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I’ve been aware that my series of posts on just intonation intervals was incomplete.  It was necessary to get the Dos, Dis, Deis, and Ducks all in a row before summarizing. 

Imagine my surprise, after carefully identifying all the small intervals, to find that there is only one minor second: it is approximately 112 cents, somewhat larger than the equal tempered minor second, and found between MI and FA (and many other places). 

In addition there are the two chromatic or augmented unisons, approximately 70 and 92 cents, as between a note and its most normal chromatic alteration (C to C#, say, with the latter having a low-tuned and a high-tuned variant). The latter, as you will see in the list below, is the sum of the small chromatic augmented unison and the 22 cent comma. In my writing, I find both necessary.

And, as we found in the Mozart Ave Verum Corpus, retuning a unison over a changing harmonic background means an intonational change in the unison of approximately 22 cents, the syntonic comma.   This comma is most commonly found in diatonic music between the low-tuned RE of the ii chord and the high-tuned RE of the V chord. 

And finally, there is the enharmonic unison, a syntonic comma less a schisma, or approximately 20 cents, between, say, a C# and a Db, with the former tuned lower than the latter.

The complete solfege scale using my modification of the chromatic solfege syllables for a given tonal center, then, might be sung in order thus:

 

DO DOI DI RA RE REI ||  REI RI ME MEI MI || MI MII FE FA || FA FI SE SOL from root to fifth,

and

SOL SI LE LA LI TE TEI TI DE DO from fifth up to octave. 

 Looking in detail at the first two steps, DO to REI (high RE). a large whole step in five subintervals, and REI (high RE) to MI:, a small whole step in four subintervals:

DO DOI DI RA RE REI

(in order, in cents) 70, 22, 20, 70, 22 (and in notes) C C#(low) C#(high) Db D(low) D(high)

REI RI, ME, MEI, MI

(in order, in cents) 70, 20, 22, 70

 

Is all this realistic?  Testimony of one:  In the privacy of my studio, I hum and groan as Beethoven is supposed to have done, and it might sound out of tune to others, but I use the above solfege syllables to keep track of where I am intonationally.

[edit] Update: recalling the Pythagorean major third, one might find another minor second between the perfect fourth and the large, Pythagorean major third. I don’t believe the third should be used melodically, or that it can be tuned, but if it were, the remaining minor second, say from high (Phyth.) MI to FA, would be a small minor second, an interval of 90 cents (the larger chromatic second less the enharmonic schisma: 92 – 2 = 90).