source file: mills2.txt Date: Tue, 27 May 1997 01:07:50 +0200 Subject: Difference pitches re: Manuel's comments on meantone & Erv. From: Charles Lucy Prompted by Manuel's recent comments on beating and difference tones I have been playing with some numbers and arrived at some odd results. It seems that Harrison's concept, when applied, generates further notes within the expected scales to very close values. Maybe someone with less vested interest can interpret the significance of these differences. LucyTuning A Major chord. (All Values in Hz.) A4 440 C#5 548.6224 E5 657.5411 A5 880 A3 220 A2 110 Difference pitches A4 to C#5 108.6224 (¢ -1.3776) * 256 176.3328 (ó - 0.1092) A4 to E5 217.5411 (£ -2.4589) * 64 157.3696 (ë3 + 0.3571) A4 to A5 440 (¤) C#5 to E5 108.9188 (¢ -1.1012)* 128 140.9536 (Û3 - 0.5545) C#5 to A5 331.3776 (ã +2.6071)* 128 333.7088 (ä + 4.9383) E5 to A5 222.4589 (£ +2.4589)* 64 157.3696 (ë3 + 0.3571) NOTES: E4 328.7706; F3 176.4420; Eb3 157.0125; Db3 141.5081 F2 88.2210 F4 352.8839 LucyTuning A minor chord. (All Values in Hz.) A4 440 C5 527.3539 E5 657.5411 A5 880 C3 131.8385 Difference pitches A4 to C5 87.3539 (ò -0.8671) * 128 110.9888 (¢ + 0.9888) A4 to E5 217.5411 (£ -2.4589) * 64 157.3696 (ë3 + 0.3571) A4 to A5 440 (¤) C5 to E5 130.1872 (Ã -1.6513) * 128 211.3626 («3 - 0.1086) C5 to A5 352.6461 (ô -0.2378) * 512 121.7536 (² - 1.0760) E5 to A5 222.4589 (£ +2.4589) * 64 157.3696 (ë3 + 0.3571) NOTES: Ab3 211.4712 B2 122.8296 So by playing A Major we generate A C# E plus approximations of Db Eb & F So by playing A minor we generate A C E plus approximations of Ab B & F which may partly explain why the Major bVIths (A C# E F) and the Minor Ninths (A C E B) also always sound so consonant in this tuning. lucy@hour.com 25 May 97. Comments welcomed http://www.wonderlandinorbit.com/projects/lullaby Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 27 May 1997 09:24 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00922; Tue, 27 May 1997 09:24:46 +0200 Date: Tue, 27 May 1997 09:24:46 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00920 Received: (qmail 2099 invoked from network); 27 May 1997 07:24:42 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 27 May 1997 07:24:42 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu