Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] Exotic harmonies



(1) It seems strange to me for someone to say that musical scales and their temperaments are tangential to the discussion of small whole-number ratios when small whole-number ratios provide the historical context for the development of the musical scale. The original topic we were asked to comment on was why we don't have ratios involving seven (8:7, 9:7, 10:7) in our scale. That means the orignal question was, in fact, a question about scales and temperaments. It was not a question about octaves, stretched or otherwise.

The simple answer to the question is that the 7th harmonic is way out of tune with the nearest note one would choose based upon the lower harmonics, which predominate and are more important.

For musical note C, the 7th harmonic ends up considerably flat of A# and way sharp of A. It just doesn't fit in a musical scale based upon 2:3, 5:4, 4:3 ratios because new notes based on 8:7, 9:7, etc. are going to beat objectionably when played with other notes in the established scale.

So the problem is not how musical notes based on the seventh harmonic will sound when played with the fundamental or with eath other, but how they would sound when played with other notes in a scale consisting of notes determined by 2:3, 5:4, etc. ratios.

If you fix musical note G as being the 3rd harmonic of C, and fix musical note E as being the 5th harmonic of C, then if you base B upon the 3rd harmonic of E and also the 5th harmonic of G, there is no discrepancy. You get the same frequency for B regardless of whether you figure it based upon E or based upon G. That's pretty amazing and is the type of thing that led to the development of the scale we use today even though we have ended up with a compromised scale.

(2) I repeat that the piano is the worst-case stringed intrument to pick in terms of "inharmonic harmonics" because of its very stiff strings. The historical context for the early scales was based upon harpsichords and organs which were not "stretched" because they don't suffer from anywhere near as bad of inharmonicities. Even the early pianoforte did not use as stiff of strings as today's pianos.

There are even worse cases than the piano, but not involving strings. One that comes to mind is carillon bells. The typical bell-shaped bell is of course very stiff and has a very inharmonic 5th harmonic (which is the basis for the third note on the musical scale). The 5th harmonic is very flat in this case, so a music-scale third based upon this harmonic is so flat it is nearly a minor third. Carillon players of traditional carillons have to be very careful to avoid note parings that beat badly. I said "traditional carillons" because a bell foundary in the Netherlands is making carillons with bells newly designed by music scientists at Eindhoven University of Technology. The bells have a modified shape and have a harmonic series that is much closer to integer multiples of the fundamental. These were featured in a Nova program called "What is Music" that originally aired in 1989. I have an off-the-air video tape of it, but it seems it is no longer availaable from WBGH (Boston). Too bad, because it is a good show.

(3) Some musicians, perhaps especially violin players and singers, do not like stretched tuning for a piano. They seem to hear and prefer octaves as not stetched and this can lead to conflicts when they are accompanied by a piano that is stretched. This means that some tuners will not stretch a piano (or not stretch it as much) if it is not being used as a solo instrument. In settings where pianos and organs are both present and played together, it can be particularly tough for two reason... (a) the organ is more true to the equal-tempered scale (not stretched), (2) the air temperature changes the overall tuning of the organ much more than it affects the tuning of the piano.

(4) When tuning a piano, you first tune the "temperament octave" (often F below middle-C to F above middle-C) or sometimes a temperament tenth (F to A) and this is tuned true, that is, not stretched. Thus, this central octave is tuned true to the equal-tempered scale (or whatever temperament you are using). Therefore this "termperment octave" on the piano is indeed based upon the small whole-number ratios which have then been altered to produce the desired temperament. After this middle section is tuned first, the notes above and below are tuned by octave relationships to the temperament octave, and this is when the stretching occurs.

Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu