Premise

The content of this page is very delicate. It is directed at musicians, however, instead of speaking their language and calling the intervals by name (minor sixth (m6), augmented fourth (A4), etc.), they will be translated into mathematical formulae, which will be incomprehensible at first sight. The mathematical structure that generates musical scales as a succession of intervals is elementary, even though it looks "repellent". It involves only four operations (actually only multiplication and division) and can be constructed step by step, which is what we will do.

How are Pythagorean scales constructed?

1. The generative mechanism of the Pythagorean scale is very simple: it can be obtained starting with only two fundamental ratios: 2:1, which is the interval of the octave, and 3:2, which is the interval of the (ascending and descending) perfect fifth. Some people also include the interval of the perfect fourth in the generative mechanism. On this page, we prefer to consider the interval of the perfect fourth as a descending fifth.
2. Then we choose a reference note, such as C, and begin to generate intervals of ascending fifths.
3. From a mathematical point of view, this equals repeatedly multiplying the initial frequency by 3/2. This procedure quickly (with the second multiplication) generates frequencies that exceed the octave that contains the reference note having a ratio with a reference frequency higher than 2. To bring these frequencies back to the initial octave, we divide the obtained frequency by ${\displaystyle 2^{n}}$ where n is the number of the octaves from the initial note to the final note. For example, starting from the note C, we obtain:

Generative rule (ascending)	...	${\displaystyle {\frac {3}{2}}}$	${\displaystyle \left({\frac {3}{2}}\right)^{2}\cdot \left({\frac {1}{2}}\right)}$	${\displaystyle \left({\frac {3}{2}}\right)^{3}\cdot \left({\frac {1}{2}}\right)}$	${\displaystyle \left({\frac {3}{2}}\right)^{4}\cdot \left({\frac {1}{2}}\right)^{2}}$	${\displaystyle \left({\frac {3}{2}}\right)^{5}\cdot \left({\frac {1}{2}}\right)^{2}}$	${\displaystyle \left({\frac {3}{2}}\right)^{6}\cdot \left({\frac {1}{2}}\right)^{3}}$	...
Ratio between frequencies	1:1	3:2	9:8	27:16	81:64	243:128	729:512	...
Note	C	G	D	A	E	B	F♯	...
Interval	…	P5	M2	M6	M3	M7	A4	...

We can even generate new notes between intervals of descending fifths. From a mathematical point of view, this equals repeatedly dividing the initial frequency by 3/2 (i.e. multiplying by 2/3). To bring these obtained notes back to the initial octave, we multiply the obtained frequency by ${\displaystyle 2^{n}}$, where n is the number of octaves from the initial note to the final note. For example, initiating from the note C, we obtain:

Generative rule (descending)	...	${\displaystyle {\frac {2}{3}}\cdot 2}$	${\displaystyle \left({\frac {2}{3}}\right)^{2}\cdot 2^{2}}$	${\displaystyle \left({\frac {2}{3}}\right)^{3}\cdot 2^{2}}$	${\displaystyle \left({\frac {2}{3}}\right)^{4}\cdot 2^{3}}$	${\displaystyle \left({\frac {2}{3}}\right)^{5}\cdot 2^{3}}$	${\displaystyle \left({\frac {2}{3}}\right)^{6}\cdot 2^{4}}$	...
Ratio between frequencies	1:1	4:3	16:9	32:27	128:81	256:243	1024:729	...
Note	C	F	B♭	E♭	A♭	D♭	G♭	...
Interval	Unison	P4	m7	m3	m6	m2	d5	...

By unifying the two cycles of ascending and descending fifths, we can obtain infinite intervals. However, the scale only makes sense if it contains a finite number, otherwise our ear would not be able to distinguish all of the intervals. The point is: where should we stop? The following paragraph will show just how delicate the answer to this question is.

Ratio	...	1024:729	256:243	128:81	32:27	16:9	4:3	1:1	3:2	9:8	27:16	81:64	243:128	729:512	2187:2048	6561:4096	...
Note	...	G♭	D♭	A♭	E♭	B♭	F	C	G	D	A	E	B	F♯	C♯	G♯	...
Interval	...	d5	m2	m6	m3	m7	P4	Unison	P5	M2	M6	M3	M7	A4	A1	A5	…

The circle does not close!

• What happens if we continue to multiply the succession of frequencies by 3/2 or 2/3? At a certain point, would we return to the same notes as if we were moving along a circle (musicians will certainly be thinking about the circle of fifths) or as if the generative mechanism was always producing new notes? The answer is that the so-called circle of fifths does not close in the Pythagorean scale. It is actually a helix of fifths. Potentially, the illustrated generative mechanism can divide the octave into an infinite number of parts making the intervals between each two notes smaller, even beyond the threshold of discrimination of frequencies of our ear.

