? Well, since the interval of 1.500 is a fifth, we can call this new note the fifth in the scale we are constructing, and the name of the fifth in the C scale is G. This is a well-known interval in music, and is technically called the perfect fifth, although as we already know, in the common Equal Temperament tuning used in the West, the fifth is off and is certainly not perfect. (It is actually 599.32, a little flat.)
Next note. 600 times 1.5 is 900 Hertz, but now we need to examine our method and make a revision. Our goal is to determine the frequencies of one octave - the white notes - from one C to the next. Since two notes that are one octave apart have frequencies that differ by a factor of two, our octave runs from 400 Hertz to 800 Hertz, but here we are, already at 900, and we're in the next higher octave. Clearly we're going to increase frequencies rather quickly.
The solution is to halve any number over 800 Hertz, thus stepping back down to the original octave. We lose nothing in this process; we are merely collapsing the series of increasing fifths into a single octave, and if we're in any doubt at all about doing this, we shall see that when we get done, this is exactly what the Sumerians did 5000 years ago. So let us proceed.
We will take 900 Hz., divide by two, and give the third note the frequency 450 Hz, and the name D, since D is a fifth above G (which was a fifth above C, where we started). So this is our program: multiply by 1.5 and divide by 2 whenever the result of the multiplication jumps up into the next octave. Table 1 shows the results for twelve such steps.
Well, the round numbers disappeared pretty quickly, and so far there are no signs of the simple fractions we saw earlier. Furthermore, we have taken twelve steps instead of seven. Why is that? But what about that last number? The notation C' means the C one octave higher that where we started. But we know that octaves differ by a factor of two, so how come C' isn't 800 Hertz?
Dividing 810.91 by 2 yields 405.46 (rounding internally, remember). If we want to compare this to 400, it will provide best to divide the numbers, as we are after universal ratios that are not tied to any particular or arbitrary frequencies. Doing this gives the number 1.013643 to six decimals. This is a well-known number in music theory, and is called the Pythagorean Comma. A musical comma, and there are several of them, means something cut off, a minute difference, even a fudge factor. As we have defined it here, it is the difference between an octave calculated by successive fifths and an octave calculated by multiplying by two.
I like to remember the words of songwriter extraordinaire Leonard Cohen: "There's a crack in everything / That's how the light gets in." Certain things don't come out even in Nature, and just as pi isn't exactly three, twelve perfect fifths in a row don't add up to a perfect octave. No one knows why the comma exists or why it has the value it does. But the real problem is what to do with it.
What we have actually done here is stack together twelve fifths, which comprise just a bit more than seven octaves - and that "bit" is the Pythagorean Comma. But in reducing the notes that these fifths make into a single octave, and then duplicating that octave across a piano keyboard, say, that "bit" appears in each octave. Since it is quite audible, it will have to be incorporated into at least one of the twelve notes.
In Equal Temperament, this problem is solved by dividing the octave in twelve equal steps. The comma is then absorbed into each note equally. This is done by making each fifth just a bit flat. Out of tune on purpose, in other words. However, this has definite harmonic disadvantages in that the disharmonies affect us in subtle, though major, ways, which was the whole point of looking for different tunings in the first place. We will find that the Sumerians had a very elegant, natural way of accounting for the Comma.
An octave is an octave is an octave
Before we get back to the Mesopotamian tunings, we will pose a question regarding this Comma. A way of visualizing twelve fifths stacked together is in a circle, and there's a name for this diagram: the Circle of Fifths.
Here again are the twelve notes, starting with F at the top this time. But if we start with G, one starting place is as good as another, we have the same notes as above. And in fact this is not a true circle, but a spiral, because the twelfth fifth, G', is not exactly seven octaves above the G we started on.
Where did we get our idea that an octave is an exact doubling of frequency? First, it sounds pretty right; we sense that two notes sounded a frequency-doubled octave apart sound like the same note, only one is higher than the other. No other nearby pitches produce this effect. Second, two is just too beautiful a number not to be the exact octave multiplier. And yet, if the octave was defined as 2.0273. instead of 2.0000 - a barely audible difference to a trained ear - the circle of fifths would come out exactly right. The number 2.0273. is defined as exactly twice the Pythagorean Comma. No one has played modern music with such an augmented octave to my knowledge. It would be an interesting experiment.
Finding Nis Gabri
Back to the twelve frequencies we constructed in Table 1. What we would like is to find some relationships among all these frequencies; after all, the numbers are quite arbitrary, because we could have started with any number besides 400. So let's look at the ratios involved between different pairs of numbers. It will be useful to start with how each successive number relates to the starting number.
