Pythagorean Scale

Pythagoras determined that whole numbers seemed to intimately tied the notion harmony in musical notes. Despite Boethius' story, Pythagoras determined by using a "monochord"-- a single string strung over a sounding box-- that one could determine which notes on such a string were harmonious.

There are certain notes, which, when played together sound harmonious-- sound like the two sounds merge together into a richer sound than either on their own.

The first of these is called an octave. Dividing a string exactly in two produce a sound which is called an octave (from the Latin for eight). If one plays such a not together with the original the two blend together so that it almost impossible to believe that there are two separate sounds. Together they sound like one sound. Sounds an octave apart are so similar to each other, that one often hears of "octave equivalence". Two singers singing an octave apart in pitch are often said to be singing the same tune. That they certainly different (one being higher than the other) the feeling is that that is unimportant. They are often felt to be really the same.

The second pair is called a fifth. Two notes a fifth apart do not sound as harmonious as an octave, but still merge together in what most regard a very pleasant, calming sound. These have a frequency ration of 3 to 2 (the frequency of one is 3/2 times the other. Or on a string, the second note is produced by a string 2/3 the length of the first. One can just hear that there are really two notes together, they merge together.

The third is the fourth, with a frequency ration of 4/3 or a string length of 3/4. While not as harmonious as a fifth, this set of two notes also sounds relatively harmonious. This is also the internal which one gets if one plays a fifth above some note, and an octave above that note together. The ratio of frequencies is 2/(3/2) =4/3. The corresponding string length is 3/4.

These three intervals (pairs of pitches) were the ones that Pythagoras took to be the most harmonious and formed the basis for a tuning and selection of sounds, of pitches, that are used in composing tunes and other pieces.


Let us start with a fifth, a frequency ration of 3/2 and go up another fifth (another 3.2), which would give us a note 9/4 above the first. Since 9/4 is over an octave, one can then go down an octave (frequency goes down a factor of 1/2) This would produce a note 9/8 (9/4 times 1/2) above the first note. This new note is called a tone above the other. If we give the original pitch the name of C, this second pitch is called a D. In terms of intervals the interval of pitch between C and D is called a second. (the interval from C to C, is called a unison). If one now goes up from D a tone ( another interval of 9/8) one gets to E, the interval of a third. This would have a frequency of 9/8 times 9/8 or 81/64. In the Pythagorean tuning neither the second nor the third is considered a harmony. If one were to go up another tone, one would have a pitch higher than a fourth, so instead one chooses the fourth as the next note, a frequency interval of 4/3. This is called the note F. and as indicated the interval is considered a fourth. The next, a fifth, has a frequency interval of 3/2 and is a tone above the fourth. 4/3 times 9/8= 3/2.

One now either goes up a tone, or one goes up a fifth from a second. (3/2 times 9/8 or 9/8 times 3/2 which are both 27/16. This is names A, and is called a sixth. Going up another tone above a sixth, or going up a fifth from a third, one gets to the note B, with a frequency ration of 243/128. Finally the next interval is the octave.

Both the interval between the third and the fourth, or between the seventh and the octave, are called a semitone. This semitone has a frequency ration of

(4/3)/(81/64) =256/243 = 2/(243/128).

Now, there turn out to be two different semitones. Consider the interval between one of these semitones and the tone. The tone is larger, and the interval from the above semitone and the tone is

(9/8)/(256/243)= 2187/2048

This is also called a semitone. It is the small semitone, and the previous semitone 256/243, is called the large semitone.

256/243= 1.0667

2187/2048= 1.051

The second semitone is almost a quarter of semitone smaller than the larger. the smaller is called the small semitone, and the other is called the large semitone.

Note that the octave has five tones, and two large semitones. Six tones is equivalent to 5 tones a large semitone and a small semitone. Thus six tones is a sightly smaller interval than is two octaves.

