source file: mills2.txt Date: Mon, 10 Mar 1997 18:19:40 -0800 Subject: Positive Linear Temperaments From: John Chalmers Here are some positive tunings which approximate just intervals or distribute the errors more or less equitably. Pythagorean , 3/2 Just 701.9550008653874 Error in Cents Errors in Squared Cents E30 E320 E51.953721 E523.817025 E73.804081 E7214.47104 E351.953721 E3523.817025 E373.804081 E37214.47104 E575.757802 E57218.28806 E3575.757802 E357218.28806 6/5 Just, 1/9-SKHISMA, 3 Sabat-Garibaldi 701.7379207778391 TF(F-T+5*OCT)/9 E3.2170801 E324.712376E-02 E5.2170801 E524.712376E-02 E76.843203 E7246.82942 E35.4341602 E3529.424753E-02 E377.060283 E37246.87654 E577.060283 E57246.87654 E3577.277363 E357246.92367 15/8 Just, 1/7-SKHISMA, 3op5 701.6758978956825 TF(5*OCT-F-T)/7 E3.279103 E327.789847E-02 E5.279103 E527.789847E-02 E77.711523 E7259.46759 E35.558206 E352.1557969 E377.990626 E37259.54549 E577.990626 E57259.54549 E3578.269729 E357259.62338 5/4 Just, 1/8-SKHISMA, Helmholtzian 701.7107857668957 TFOCT-(T+3*OCT)/8 E3.2442151 E325.964101E-02 E50 E520 E77.223093 E7252.17307 E35.2442151 E3525.964101E-02 E377.467308 E37252.23271 E577.223093 E57252.17307 E3577.467308 E357252.23271 7/4 Just 702.226720966491 TFOCT-(S+5*OCT)/14 E3.2717201 E327.383181E-02 E54.127481 E5217.0361 E70 E720 E354.399202 E35217.10994 E37.2717201 E3727.383181E-02 E574.127481 E57217.0361 E3574.399202 E357217.10994 7/6 Just, 7702.2086062930842 TF(F-S+9*OCT)/15 E3.2536054 E326.431571E-02 E53.982564 E5215.86082 E7.2536054 E726.431571E-02 E354.23617 E35215.92513 E37.5072109 E372.1286314 E574.23617 E57215.92513 E3574.489775 E357215.98945 21/16 Just, 7op3 702.2476225127298 TF(9*OCT-F-S)/13 E3.2926216 E328.562743E-02 E54.294694 E5218.4444 E7.2926216 E728.562743E-02 E354.587316 E35218.53002 E37.5852433 E372.1712549 E574.587316 E57218.53002 E3574.879937 E357218.61565 7/5 Just, 7702.9146345659516 TF(T-S+4*OCT)/6 E3.9596337 E32.9208968 E59.630791 E5292.75212 E79.630791 E7292.75212 E3510.59042 E35293.67302 E3710.59042 E37293.67302 E5719.26158 E572185.5042 E35720.22121 E3572186.4251 35/32 Just, 7op5 702.0391081666382 TF(14*OCT-T-S)/22 E3.0841073 E327.074038E-03 E52.626579 E526.898918 E72.626579 E726.898918 E352.710686 E3526.905993 E372.710686 E3726.905993 E575.253159 E57213.79784 E3575.337265 E357213.80491 LST35 701.714542922257 TF(2*F-16*T+80*OCT)/130 E3.2404579 E325.782002E-02 E53.005724E-02 E529.034378E-04 E77.170493 E7251.41597 E35.2705152 E3525.872346E-02 E377.410951 E37251.47379 E577.20055 E57251.41687 E3577.441008 E357251.47469 LST357 702.099167047429 TF(2*F-16*T-28*S+332*OCT)/522 E3.1441662 E322.078389E-02 E53.10705 E529.653761 E71.785755 E723.18892 E353.251216 E3529.674545 E371.929921 E3723.209704 E574.892805 E57212.84268 E3575.036971 E357212.86347 LST35711 702.4344609176296 TF(2*F-16*T-28*S-36*E+728*OCT )/1170 E3.4794601 E32.2298819 E55.789401 E5233.51717 E72.908359 E728.458554 E356.268861 E35233.74705 E373.387819 E3728.688436 E578.697761 E57241.97572 E3579.17722 E357242.2056 LST3571113 702.6053399022005 TF(2*F-16*T-28*S-36*E-42*TH+ 1274*OCT)/ 2052 E3.650339 E32.4229409 E57.156433 E5251.21453 E75.300665 E7228.09705 E357.806772 E35251.63747 E375.951004 E37228.51999 E5712.4571 E57279.31158 E35713.10744 E357279.73453 This was a theoretical study; I do not think the 15/8, 21/16, and 35/32 to be important harmonic intervals, but I do think the result of setting the errors of the harmonics equal and equal/opposite to be interesting. The methods are the same as in the earlier post on negative systems. The functions are (F-X), (T+8*X-5*OCT), (S+14*X-9*OCT), (E+18*X-11*OCT) and (Th+21*X-13*OCT). The latter two are 11/8 and 13/8. I have not included the 6/5 or 5/3 in these calculations. See my paper in XH2. Doubly postive systems resembling 22-tet are also possible. I will post some such tunings at a later date. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 11 Mar 1997 03:33 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01120; Tue, 11 Mar 1997 03:33:31 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01118 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id SAA17391; Mon, 10 Mar 1997 18:32:04 -0800 Date: Mon, 10 Mar 1997 18:32:04 -0800 Message-Id: <3389a501.891562562@kcbbs.gen.nz> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu