<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div>On Oct 14, 2008, at 5:06 PM, Porritt, David wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Consolas; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; ">I was pointing out the small difference in the 12<sup>th</sup><span class="Apple-converted-space"> </span>root of 2 and the 19<sup>th</sup><span class="Apple-converted-space"> </span>root of 3.  Differences which are mathematically significant (if you’re looking at decimals to the 5<sup>th</sup><span class="Apple-converted-space"> </span>digit) but insignificant if you’re tuning real pianos with real inharmonicity with contemporary tools.  The pitch change from start to 4 seconds later is much wider than that.  When in the decay cycle are you measuring this to 5 digit accuracy? </span></blockquote></div><br><div><div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><span class="Apple-tab-span" style="white-space:pre"> </span>The difference is very small when applied to individual half steps. It becomes somewhat significant when applied across the entire range of a piano. How significant? Well, if you start both "base 2" and "base 3" at the same note, the divergence is approximately 2 cents when you reach a 12th. Expand that over the entire range of the piano, and it becomes about 9 cents. "Somewhat significant." But you only have to change base 2 to base 2.0125 or thereabouts for the convergence to be absolute.</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>In any case, this is nowhere near the "stretch reality" of the real inharmonic piano. I think that we can agree that the spread from top to bottom of a fairly conservative tuning would be in the neighborhood of 50 cents. (It is difficult to measure the first partial of notes as low as A0, so I am estimating and extrapolating a bit, but I think we can agree that this is a reasonable estimate). For those of us who stretch more, that divergence can easily be 100 cents.</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>So to my way of thinking, the 19th root of 3 is irrelevant. In real tuning, one might take an expanded 3 as the basis, a 3 that corresponded to the measured distance between the 1st and 3rd partial of a sample note. This is simply, to my mind, nothing more or less than selecting some degree of stretch to use.  If your choice for stretch is to accept a 3:1 partial match, that's a perfectly legitimate choice. So is a 4:1. Or a 4:1 expanded by 0.5 cents. Or 3:1 expanded by 0.5 cents. All of these will "account naturally for the inharmonicity of the piano." All "fall along a continuum" of increasing or decreasing stretch.</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>In an inharmonic real piano, base 2, base 3, base 4, or any other whole number doesn't work except as a starting point. In any of these cases, it will be necessary (for purposes of calculation) to establish an expanded whole number: 2.15, 3.1, 4.06, something along these lines (these are imaginary numbers, not based on experience or calculation, chosen for example purposes only), on the basis of which to make the division of the octave. The same principle is involved: a single proportional multiplier to produce proportionally equal semitones. (And if you want to do unequal, this provides a very useful basis for WT offsets). There are other mathematical paths to the same result, but the result is identical.</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>I am not for a minute arguing for or against 3:1 as a "stretch basis." I think, personally, that it is quite acceptable for the mid range. I prefer a stretch based on wider intervals, either 6:1 or 8:1 for the outer ranges. That is simply my own personal preference based on my own taste and experience. I don't argue against the tastes of others. (I do think that most of us would agree that a bass tuning based on 3:1 would be too narrow).</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>What I am trying to do is to point out that, IMO, "there is nothing magic" about the 19th root of 3 as a basis for tuning. It is simply indistinguishable from other mathematical ways of establishing equal half step relationships in the real, inharmonic world of piano tuning.<br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Stopper argues otherwise (see his article, referenced in a post I sent previously). I don't find his arguments at all compelling. Others may. He makes the 19th root of 3 division the basis for the "Stopper comma," which he makes great claims for. He does say that to the "additional stretch" produced by beginning with a pure 12th must be added the inharmonicity of the piano, though his explanation of how this is done is VERY vague, and doesn't demonstrate a very good grasp of the complexities involved. An example of his explanation of inharmonicity and tuning:</div><div> "The inharmonicity itself pushes the whole scale away from the theoretical frequencies derived by the scale functional formula. The inharmonicity is already considered when tuning aurally, since the ear makes an integration of the harmonics to a “virtual pitch.”  If an aural tuner tunes a slight beat-rate-narrow fifth, that fifth remain about the same amount beat-rate-narrow in instruments with different inharmonicity, wheras the absolute frequency deviation is up to some cents on stiff strings in the treble." </div><div><span class="Apple-tab-span" style="white-space:pre"> </span>He claims "the recent discovery of the Supersymmetry between the beats and the frequencies" based on his tuning. Perhaps if it is demonstrated to me, I will be blown away. I am skeptical. Actually, he seems more focused on electronic and other "essentially harmonic" instruments than on acoustic pianos. </div><div>Regards,</div><div>Fred Sturm</div><div>University of New Mexico</div><div><a href="mailto:fssturm@unm.edu">fssturm@unm.edu</a></div><div><br class="webkit-block-placeholder"></div></div></div><br><div></div></div></body></html>