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<< that Pianos are not tuned in what I would call a pure octave based fashion. And if a piano was tuned, so that say each C, was a pure 'mathematical' octave higher than the previous one - then playing peices of music which had a broad range i.e. hands far apart, would sound out of tune. >>
Greetings, The various frequencies emanating from a vibrating string are in a specific order. The lowest is the fundamental, the next is theoretically twice that, forming an octave tone 'over' the fundamental. Thus we get a confusing use of 'overtone' and I will herewith refrain from using that word. The more useful term is 'partial'. The fudamental is the 1st partial, the octve is the 2nd. In a string, this second partial is slightly more than double the fundamental, due to inharmonicity of the string. When you tune the note an octave above the first note, you are matching its fundamental with the 2nd partial of the lower note. So, if the second partial is slightly sharp of the fundamental, the resulting octave is going to be slightly wide from a perfect match. As you go up the scale, tuning octaves from octaves, the cumulative error of inharmonicity begins to require that more and more sharpness enters into getting a clean octave. This is the source of 'stretch' in a tuning. A tuning with pure octaves all the way through will sound flat in the treble to our ears. At some point, the single octave will begin to have a beat in it, just so that the triple octaves will not sound flat. The same holds true in the bass, just in the other direction.
<< In the 16th/17th century spinnets (etc) were tuned in this manner and they found that only peices played with the hands close together in the mid-range sounded harmonious to the ear. >>
Hmm, I don't know about this. The keyboard tuning of the 16 and 17th centuries was meantone, in various forms. This tuning has pure thirds in it, and tenths and 17ths are as clear as thirds. The thirds do begin to sound dissonant if played too low, as the non-communal partials approach near enough to each other to create beating. Plomp and Levelt's work in the sixties offer the 'critical band' concept to explain this.
<< Consequently I am told, pianos and modern keyboard instruments are tuned in a more 'relative' manner up the keyboard using fifths. >>
Modern keyboards are being tuned in a wide variety of ways these days. The whole gamut of tunings have become available and temperament variety is being introduced in many places around the country,(and world, if my foreign contacts are to be believed) Regards,
Ed Foote Nashville, Tennessee Coming this fall from the Temperament Revival, two new well tempered CD's; 'Six Degrees of Tonality' and 'Menage A Mozart'. more info later!
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