Secor: Difference between revisions

[[File:Derivation of the secor.png|thumb|600px|A diagram taken from George Secor's article "The Miracle Temperament and Decimal Keyboard" which was published in Xenharmonikôn 18 (2006). This version includes minor revisions for clarity, done by Douglas Blumeyer on Dave Keenan's request. <br><br>This diagram visually demonstrates how the secor is found as the interval that nearly-equally subdivides all six of the smallest odd harmonics — 1, 3, 5, 7, 9, and 11 — where the width of the error band is narrowest, thus minimizing the maximum error of any interval in the 11-odd-limit [[tonality diamond]]. <br><br>Note that the method here is not to minimize the ''absolute'' deviation from 0{{cent}} in each harmonic individually, but to minimize the ''relative'' difference between the errors of the harmonic with the greatest positive error and the harmonic with the greatest negative error. In other words, the method is not to minimize the maximum distance of the diagonal harmonic lines from the horizontal 0{{cent}} axis, but to minimize the width of the band between the highest harmonic line and the lowest harmonic line at any given vertical slice through the chart. <br><br>Interestingly, the case of the secor, both methods would happen to give the same result. This is because at the point where the difference in errors of the harmonic is least here, all errors are non-positive. So the odd harmonic with the most positive error is actually 1, with an error of 0{{cent}}, because it is purely tuned of course, and therefore its harmonic line is identical with the horizontal 0{{cent}} axis. And the odd harmonic with the most negative error is a tie between 5 and 9, both of which have an error of -3.323{{cent}}.]]