• The reason for the missing "closure of the circle" is simple: a circle closes when it returns to its initial point. For this to happen, the generative mechanism should allow us to produce a note, the frequency of which has a ratio with the initial note of ${\displaystyle 2^{n}}$ (at this point, to return to our initial point, all we have to do is divide the frequency by ${\displaystyle 2^{n}}$!). However, there is actually no way to choose integer n so that it gives ${\displaystyle (3:2)^{n}}$ equal to ${\displaystyle 2^{n}}$.

comma-pitagorico.mp3 C♯ and D♭ in the Pythagorean scale of C

Observe that the missing closure of the circle causes the notes C♯ (ratio 2187:2048) and D♭ (ratio 256:243) to not coincide, which happens in modern scales based on equal temperament, in which they are considered homophone (or enharmonic) sounds (i.e. sounds of different names with the same intonation). In the Pythagorean scale, they differ by only 23.46 cents. This difference is the so-called Pythagorean comma, which is about one quarter of a temperate semitone. If we decide to include C♯ as a distinct note from D♭ in our musical system, we would have to double the size of the black keys on the piano. However, in modern equal temperament D♭ and C♯ do coincide and the black keys are used in the same way for sharp ♯ and flat ♭.

The diatonic Pythagorean scale

The solution for the construction of a scale is obviously to truncate the circle at a certain point. The notes to choose may appear absolutely conventional, however, they must actually satisfy precise aesthetic needs (consonance) and facility of intonation (uniformity of the consecutive steps of the scale). A possible choice is to only consider the seven central notes of the last table (from F to B). This choice leads to the diatonic scale.

Diatonic Pythagorean scale Note Ratio Frequency (Hz) Cents C 1:1 261.6 0 D 9:8 294.3 204 E 81:64 331.1 408 F 4:3 348.8 498 G 3:2 392.4 702 A 27:16 441.5 906 B 243:128 496.7 1110 C 2:1 523.2 1200

Advantages From the table on the left, it is immediately clear that this scale contains only two types of intervals between its consecutive steps. They are the Pythagorean tone (e.g. D-E, which is equal to about 204 cents) and the Pythagorean semitone, or limma (e.g. E-F, which is equal to about 90 cents). All the octave and fifth intervals contained in the scale (the B-F# fifth is not there because it requires an altered note) are perfectly consonant because they coincide with the simple ratios 3:2 and 2:1 of the natural scale. This comes as no surprise because the fifth interval is the "seed" from which the entire scale was generated. Pythagorean semitones will change position within a scale depending on the note used as a starting point (e.g. we could have started and ended the scale with the F note) giving each scale a different "flavour". Using only seven notes, the Greeks had already intuited the expressive possibilities provided by the "translation" of the starting point of the scale and had developed a theory of musical "modes".

To tell the truth, more than reasoning in terms of translation, the Pythagoreans' music theory used the tetrachord (tetraktys) as the foundation of various modes. A mode was made up of two consecutive descending tetrachords. The tetrachords were often also homologies; i.e. having the same succession of tones and semitones. For example, the mode with a scale starting from C was generated by the two tetrachords C-B-A-G and F-E-D-C, in which the succession between tones (T) and semitones (S) was S-T-T-T. The names of these modes were derived from the regions in ancient Greece from where they originated.

Modes of Greek music Name 1st tetrachord 2nd tetrachord Initial note Ionic C-B-A-G F-E-D-C C Dorian D-C-B-A G-F-E-D D Phrygian E-D-C-B A-G-F-E E Lydian F-E-D-C B-A-G-F F Mixolydian G-F-E-D C-B-A-G G Aeolian A-G-F-E D-C-B-A A Locrian B-A-G-F E-D-C-B B

Disadvantages

1. Intervals of the third and the sixth are not consonant. Moreover, they are expressed by complicated ratios that involve rather large numbers. If, using pythagorical reasoning, the criterion of consonance is that of "simple ratios", then these intervals sound dissonant (for the physical aspects of consonance of simple ratios, visit this page on Galileo). In the playing of music, particularly after polyphony became common practice, an interval of a third with a much simpler ratio (5:4) became increasingly important because of its greater consonance. This choice teaches us that "musicians" do not get caught up in abstract numerological speculations; they choose consonant intervals by ear. The choice of more consonant thirds leads to the natural scale. However, it is still true that many violin soloists prefer to execute their pieces in the Pythagorean scale, affirming the centrality of the interval of the fifth in music harmony (string instruments are often tuned in fifths).
2. As with all diatonic scales, the limited number of notes offers a limited range of melodic possibilities.