First, though, we need to sort the table according to ascending frequencies. This we have done in Table 2. The notes now correspond to a piano keyboard or guitar fretboard.
For the first interval, 427.15 divided by 400.00 is 1.0679. (All our calculations will be rounded to four decimal places.) That number doesn't mean much, so let's go on the next one. 450 divided by 400 is 1.1250, which converts easily to the fraction 9/8. Hmmmm. Riding a hunch, we calculate the ratios for the unsharped notes from C to B, and find the fractions they represent. (The fractional values are exact, not rounded.)
We pause to note two things here.
- These seven fractions match one of the Mesopotamian tunings as found on the original Sumerian clay tablets. This tuning is Nis Gabri. The five sharped notes do not appear, leading me to believe the Sumerians used a seven-note scale, as opposed to our twelve-note scale. However, there is clear evidence from the numbers, as we shall see, that they knew about the other notes. In fact, these seven fractions depend on their existence, and knowledge of their existence.
- The octave as written, using 810.91 Hertz, doesn't come out to be 2/1 as we have seen, so we're going to have to face the issue of the Comma right now. Just as a note, though, the ratio associated with this frequency is 531441/262144. This number is exactly twice the Pythagorean Comma.
Regarding the octave now, this tuning was actually written starting with 1/1 and ending with 2/1, so we know the Sumerians used double-frequency octaves, so we will do the same. We will define C' as 800 Hertz and write its fraction as 2/1. We will see that this seeming sleight of hand is the necessary and correct step, and that the Comma is still "in there." What we are doing with this step is defining one octave that we can replicate up and down the keyboard, instead of having to deal with octaves that are slightly different from each other by the Comma amount. In our defense, this is how musical instruments would be tuned by ear, either by ourselves or by the Sumerians.
Table 4 shows our progress so far.
The new column on the right shows the intervals between the fractions or ratios. These numbers are the ratios of the ratios, so to speak. I wanted to see if the fractions had any pattern or regularity, and I found that they did. Each interval fraction was found by dividing the original fraction in the same row by the fraction above it. So 81/64 divided by 9/8 is 9/8 again. (If you're calculating along, it's easier to divide 1.2656 by 1.1250. If you're fractionating, remember that to divide by a fraction, you invert the divisor and multiply. 81/64 times 8/9 is 9/8.) Another example is 3/2 divided by 729/512 equals 256/243.
What struck me, and why I bring these seeming details up at all, is the pattern. If we denote 9/8 by A and 256/243 by B, then the pattern of intervals is
A A A B A A B
This is only noteworthy when it is compared with the patterns in the other six tunings. All the patterns are the same, except that the pattern is rotated by one or more letters. For instance, the pattern of intervals for Mitum is
A A B A A B A
The first A has been chopped off and stuck on the end. The seven patterns are shown in Table 5. Each row has five A's and two B's, and the B's form nice diagonals down the table.
|Ishartum||B A A A B A A|
|Embulum||A B A A A B A|
|Nid Murub||A A B A A A B|
|Qablitum||B A A B A A A|
|Kitmun||A B A A B A A|
|Mitum||A A B A A B A|
|Nis Gabri||A A A B A A B|
The ratios for all seven tunings are summarized in. Although some of the fractions look quite odd, the patterns still show only two distinct intervals between notes, which we are calling A and B. It's time we looked at where the fractions came from.
We know that the seven fractions in Nis Gabri come directly from the Circle of Fifths, because we went through the steps above. Just as I wanted to find patterns in the intervals between notes, I also wanted to find any patterns between tunings. Playing around, I happened to multiply all the ratios of one tuning by the inverse of the second fraction. Boy, did I get a surprise!
Try it yourself. Take Nis Gabri as a starting point. Multiply each fraction by 8/9. To make a whole octave, you'll have to drop off the starting 1/1 fraction and add a new 2/1 at the end to complete your octave.
So 9/8 times 8/9 is 1/1, keeping similar notation.
Next, 8/9 times 81/64 is 9/8, the same as we had.
Once more, 8/9 times 729/512 is 81/64. And again, 8/9 times 3/2 is 4/3. Here's a new fraction that doesn't exist in Nis Gabri.
8/9 times 27/16 is 3/2.
8/9 times 243/128 is 27/16.
Lastly, 8/9 times 2/1 is 16/9.
We tack on a 2/1 to complete the octave, and what have we got?