Cycle of fifth's

As we saw above, if we go up a fifth from C, we get G. Another fifth higher gets us to a note tone above C (if we also come down an octave), namely D. Going to another fifth, brings us up to A. Another fifth (and down an octave), gives us a tone above D, namely E. another fifth up brings us to B, another fifth (and down an octave) brings up to a note between E and F, called E# (E sharp). This is a note a small semitone above E. When you sharpen a note, you go up from that note by a small semitone. A double sharp brings you up by two small semitones, which is smaller than a tone by about 1/4 of a semitone.


where B## is two small semitones above B. This is almost but not quit the same as C. The difference is called the Pythagorean comma (comma meaning a difference originally) It is about a 1/4 of a semitone (actually about 1/5 of a large semitone, and 1/4 of small semitone.).

While the difference between these numbers and 1 seem absurdly small, the human ear is so so sensitive to difference in frequencies or pitch, this difference is easily recognizable.

In fact a trained ear (ie the hearing of a trained musician) can recognize a difference of about 1/100 of a semitone-- 1.0006 frequency ratio. That is less than .06% in frequency. This is far far smaller difference than any other sense can detect.

One can also go down-- Ie start at a note and go down by a fifth successively. One gets

C->F->Bb->Eb->Ab->Db->Gb->Cb->Fb->Bbb->Ebb->Abb->Dbb where each of the b indicates going down by a small semitone from the note. Dbb is almost the same as C (it is about 1/4 of a semitone sharper than C). In each case, on the modern piano, the double flat on a note brings it down by two semitones. In Pythagorean tuning that is two small semitones, which is smaller than a tone by about a 1/4 of a semitone. In modern equal temperament, it brings it down by a tone.

In each case we have what are called harmonically equivalent notes. F# and Gb, C# and Db which are not equivalent.


The above assigns names to each note in the sequence of notes, starting with some fixed note. In ancient times, the lowest note of use say in Gregorian Chant was called Gamma (the Greek letter G). But what if one wanted to talks about a sequence of notes which had the above relation but not have to worry about the where the first one was. Guido d' Aretzo, in about 1000, trying to teach his students the chants, came up with a naming scheme based on hymn to Saint John.

Ut queant laxis
resonare fibris
mira storum
famuli tuorum
solve polluti
labii reatum
Sancte Joannes!

set to a tune in which each successive line was tuned higher by the TTSTTT form. Ie, this was just like Julie Andrews in Sound of Music where she sings the song "Doe a deer, a female deer...". Whether this was already a standard hymn or whether Guido made it up for teaching his choirboys, just like in the movie, is unknown. The last line goes back to the original note that Ut sounds in the sound, so was not part of the system.

Note that in about the 17 century, the Ut got replaced by Doh, as being easier to sing, and a final entry was added (with names variously given by Sa (the first syllable of Sancte) or Si (certainly in Italy) or Ti in England and the Us. to give the standard Doh Re Mi Fa So La Ti Doh we now use.

In additions, numbers were added to indicate jumps from one note to the other. Thus a jump from Doh to Doh is called a First (1). from Doh to Re a Second, from Doh to Mi a Third, from Doh to Fa a Fourth, from Doh to So a Fifth, from Doh to La a Sixth, from Doh to Ti, a Seventh, and from Doh to the higher Doh and Octave. This is the standard seven note Major scale which we still in general use.


In the 13th century, musician's "discovered" that instead of being a dissonance, a note used in music to go over or through to finally get to Harmony, the third could be regarded as a consonance, a harmony, if one very slightly changed the tuning of the third. Recall that the Pythagorean third is 81/64. If one flattens this slightly, one gets 80/64=5/4. This is another ratio of small numbers. While Pythagoras did not believe that this was important or a harmony, the musicians of the 13 century felt that it sounded more like a harmony than a dissonance if it was slightly mistuned like that. This completely threw the cat amongst the pigeons. The practical musicians ran with this new idea and played and composed pieces in which this note was treated harmoniously. Now one could tune the notes in the octave as

Doh-Tp-Re-Tj1- Mi-S-Fah-Tp- So-Tj2-La-Tp-Ti-S Doh

where Tp is the Pythagorean fifth (9/8) Tj1 and Tj2 are two different Just tone (10/9) and (6/5) and S is a semitone which is different from either the Big Semitone of Pythagoras, or the Small Semitone of the Pythagorean system. and has a frequency ration of 15/16. This results in the frequency ratios of the notes being
Re- 9/8
Mi- 5/4
Fah- 4/3
So- 3/2
La- 5/3
Ti- 15/8

This is clearly much more complex a tuning system. It is also unstable. As Huygens, a physicists from Holland in the 17th century (he promulgated the wave theory of light in contrast to Newton at about the same time who promulgated the particle theory. He also invented one of the first clocks which had second hand, and had an accuracy approaching seconds per day), pointed out, if you have a set of notes where you go down by a fifth down by a third, then up by two fourths, you end up one syntonic comma (124/128) lower than the the original first note. Thus if one would do this a number of times in a piece, the whole pitch would steadily go down.