The chromatic Pythagorean scale

The latter disadvantage can be overcome by increasing the number of notes that make up a scale. Obviously, the new added notes, which can greatly enrich the melodic possibilities, must not compromise the advantages of the diatonic scale. They must:

1. continue to guarantee the consonance of the intervals of the octave and the fifth;
2. make the consecutive steps of the scale as uniform as possible;
3. not be of an excessive number, so as not to have frequencies too close together.

A possible comprise solution can be obtained by considering the 12 central notes (from E♭ to G#) listed above. In other words, to include in the scale:

• C♯ but not D♭
• F♯ but not G♭
• G♯ but not A♭
• B♭ but not A♯
• E♭ but not D♯

Thus, the following scale is obtained:

Chromatic Pythagorean scale

Note	Ratio	Frequency (Hz)	Cents
C	1:1	261.6	0
C♯	2187:2048	279.4	114
D	9:8	294.3	204
E♭	32:27	310.1	294
E	81:64	331.2	408
F	4:3	348.8	498
F♯	729:512	372.5	612
G	3:2	392.4	702
G♯	6561:4096	419.1	816
A	27:16	441.5	906
B♭	16:9	465.1	996
B	243:128	496.7	1110
C	2:1	523.3	1200

Observe that, if along with all the altered notes, we also had their (enharmonic) alternatives, the scale would have 17 steps instead of 12; with five that differ from the Pythagorean comma by about only 24 cents.

Advantages

This choice maintains the consonance of the intervals of the octave and the fifth and all the fifths in the same scale are intonate (except one, see the disadvantages below). The consecutive steps of the scale are sufficiently uniform: this uniformity was obtained by inserting the altered notes almost halfway between the Pythagorean tones. The chromatic scale has only two intervals, the limma and the apotome (e.g. C and C#) (see the table), the sum of which gives the Pythagorean tone and the difference of which gives the Pythagorean comma. The frequencies between two consecutive steps are not too close; despite the increase in the number of notes, the interval of minimum amplitude is still the limma.

Intervals between the steps of the chromatic Pythagorean scale Interval Ratio Cents Limma 256:243 90 Apotome 2187:2048 114 Pythagorean tone 9:8 204

Disadvantages

The fifth G♯-E♭ is out of tune (the fifth in tune would be G♯-D♯ but D♯, as we said, was excluded from the scale). The lack of consonance of the fifth G♯-E♭ in the Pythagorean scale of C is only an aspect of a more general issue: the problem of the change in tonality. If an instrument is tuned according to the Pythagorean scale to play in a certain tonality (e.g. C), it could be out of tune when playing the same scale in another tonality "far" from the previous one. Intervals of the third and the sixth continue to be not very consonant.

quinta-pitagorica-dissonante.mp3 Fifth G♯-D♯ and G♯-E♭ in the Pythagorean scale of C. The first cannot be part of the 12-tone scale. The second can but it is dissonant.

Visit the section Comparison of scales, to listen to several examples that highlight these advantages and disadvantages.

quinte-pitagoriche.mp3 All the fifths of the Pythagorean scale in C. Only the penultimate (B-F♯) is dissonant.

Pythagorean scale and consonance

The lack of consonance in the intervals of the third and the sixth is a problem not only for the correct intonation of these intervals but also because it can cause unpleasant beats during the simultaneous execution of bichords (e.g. C-E); especially, when instruments rich with upper order harmonics are being played. For example, the fifth natural harmonic of C is very close in frequency to the fourth harmonic of the Pythagorean E. For example:

• if we suppose that the frequency of C is 261.6 Hz, then its fifth natural harmonic will have a frequency of 261.6 •5=1308 Hz;
• however, the fourth natural harmonic of the Pythagorean E (with a frequency of 331.1 Hz) will have a frequency of 333.1• 4=1332.4 Hz;

• the beat frequency is given by the difference between two calculated frequencies and is about 25 Hz. This value, according Von Helmholtz's theory of critical bands, will fall right within the interval of values that give a sharp and unpleasant character to sound. For more on this, visit our virtual laboratory, where you will find a guided experiment on beats. By modifying the default frequency values, you can evaluate the sharpness of the sound obtained by overlapping the above calculated frequencies. As well, visit the page on the theory of beats.

We are probably overestimating the role of dissonance in the intervals of the third and the sixth in ancient Greek music for two reasons:

• their music had a melodic character and the overlapping of voices that were not in unison or separated by an interval of a fifth was rare;

• the instruments they used (e.g. the flute) were poor in upper harmonics.

In-depth study and links

• For an in-depth study of the mathematical aspects used in determining intervals, visit the page How is a musical scale constructed?;
• For a brief historical background on the origin of various musical scales, visit the page From monochords to musical scales;
• To compare the Pythagorean scale with other scales, visit the page Comparison of scales.

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