We've got Mitum! We diddled the numbers so that a new scale started on the second note of our first scale. All we did was simplify fractions, dividing through by 9/8 (or multiplying through by 8/9). I couldn't believe my eyes, so I repeated the whole process on Mitum, and got Kitmun. The pattern holds true for all seven tunings. You read up each row in Table 6, and when you get to Ishartum, just cycle back down to Nis Gabri and start again. Each tuning is related to each other tuning according to some underlying schema that I didn't see yet. All I knew was the tunings turned into each other in the simplest manner possible.
Orbits, cycles and the missing notes
Think of a huge Hula Hoop that somehow got bent out of shape so that it's no longer round. At one point on this hoop is attached a bracelet with seven charms. At another point nearby, there is another bracelet, which has some of the same charms and some new ones. As we progress around the hoop, we find seven bracelets, each of which has seven charms attached, but they are never the same seven - they seem to be picked from a pool, some often, some only once.
The bracelets, of course, represent the seven tunings, and the charms represent the ratios or fractions in each tuning. The rule for selecting the charms to use says start with the single interval 3/2 or 1.5, then pick seven notes. This determines the first set of charms. The other sets are determined by multiplying through by the inverse of any one ratio; this immediately jumps you to another bracelet and determines its charms.
But what does the Hula Hoop represent? What is the grand pattern that determines what the bracelets are, and even how many of them there are. To do this, we have to go back and fill in the missing notes.
Table 4 is repeated below.
Nis Gabri. Table 4.
We more or less arbitrarily skipped over the notes marked with a # sharp, so as to correspond with the "white notes" on a piano. And initially, there seemed to be no reason to go further. Eventually, though, I got thinking about the Circle of Fifths. There are twelve steps, and there are twelve notes in the modern chromatic scale - both the white and black notes. So I continued multiplying by 3/2, and finally calculated all the ratios and all the fractions, as shown in Table 7.
The Twelve Notes of Nis Gabri. Table 7.
The fractions get ugly fast, as no amount of multiplying powers of three in the numerator and twos in the denominator will ever reduce to a simple fraction. (Oh, yeah. Did you notice? All the numbers in all the fractions in all the tunings are a power of two or three.)
But here I was anyway, again looking for patterns. Lo and behold, here's another one! And this is the Queen of them all, hiding the lost Pythagorean Comma, no less.
The first interval, that between C and C#, is 2187/2048. Expressed as a decimal this is 1.0679 (rounded). Call this number I.
The next interval is between C# and D, and is 1.0535 (rounded). Expressed as a fraction, this is 256/243. Call this number J.
But the interval between D and D# is again I. In fact, all twelve of the intervals are either I or J. I occurs 5 times; J occurs 7 times. And as a little math check to make sure we haven't made a mistake somewhere,
I5 * J7 = 2.0000
In other words, multiplying all the intervals together should equal one octave exactly, which they do.
The master tuning
Pattern Man was instantly interested yet again. And guess what I found? If you divide interval I by interval J you get the Pythagorean Comma.
(In Table 7, if you switch the very last interval J for an I, you get a C' that represents the original 810.91 Hertz from Table 1.)
The Comma is hidden in this twelve note tuning exactly seven times, which is the same number of times we had to reduce the octave way back in Table 1.
The twelve ratios in Table 7 represent the Master Mesopotamian tuning, because from it the original seven can be derived by just discarding the "black notes." But if you have looked closely at Table 7, you may have noticed something fishy. The ratio on the F# row originally was on the F row. I had to move it down one, because one of the new intervals displaced it. So we have an F# in our scale, instead of an F.
Any of you musicians see what's happened? In a twelve-tone octave, we are playing in the key of G, which has one sharp - F#.
In fact - and perhaps you see it coming - this master tuning can be turned into 11 other tunings by the same process of multiplying through by the inverse of any fraction, yielding "scales" in every key. So what we have been calling "white keys" and "black keys" are completely arbitrary and completely interchangeable.
So you can play these tunings as either seven-note scales or twelve-note scales. The music on Memories of Home is all played with seven-note scales, except "Trinity and a Soul," which uses two of the "sharped" notes.
Equal Temperament has been called Equal Tampering, because the original harmonic intervals were "adjusted" out of tune. True, this was done for good reasons (which are described elsewhere), but the advantages of the originals was forgotten, as nearly was the fact that a "readjustment" was applied at all. Only music teachers, students and textbook writers seem to remember this; the general musician does not.
We shall call this master twelve-tone tuning Mesopotamian Temperament, MT for short. It has many of the advantages of Equal Temperament while preserving the subtle, but important, harmonies that ET has completely lost. You can pick a particular tuning to express a certain mood or purpose, and you can modulate from one key to another while still retaining exact harmonic relationships.
