This is a multi-part message in MIME format. ---------------------- multipart/alternative attachment Yes, I would agree with you about the oddity of this temperament, but I am pretty sure that I know what happened. To me it sounds like this is a kind of Kirnberger II temperament, with some modifications and some misunderstandings all mixed together: The temperament should have started on C instead of A. Unfortunately, the only way to make the directions you were given make sense is to use A as a starting note for the temperament. This is not such a huge deal, but it may have caused some confusion in the transposition. With a temperament based on a cycle of fifths, there should be alterations that total a -23.46 cent comma. Now in theory, this amount can be distributed, either equally or unequally, among any number of intervals (1-12). Kirnberger II has two unequal tempered intervals. The following is Kirnberger II transposed with A as the starting note: A 440.00 D 586.66 [P5 down] G 782.22 [P5 down] C 521.48 [P5 down] F 695.31 [P5 down] Bflt 463.54 [P5 down] (A 440.00) E 660.00 [P5 up] B 495.00 [P5 up] (A 440.00) C# 550.00 [M3 up] G# 825.00 [P5 up] D# 618.75 [P5 up] F# is tempered with an equal beat rate between D-F# [M3] and F#-B [P4] D-F# as a Just M3 would make F# 733.32 F#-B as a Just P4 would make F# 742.50 To get an equal beat rate add the two frequencies and divide by two: F# 737.91 [Equal Beating] The "Other" temperament Tune by Just 5ths E should be tempered A and B [mean tone?] F# should be a Just M3 with D (First tune the cycle without the tempered E) A 440 E 660 B 495 F# 742.5 C# 556.87 G# 835.32 D# 626.48 A# 469.86 F 352.40 C 528.60 G 792.89 Next tune D as a Just M3 to F#: D 594 I am not sure I understand exactly how to make E a meantone between A and B. Generally a meantone temperament uses narrow 4ths and wide 5ths. The only thing that I can think of is that that you tune A - E - B, and then lower E to make a widened 5th between E-B (a widened 5th between A-E would be reverse temperament, so that will not work). But how much should it be lowered? How do you tell? The only way to make this work [with only two tempered intervals] is to let E absorb the rest of the comma (-21.576): E 651.83 (-8 bps) After you follow through with the rest of the adjustments, the frequencies should be as listed below (hopefully I haven't made too many calculation errors): Kirnberger II "The Other" A 440 440 A# 469.9 469.9 B 495 495 C 528.6 528.6 C# 556.9 556.9 D 594.67 594 D# 618.7 626.5 E 660 651.83 F 704.8 704.8 F# 737.9 742.5 G 792.9 792.9 G# 825 835.32 I told him that there weren't going to be any thirds tuned "pure" in the Kirnberger. Well, Kirnberger II does have one M3, which normally falls on C-E. The M3 is used to lower the amount of the comma by -21.5 cents. Remember the total comma is only -23.462 cents. And if any of the historical temperaments would have a pure third, it would be C-E. Was I wrong? Not at all . . . well a little bit. The general understanding is that well temperaments will start on C and then move away. The result is that the closer you remain in relation to C, the more harmonious the key area will sound. But there is nothing stopping a tuner from starting on C# instead! One last interesting note: in working through the calculations of Kirnberger II, I found that the comma never really gets properly absorbed. The M3 accounts for -21.5 cents, but the interval tempered with equal beating rates between M3 and P4 add an additional -10.7 cents; this means that this last interval was tempered too much (+8.7 cents). To officially close the loop, set the intervals to beat equally, and then raise the tempered interval slightly. Bradley M. Snook ---------------------- multipart/alternative attachment An HTML attachment was scrubbed... URL: https://www.moypiano.com/ptg/pianotech.php/attachments/66/51/4f/4e/attachment.htm ---------------------- multipart/alternative attachment--
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