4. Ring the Bells – a crack in the universe Ring the bells that still can ring Forget your perfect offering. There is a crack in everything That's how the light gets in. Leonard Cohen 'Anthem' Consider the harmonic series again. One notices that this simple arithmetic series (1, 2, 3, 4 … ) refers to a series of multiplications of a base frequency, and within this lies an even deeper identity: the series 1, 2, 4, 8, 16… , the octaves above the fundamental. We can also express this as a power of 2 series 2^0, 2^1, 2^2, etc. This relationship based on the power of 2’s is so strong in music that it provides us with a definition of a ‘note’ that is, a name of a pitch or specific frequency irrespective of octave displacement. Any frequency in a relation of a power of 2 to another is called by the same name. 220 440 880 1760 etc. are all the note ‘A’. f*2^n n={ …2, 1, 0, 1, 2…} We can now construct a second series, the power of 3 series. It gives the fundamental, then 3 * f, 9 * f, which starts to get very high in frequency quickly but we can transpose downward using the octave identity, or power of 2 series. 3, 3/2, 3/4, 3/8 are all E We have already seen that 3f, the 3^{rd} harmonic, is referred to as the musical fifth one octave above the fundamental. Thus 3/2f is the same note by the octave transposition identity f * 3 * 2^1 I begin to build a set that looks like this: G D A E B F# C# etc. Perhaps we have waited too long to travel backwards in time to meet Pythagoras. This is his series after all. Certainly, world musics have understood instinctively these relationships and base their music upon them – the octaves, fifths, building 5 note, 8 note scales, etc. and these basic elements appear in most musics of the world – but credit for elucidating the numeric relationship between these fundamentally appealing and harmonious sounds is given to Pythagoras. The story goes that Pythagoras passed by an iron workshop hammers were striking anvils. Some sounds were pleasant and in harmony with each other, others were not. After testing various hypotheses, he found that the difference was due to the relative weight of the hammers. Finding weights in proportion 6 8 9 12 Pythagoras identified the numeric relationships of the octave [6:12] the fifth [6:9] the fourth [8:12] and the second [8:9]. This beautiful set just happens to be the lowest integer relation of its kind, [perhaps a finding as beautiful as the first perfect right triangle 3 4 5]. We can reduce and transpose the relationships: 1 1.33 1.5 2 1/1 4/3 3/2 2/1 I IV V I or .75 1 1.125 1.5 3/4 1 9/8 3/2 V I II V or .667 .889 1 1.333 2/3 8/9 1 4/3 IV bVII I IV Pythagoras was duly impressed [as am I] by this and began to extrapolate upon the principle. Seeing no need to go beyond the integers 1, 2, 3, 4 – the basis of his holy Tetrakys – Pythagoras devised what will be termed here, the Pythagorean series. P3/2(n) = f*(3/2)^n n={0,1,2,3…} Notice that this is not a scale but a series of interval leaps of a fifth. The inversion gives a descending series of musical fourths. P2/3(n) = f * (2/3)^n n={0,1,2,3…} To create a scale then, we would need to apply our octave identity to keep the series within a specified range. This is usually done by brute force, transposing as needed, enabling the creation of various modes. Historically the Pythagorean lyre had 8 strings tuned as: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1 In general, I will use here a function called “ratio_limit”, that will bring all ratios > 2 to between 1 and 2, and ratios < 1/2 back up to the 1/2 to 1 range: RL(n) = pow(2, (log2(n)  trunc(log2(n)) [N.B. ] Now here’s the rub: if we continue with the series (not the scale) up to n=12 we have traveled up 7 octaves, but (3/2)^12 ≠ 2^7 P(0) 1.5 69. A3 P(12) 1.520465 69.234406 A3 Tantalizingly close. Here is the first obvious crack in the universe! [As always – more perfect than perfect]. This small difference, known as the Pythagorean comma has been the source of great trouble both theoretically and philosophically. In fact, it is said that Pythagoras knew of this comma but only revealed the fact to his closest initiates. Telling others of this secret, including lesser initiates, was punishable by drowning! In observing the Pythagorean series limited to an octave, after 12 luscious notes, each a glorious perfect fifth away, we find a unison that wobbles, or alternately an octave that is stretched slightly. The sensation is one of almost of not being able to breathe because they’re suddenly so much more oxygen, much more room. Fill this plussized octave in with a biting Pythagorean triad however, and one begins to feel like a cat being skinned alive for gut. Certainly, no serious musician would abide such caterwauling. The natural tendency is to fix this interval to the perfect 2/1 octave. For now, enjoy the expanse and the throbbing in the room [and possibly in your head]. Resist the temptation to reorder the series for now. See how the cluster of major seconds on the keyboard produce the first notes of the scale, that the major third {K:M3} produces the socalled Pythagorean third {P:M3} (12) 7 9 11 6 8 10 (locrian) (Lydian) p +1 +4 major +10 dom7th\ p +1 +9 minor p +2 +4 +1 +3 After taking time to familiarize yourself with this series, we’ll do a few more comparisons. See if these features stand out: * the Pythagorean third is higher than the third of the harmonic series. * the Pythagorean sixth is awkwardly high as a melodic step – [at least, this is what it says in most textbooks]. * compare again the triads in ET, the harmonic, and Pythagorean series. Mathematically speaking, we note that while the harmonic series is multiplicative and linear, the Pythagorean series is a power series [logarithmic]. Most contemporary textbooks speaking from a modern bias describe the stretched octave [or wobbling unisons], the high thirds and ‘difficult to sing’ sixths as “problems”. Here we see them simply as features of the series. Certainly, it would be nice to construct a Pythagorean scale with a perfectly closed octave, but this cannot be done using any method. This mathematical fact has not stopped people from trying, however, and failing that, going to extreme measures to fix ‘the problem’. Moreover, there is another complication: that of the harmonic series. People just seem to like the sound of 5/4 at times and at other times the sound of 81/64. For fun, we use modern computer to crunch away forever and never come back to our beloved 2:1. A simple algorithm for generating the Pythagorean spiral of fifths is given. This algorithm uses a conditional statement rather than the log function used above. The result is similar. On a technical note, this bit of code satisfies the formal definition of an algorithm, a process with a beginning and an end. The beginning here is t=1 or the first step from 1 to 1.5. The end occurs when the algorithm returns to 1.0. I turned my version of this patch on in 2008/7/11 and it’s still running. Let me know if your version should ever stop! I have sometimes been tempted to work out the pattern, but as usual I get sidetracked by the multitude of variations and complexities implied by this method and must leave that bit of math for another time to work out. I will offer some preliminary observations: first, that our 12 notes migrate upwards by the distance of the Pythagorean comma at each turn of the wheel. Given ‘D’ as the center and keeping the note names constant, eventually after [x] number of turns, C# is closer to the ratio 1 than D now is, which is now close to Eb, etc. But this migration misses being exact by some amount I don’t have the patience to calculate. This series never repeats. If we turn up the tempo to a ridiculous number [480 BPM or so], I hear a pulsing firebrand of consonance whirling throughout spacetime. How about you? Going further we find that there is no whole number ratio (a/b)^n which will divide the octave evenly. The proof of this is related to Fermat’s Last Theorem and thus need not be discussed here. I’ll write it in the margin later.