How do you tune a modern-day instrument to play these tunings? To visualize the problem, consider a guitar fretboard. The equal intervals of ET are expressed by the uniform spacing between frets. (The steadily decreasing fret spacing from bridge to nut is an artifact of logarithms - don't worry, we're not going to get into it.) To play Kitmun, say, the frets would have to be spaced differently. To play Ishartum, they would have to be spaced differently yet. A piano could play the tunings, but would have to be re-tuned first; all the notes, except one set of octaves, are different.
We are fortunate today to have electronic synthesizers, because many of these can be tuned to scales other than Equal Temperament. But - hang on - how did the Sumerians tune their instruments, supposing they didn't have synthesizers? One way is to build a stringed instrument like a modern sitar. The frets are moveable, and can be locked down to produce only certain intervals. Or a stringed instrument could be used that didn't have frets, like a violin or fretless bass for the low notes. And it would be easy to build seven different woodwind or flute-like instruments, each with different hole spacing. Or an instrument like Pan pipes could be used, where individual pipes of the correct length were selected to make up one tuning. No one knows how they did it, as no pictures of their musical instruments seem to have survived.
Whatever they used, it was easier for me to use my trusty synth.
As it is not possible to just dial in fractional ratios, at least in Kurzweil synths like mine, a little math was required to get from these ratios to the semitones and cents the synth can handle - an attribute of ET. An ET octave has twelve notes or semitones, equally spaced. Each semitone is further broken down into 100 steps called cents, much as there are 100 cents in a dollar. So what we need is a formula that converts a decimal ratio (which is easier to use than all those fractions) into the equivalent number of semitones and cents. Then the departure from the standard number of ET cents can just be dialed in and saved for future playing.
Without any of the gory details, here's the formula:
where ratio is the desired MT ratio for each note. For example, to use Nis Gabri again, the first ratio is 9/8 or 1.1250. Plugging this in gives 204 cents. On my synth this process must be done in the key of C (which can be adjusted to give the correct intonation later). So the first note, C, is 0 cents. The second note, C#, is 100 cents, but we're skipping this note since Nis Gabri is a seven-toned scale. The next note is D, which is 200 cents in ET, but now our formula calls for 204 cents, so the adjustment +4 is given. I just enter the number 4 in the slot for the note D.
The other intervals are entered in the same way, as well as the other six tunings. Each tuning can be saved separately, and affects the entire keyboard whenever it is called up. Voila.
is the master Mesopotamian Temperament. The cents adjustments are entered in the same way, but this time using both the black and white keys.
Different musical tunings produce different effects on us humans. Since most of us have only heard music played in the ubiquitous Equal Temperament tuning, we really have no idea what these effects might be, other than the "normal" ones: certain, mild emotional "tendencies," and certain tastes and distastes we have developed based on treatment and genre. And of course the musical milieu in which we grew up. The point that music has other, stronger effects is missed by most of us because our experiences don't include other tunings. It's as if we were asked to comment on the color red, having only seen greens, blues and grays all our lives.
Equal Tempered means Equal Tampered. Consonant intervals were substituted with dissonant ones, and the subjective effects ancient music had were destroyed.
The earliest known musical tunings are attributed to Pythagoras in the 6th century BC. However, he is known to have learned what he knew in Egypt, before establishing his Academy in Greece. There is very little in the Egyptian archeological record about music. We have to go farther back yet to their antecedents, the Sumerians, which is the only known, mature civilization to predate them. (Unless it was the early civilizations of the Indus Valley, where the science and mathematics of music has come down to us in a very developed form, but there is as yet no firm descent from the Indus Vally to either Mesopotamia or Egypt, so all we can do is speculate.)
So it is reasonable to assume that what Pythagoras taught was actually the Mesopotamian Temperament, although possibly in a simplified form. For if the mathematical knowledge behind exact ratios such as 729/512 was lost, as it appears to have been with the Greeks, approximate substitutions would have been made. Indeed the seven Greek modes, as their tunings are called, are each a parallel of one of the seven Mesopotamian tunings.
Indeed, it was with Pythagoras that musical thirds and sixths were introduced, which have no counterpart in these tunings. They are appealing because they have small integer fractions, such as 5/3, but they introduce a different set of harmonies, and are therefore not as pure as the Mesopotamian tunings for certain purposes.
In the process of unlocking the lost, forgotten and suppressed secrets of the Ancients, we are indeed fortunate to have these seven wonderful tunings.