This also means that when one plays two notes together there are lots of harmonics which could beat with each other. One can hear these as slow of fast wavering in the sound of the notes together. But it is also clear that the tone colour of the two notes is very different. For example, if one listens to the Just Major third vs the Pythagorian Major third, while the Just is "harsher" in its sound than say the Just perfect fifth, the tone colour of the Pythagorian is quite different from that of the Just Major third, in addition to having very rapidly flucuating warbling in its sound.

In the following in each case I have also included the intervals with each note being a pure sinusoid. There are no harmonics. It is very hard to tell the difference between the various tunings of the intervals for the sinusoidal pitches. There is no beating between harmonics in this case, and even intervals which are significantly different from the whole number relationship do now sound discordant togeter. It is the beating between the harmonics of the two notes which gives the fluctuating tone colour or warbelling of the pitches in the case of the sawtooth wave.

In all cases the lower note is the same, at A 220Hz (the A just below
middle C on the Piano).
To play these notes, your browser needs to be HTML5 compatible, and play .wav
files. This is true of all modern browsers except possibly Internet Explorer.

Saw
Sine

Below we have various intervals, in various temperaments based on this "tonic" note. In each case the sound has this 220Hz note plus an additional note of higher frequency in combination.

Octave with frequency ratio of 2-1

Saw
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Saw Sine

Equal tempered perfect fifth freq ration of 1.4983

Saw
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Just Major third (the interval which gives most of the trouble) Frequency
ration of 5/4=1.25

Saw
Sine

Pythagorean Major third with frequency ration of 81/64= 1.2656

Saw
Sine

Equal Temperament Major third with frequency ration of 1.25992

Saw
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Finally here are the sounds of the major chord (unison, maj third, perf
fifth) in just temperament (220-275-330 Hz)

Saw
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and the same chord in equal tempermant.

Saw
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There have been recent proposals by some people that perhaps one could make an equal temperament (ie all semitones are the same ratio) but such that the Octave not have exactly a 2:1 ratio of frequencies. I dicuss this more at the bottom of this page, but present them here so that they can easily be compared to the standard octave chord above.

Hinrichsen's octave with semitone being 1.0005 larger than equal Temperament.
This makes the octave sharp by about .6 cents.
See
arxive.org/pdf/1508.02292v3.pdf

Saw
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/

Cordier's octave with semitones defined so that 7 equal semitones equals a Just
Perfect Fifth. (See S. Cordier, Equal Temperament with Perfect Fifths, paper
presented at the International
Symposium on Musical Acoustics, Dourdan, France (1995)). The octave is larger
than the 2:1 ratio by about 3 ceents. See also
http://www.pykett.org.uk/impureoctaves.htm who discusses a variety
of temperaments with impure octaves.

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The advantages of the above schemes (almost completely mathematical rather than musical in the case of the Hinrichsen) are obscure. By destroying the purity of the octave they make some intervals (eg the perfect fifth) slightly better in their purity, but at the expense of making the major third much worse. Both stretch the octave (I.e., make the octave ratio slightly larger than 2:1). However, both the Hinrichsen and Cordier proposals, by making the semitone larger than in equal temperament, also make the Major third worse (further from the harmonious Just Major third) than in equal temperament. Since the whole point of temperament was to make the major thirds better so that they could play the role of a harmony, rather than a dissonance, and they also now make the octave slightly dissonant (the tone colour of the two notes an octave apart changes with time), it would seem to me to be pretty unclear why one would choose them musically.

Mind you in the early 1970's, William Benjamin (now an emeritus prof of Music at UBC, then a graduate student in composition at Princeton) composed a piece in which he tried to see if he could make the major Seventh (one of the most dissonant intervals) play the role of the octave in the temporal development of the piece (eg, such that listeners would feel that finally landing on the seventh would provide a sense of completion in the piece, like an octave does in most music). He also composed another in which he had the ninth (an octave plus a tone) play the role of the octave. Ie, in the right hands one might be able to use the wide octaves of those temperaments effectively. Note that it was discussion with him that led to my development of the "octave equivalent piano" (when you play any note, all of the octaves, from lowest to highest in the range of hearing, of that note also play at the same time), which I will demonstrate in the last lecture of the course.

In the 17th century a propsal to make a 19 step equal tempremant was floated (ie there would by 19 "semitones" in an octave, each semitone being the same size-- ie the same ratio. These semitones have little to do with what in music is called a semitone, although it is very close to the small (Just chromatic) semitone (between the just tuned minor third and the just tuned major third). In that case a major third would be 6 of these semitones, a perfect fourth would be 8, a perfect fifth would be 11. The perfect fifth would be about 6 cents flat of the Pythagorean fifth, and the major third would be about 7 cents flat of a just major third (ie much less than in either equal or Pythagorean temperament).

This is what the two notes-- 220Hz and 273.83 Hz-- the major third in this
temperament-- would sound like.

And this is what 220Hz and 328.63Hz ( the perfect fifth in this temperament)
would sound like together.

Although this temperament produced slightly worse fifths and much better
major (and especially minor) thirds than our temperament, trying to learn to play an instrument with 19 keys
per octave was just too difficult and did not offer sufficient advantage over
the standard equal temperament. However, it is still occasionally advocated,
and guitars have even been built so that their frets follow this 19-equaltoned octave.
.

Again, the fact that all thirds are the same, all fifths the same, etc. makes this, as well as any equal temperament more boring than some of the unequal temperaments from the 17 and 18 centures.

As with the standard equal temperament, the inharmonicities in the piano because of the stiffness of its strings, and the use of vibrato on stringed instruments and voice, make the differences between any temperaments disappear.

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Copyright W. Unruh 2015 (includes all the sound examples)