Positive Linear Temperaments

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


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