This was called Just tuning. Things like Benedeti;'s puzzle meant that it was extremely difficult to keep a tuning in which the Just intervals were always kept. However, by deviating at places in the music where such deviations were masked by other things going on, one could play and sing pieces in Just intonation, producing sometime ethereal beauty in the pieces because of the harmony of the thirds.


Kepler got involved in music theory and in trying to understand what made some notes be harmonious and some dissonant. His approach, like his approach to the motions of the planets, was mathematical and geometrical. Unlike his work on the planets, his musical work had little effect on others (but is still quoted by some today for its near mystical quality regarding the link between music and the motions of the heavens, especially the planets. Although we could not hear the planets, he felt that the reason the planets had the features they had, had the eccentricities they had was because this gave their velocities around the sun a mathematical relationship which reflected that of the relationships of the notes used to make music. The planets were making mathematical music unhearable by humans, but which could be appreciated as symbols of the mathematical relationship of God to his universe in the planets as in music.


He is interesting as he was one of the first to suggest that the proper tuning for music is equal temperament. This is a tuning which makes all tones the same, and makes the semitones such that two semitones exactly make up a tone. Contrary to the opinion of all musicians, he said that the this did not alter the harmonies of the music. This battle between equal temperament has continued to this day. (For an indication of the battle, even today, see the books Isacoff "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization" who is strongly in favour of equal temperament, Duffin "How Equal Temperament Ruined Harmony: And Why You Should Care" whose title gives a feeling for where his sympathies lie.

Vincezo Galilei.

Vincenzo Galilei was the father of the better known (now) Galileo Galilei. He was one of the foremost music theorists of his day, expounding what was then the New Music. One of the things he is best know for is his experiments. Since Boethius, the lore had been that things sounded at a certain pitch because of some property. Thus, if you had a string and hung a weight from the string to put tension into the string, the pitch would rise. From Boethius it was assumed and almost universally represented that if you doubled the weight the pitch would go up an octave. Now string instruments have pegs that the string wraps around and to increase the tension, you wind the peg. Thus there is no indication as to how much tension there is by this procedure. He hung weights from the ends of the strings, and instead of doubling he found that he had increase the weight by a factor of 4. If he wanted to increase the pitch by an octave and a fifth (2 times 3/2 in pitch), you would have to increase the weight by a factor of 9, not 3. (of course they could not measure tension, so he would say that to increase the pitch by an octave you would either have to decrease the string length by a factor of 2 or increase the weight by a factor of 4.

His son, Galileo Galilei, is usually given the credit of being the first modern experimentalist, subjecting a theory to actual physical tests. It seems that the son learned the importance of experimentally verifying theoretical ideas from his father. It was Vincenzo Galilei's books that Kepler read when heading out to work with Brahe, not his son's.

Galileo Galilei

One of the key inventions related to music was the pendulum. The story goes that he was watching the chandeliers in the cathedral during a service and happened to time one with his heartbeat at the beginning after they had been lit and swinging. At the end of the service he did so again. The time for a swing was the same. (My suspicion is that he actually timed it with the beat of some chants sung in the cathedral at the beginning and end of the service. The human ability to keep time in music is far better than the human heart beat.) The independence of the timing of a pendulum from the amplitude was thus first noticed by him. This became the basis for pendulum clocks (the first highly accurate clocks, primarily further developed by Christian Huygens in Holland. He also extended the ideas of his father that sound was a succession of pulses produced by a moving string, and if the pulses of two notes hit the ear drum at the same time, then the ear felt the sound was harmonious. This idea had many problems, but was a sort of precursonr to the idea that sound is a wave in the air, and that the "lining up" of waves was somehow related to harmony.

copyright W Unruh (2018)