Given a musical sound , its pitch is characterized by the frequency of its sound wave. For example, if we regard the pitch as a musical note, then the pitch of frequency corresponds to what is called an octave higher above ; so we have had this feeling of consonance in between two frequencies differing by a multiple of . Now a thesis here is that our minds’ ears are musically interested in the ratios of frequencies, and this is the basis of the usual octave division in western music: if you are going to divide pitches between the note and its higher octave into notes, then it might be better to put this notes in a geometric progression, i.e. the -th note has frequency . In this manner, the ratio between any two consecutive notes is equal to a universal amount, . Pianos, classical guitars, saxophones, and other instruments all obey this tuning, known as 12 equal temperament (12-ET) tuning.

Yet, in the meantime you may have heard of the consonance quality of the perfect fifth interval, e.g. when you play the notes and of an octave, that the note in an octave has the frequency for the note in the same octave, and it’s said that it is because of this simplest ratio of in between them that the two notes sound so resonant to our ears (I checked with Google Gemini now, it said it too). But is the ratio actually equal to ? Doing the math from the above paragraph, we have that the perfect fifth interval is divisions (called semitones officially) higher than , we have in the above formula and

so the actual ratio is not ! What’s happening here?

Well first off, the numbers is so close to so that and are sonically indistinguishable to our ears (or you are blessed with divine ears), hence it might be the case that actually is the number of magical quality providing the consonance rather than . We still haven’t ruled out the saying “simple ratios (like ) of notes consonance between the notes”, but we have to further investigate it.

Simple ratios indeed make physical sense through the harmonic series and overtones: If we play a note, say , on the piano, it generates various sine waves (also called harmonics) of frequencies for all , each of which with some loudness (the loudness of these auxiliary waves depends on the shape of the original wave, determining the timbre of the instrument). So we are hearing all of the frequencies to some extent, and in most natural sounds, the loudness of the present frequencies tend to decrease as we go further in this sequence. Now notice that the auxiliary wave of frequency is away from the wave of frequency , i.e. we are hearing an interval of perfect fifth just by playing one note on the piano. Similarly, our mind is hearing a ratio of through the third and fourth frequencies. And a ratio of , etc. So if we agree that a note on piano sounds pleasant, then these observations might lead us to consider as the correct ratio for the consonance of the perfect fifth, rather than , and further to define a simple ratio as “a ratio whose numerator and denominator appear soon in the sequence “. Pure intonation, also called Just intonation (JI), is a tuning system based on this philosophy. Here’s a table of the justly tuned major scale.

Note	C	D	E	F	G	A	B	C
Ratio from C	1/1	9/8	5/4	4/3	3/2	5/3	15/8	2/1
Cents	0	204	386	498	702	884	1088	1200

where cent is just computing the power of in the geometric series approach by . So now we can compare the last row and see the difference between the 12 ET system where the cent values for the notes in the major scale are respectively and . Here’s Claire de Lune played in just intonation.

In fact, harmonic-series-based approaches are among the earliest ones used for tuning, dating back to Pythagoras at the latest (see Pythagorean tuning which is similar in idea to just intonation). The dots on the soundboard of the Chinese instrument Guqin indicates some of the harmonic positions. The just intonation and Pythagorean tuning where widely used in western music even in the 15th century. Considering the trajectory of the discussion, we might now be turning the unjustifiability objection towards the 12 ET system instead of just intonation.

Here’s a major issue with just intonation. Suppose we started with the root note and formed our intervals based on the table above, and let’s say we want to play in the key of major. As before, the perfect fifth interval humanly has an important quality to our scale, which is now the note . What is the ratio of to its perfect fifth in this tuning? Using their ratios with we can compute the ratio as ; i.e. in just intonation, the perfect fifth of has ratio with it (by construction) whereas the perfect fifth of has ratio with . Human perception of pitch is roughly based on the logarithm of ratios, e.g. based on cents, with about 5-10 cents threshold for identifying pitch change (not a universal number though; it depends on the interval too, e.g. for the octave the threshold is about 2 cents for a trained ear), so converting to cents we have versus , whose difference is over cents—way bigger than the threshold of ! This means that if we are to use various scales in our just intonation system, the various perfect fifth intervals don’t provide a uniform sonic experience. On the other hand, clearly the 12 ET system, or any other equally divided octave system for that matter, doesn’t have this issue by construction. In fact equal temperament systems came to the scene to remedy the inconsistent interval experience. Back in time not much key modulation was felt to be necessary; the black and white alignment of the piano keys based on the major scale to me resembles an era of commitment to single scale music. Bach seems to be among the first to use equal temperament tunings to explore key changes. A reason that is a good number for an ET system in my view is that it contains notes with ratios pretty close to simple ratios with the root frequency, i.e. we have the (the perfect fifth) and (major fourth) by only cents off.

One last thing I would like to point out is that 12 ET has been practiced prevalently for no more than 500 years, and before that time, it was considered dissonant and unpleasant. When I now listen to a major third in just intonation, it sounds strange and off to me; you may also have felt the same in listening to the Claire de Lune above. This demonstrates that human apprehension of sonic beauty is shaped by our cultural influences and our experiential memory. If you are interested in other tuning systems with different philosophies, check out Xenharmonic.