The note at the top of a stack of 12 pure fifths is different from the note at the top of a stack of 7 octaves.
Pythagorean Comma, defined approximately 2,500 years ago.
W. A. Mathieu, “Harmonic Experience” 1997
“Since modern Western musicians are so used to ‘the circle of fifths’ as a theoretical concept as well as a practical learning tool, it may be surprising to some that twelve perfectly tuned fifths stacked from from C do not equal some distant octave of C, and that there is hence no circle of perfect fifths.”
I would like to name two paths forward: “intonalistic serialism” and “intonalistic pitch class sets”.
For the first, I can imagine a new serialism where a) there are more than 12 notes, b) certain intervals are allowed (those that can be tuned) and others disallowed (those that can’t be tuned). The question of the fifth to minor seventh (untunable as used in traditional dominant-tonic harmony) is left for later.
For the second, it would be great to develop some of the tools used in manipulation 12-member pitch class sets (and sets with fewer members) with regard to the tuning of the individual elements, and perhaps extend to sets of greater numbers. As a first step, perhaps, the 8 note diatonic set of tonic, low supertonic, high supertonic, mediant, subdominant, dominant, submediant, leading tone. Then the 10 note set of tonic-dominant harmony. usw.
I attempt to describe better what it is that I do:
Each and every note for orchestra and chorus is assigned a specific intonation, such that through a major work like my Moses at the Jordan River, approximately 50-60 distinct pitches are used. As a rapid summary example, for the note C# on a piano, the score might use a high Db, a low Db, a high C# quite distinct from the Db, a low C#, and a two-comma flat double low C#. For the most part, however, these pitches are tuned in pure relation to fairly traditional harmonic structure and come naturally to the naive singer and to a good musician. The special rules of Intonalism guide the composition of every melodic line such that every interval may be tuned perfectly. This is not to imply that a normal variation in pitch and even outright mistakes during performance will spoil the work in any way, more than they would spoil any other composition.
As an example of how one might combine an exercise in sight reading with an exercise in tuning, I made this harmonization of the Agnus Dei from Mass XVIII chant. Needless to say, if you listen, it is a synthesis. But it is tuned correctly.
Symbol expanations may be found in an earlier post, but briefly the heavy down triangle is a comma-low modal scale note; the circle+ tonic; the light up and down arrows dominant and sometimes supertonic over dominant, and subdominant; the double heavy down triangle two comma low implied secondary dominant third.
I intend to make this available free, but with each use requiring an explicit license stamp allowing duplication and performance. Contact me directly for such a license. For perusal, naturally, no license at all is needed.
It has been asserted that a dimished seventh chord cannot be tuned in just intonation. While a composer may well want the sound of the equal tempered blur of a diminished seventh chord on the piano or organ, I believe as used by Mozart and Copper, and maybe Scriabin … one day i’ll look into that music more …, it can and should be tuned as shown. Not all composers spell the chord correctly even when used as in my example as a clear substitute for a secondary dominant.
There is only one untunable interval, the Pythagorean minor third, the same interval that falls in the dominant seventh chord between fifth and seventh. The other intervals, including the resolution major and minor seconds, have clear and unambiguous tuning.
The first of many examples taken from standard harmony books. This one, chapter 12 of Aldwell and Schachter, “Harmony and Voice Leading”, 1989. A good harmony text over all, but with this and a number of other intonational problems.
Each of the excerpts has the same problem: an untuneable fifth in the bass, identified with a bracket in the intonation diagram.
There are two major thirds used in Intonalism, and in most great music.
One of the major thirds is tunable, the pure major third of the harmonic series. If you play the open C string on a viola, and listen carefully, you can hear that third (two octaves and a third higher). That is the fifth overtone or fifth harmonic in the harmonic series. If you sing the third of a major triad in pure intonation it should be the same pitch as that harmonic.
The other major third, often called the Pythagorean third, is not directly tunable. It is much sharper than the pure, tunable major third. However, it is used in tonal modulations along the circle of fifths. It is the result of a stack of pure fifths. From the same open C string on a viola, each successive string is tuned a pure fifth higher: C – G – D – A on the viola. A fifth above the viola’s A string is the violin E string, another pure fifth higher. So the violin E string is two octaves and a Pythagorean major third above the viola C string.
If a portion of music were to modulate from C to G and on to D and A, and finally to E, it would be properly tuned to that violin E string, a Pythagorean third away. If a different portion of music were to modulate from C by means of a pivot on the third scale step, it would be properly tuned to the harmonic third, and the violin open E string would sound badly out of tune.
The third on the organ or piano, in equal temperament, must fulfill both purposes, so as a reasonable compromise it is somewhere in between in tuning, rather closer to the Pythagorean third than to the Harmonic third.
The audio example gives all three over a viola open C string.
In relation to the equal tempered third, the Pythagorean third is approximately 8 cents sharper and the Harmonic third is 14 cents flatter. (A cent is a rather artificial measurement, meaning 1 / 100 of an equal tempered semitone).
The first, the harmonic third, is played lightly by a flute, and blends almost undetectably into the viola sound.