Reply to PaulE/Lst Sqr tunings

Paul: I agree that most of my lst sqr tunings have slightly worse
major sixths than your two. However I'm still not completely convinced
that the intonation of the 5/3 and 6/5 are as important as the
that of 3/2 and 5/4, but even if it is, there are not independent
intervals. In triads, one has both a major third and a minor third and
adjusting either the fifth or the major third adjusts the minor third
as well. There are only two degrees of freedom, not three, which reduces
to one in meantone-like tunings or their positive analogs.

Since the differences between the fifths and thirds of these tunings are
measured in fractions of a cent, the question is rather moot for
synthesizer users at the moment, though Csound and Kyma may have the
required resolution.

Another question is whether the sum of the squared or even absolute
errors
is the proper measure of out-of-tuneness. Errors in the flat direction
may be more serious than sharp ones, but both our techniques conflate
the two.

I thoroughly agree that the triads of 15-tet are more consonant than
those of 10-tet, but this is not relevant. We were both talking about
meantone-like tunings which form their major thirds as 4 fifths up,
their major sixths as 3 fifths up and their minor thirds as the difference
between the fifth and the major third, which is equivalent to taking
the minor third as 3 fifths down. If the fifth is greater than 700 cents,
these relations no longer hold in any meaningful sense.

In the case of 10-tet, the major third is 360 cents and is non-cyclic
with respect to the fifth. It is also the same as the minor third. In
15-tet, there is a non-cyclic minor third of 320 cents and a n-c major
third of 400. The cyclic "thirds" of both tunings are 240 and 480 cents,
a whole tone and a fourth, so neither would be used in triads.

There is also the question of how one weights the intervals, whether or
not one includes the major sixth/minor third in the mix. By varying the
weights, one can bias the tuning towards any of the intervals, though I
think both mine and Paul's can be defended.

I've appended a list of my tunings with their error analyses along with
Paul's and a few novel and historical ones.

Tuning/Errors: Squared Errors (EX2):

JC LST35 again696.8946974308664
TF(2*F+ 8*T+ 16*OCT)/34
E35.060303 E3225.60667
E51.265076 E521.600417
E5/36.325379 E5/3240.01042
E356.325379 E35227.20709
E355/312.65076 E355/3267.21751

JC WLST35 4var697.3737794128327
TF(50*F+ 72*T+ 144*OCT)/338
E34.581222 E3220.98759
E53.181404 E5210.12133
E5/37.762625 E5/3260.25835
E357.762625 E35231.10892
E355/315.52525 E355/3291.36727

JC WLST35 3var696.6967393502981
TF(18*F+ 200*T+ 400*OCT)/818
E35.258262 E3227.64931
E5.4732435 E52.2239594
E5/35.731505 E5/3232.85015
E355.731505 E35227.87327
E355/311.46301 E355/3260.72342

JC WLST35 2var697.0856522774519
TF(10*F+24*T+48*OCT)/106
E34.869349 E3223.71056
E52.028895 E524.116416
E5/36.898244 E5/3247.58577
E356.898244 E35227.82697
E355/313.79649 E355/3275.41274

JC WLST35 1var696.7727624083477
TF(6*F+40*T+80*OCT)/166
E35.182239 E3226.85559
E5.7773358 E52.6042509
E5/35.959574 E5/3235.51653
E355.959574 E35227.45985
E355/311.91915 E355/3262.97637

JC LST 4var696.5866874099862
TF(162*F+1800*T+150*MS+3750*OCT)/7812
E35.368313 E3228.81879
E53.303577E-02 E521.091362E-03
E5/35.401349 E5/3229.17457
E355.401349 E35228.81988
E355/310.8027 E355/3257.99445

JC LST 3var695.2732026499269
TF(50*F+72*T+486*MS+630*OCT)/1796
E36.681798 E3244.64643
E55.220903 E5227.25783
E5/31.460895 E5/322.134214
E3511.9027 E35271.90426
E355/313.3636 E355/3274.03848

JC LST 2var695.6957076245525
TF(10*F+24*T+54*MS+102*OCT)/268
E36.259293 E3239.17875
E53.530883 E5212.46714
E5/32.72841 E5/327.444221
E359.790176 E35251.64589
E355/312.51859 E355/3259.09011

JC LST 1var695.829791803032
TF(18*F+120*T+150*MS+390*OCT)/948
E36.125209 E3237.51818
E52.994547 E528.96731
E5/33.130662 E5/329.801047
E359.119756 E35246.4855
E355/312.25042 E355/3256.28654

My Variant of Paul's 696.0743748037858
TF(6*F+40*T+30*MS+110*OCT)/256
E35.880626 E3234.58176
E52.016215 E524.065122
E5/33.864411 E5/3214.93368
E357.896841 E35238.64688
E355/311.76125 E355/3253.58056

PAUL'S LST 2696.0187221912154
TF(18*F+200*T+150*MS+550*OCT)/1268
E35.936279 E3235.2394
E52.238825 E525.012338
E5/33.697453 E5/3213.67116
E358.175104 E35240.25174
E355/311.87256 E355/3253.9229

PAUL'S LST 1696.1648459739642
TF(2*F+8*T+6*MS+22*OCT)/52
E35.790155 E3233.52589
E51.65433 E522.736808
E5/34.135825 E5/3217.10505
E357.444485 E35236.2627
E355/311.58031 E355/3253.36775

17th root of 21 in 696.6513548
E35.303646 E3228.12866
E5.2917053 E52.085092
E5/35.595351 E5/3231.30796
E355.595351 E35228.21375
E355/311.1907 E355/3259.52171

16th root of 35 696.1740565
E35.780944 E3233.41932
E51.617488 E522.616267
E5/34.163456 E5/3217.33437
E357.398432 E35236.03558
E355/311.56189 E355/3253.36995

1/4-comma Meantone, 5/4 Just696.5784284662087
TF( T +2*OCT)/4
E35.376573 E3228.90753
E50 E520
E5/35.376573 E5/3228.90753
E355.376573 E35228.90753
E355/310.75315 E355/3257.81506

LST35696.8946974308664
TF(2*F+8*T+16*OCT)/34
E35.060303 E3225.60667
E51.265076 E521.600417
E5/36.325379 E5/3240.01042
E356.325379 E35227.20709
E355/312.65076 E355/3267.21751

LST37696.9328125302637
TF( 2*F+20*S +100*OCT)/202
E35.022188 E3225.22238
E51.417536 E522.009409
E5/36.439724 E5/3241.47005
E356.439724 E35227.23178
E355/312.87945 E355/3268.70184

LST57696.8406372426774
TF(8*T+20*S+116 *OCT)/232
E35.114364 E3226.15672
E51.048835 E521.100055
E5/36.163199 E5/3237.98502
E356.163199 E35227.25677
E355/312.3264 E355/3265.24179

LST357696.8843497522732
TF(2*F+8*T+20*S+116*OCT)/234
E35.070651 E3225.7115
E51.223685 E521.497405
E5/36.294336 E5/3239.61867
E356.294336 E35227.20891
E355/312.58867 E355/3266.82758

Just 7/4 NEG696.8825906469125
TF( S +5*OCT)/10
E35.07241 E3225.72935
E51.216649 E521.480234
E5/36.289059 E5/3239.55226
E356.289059 E35227.20958
E355/312.57812 E355/3266.76184

Weighted LST35 697.0856522774519
TF(10*F+ 24*T+ 48*OCT)/106
E34.869349 E3223.71056
E52.028895 E524.116416
E5/36.898244 E5/3247.58577
E356.898244 E35227.82697
E355/313.79649 E355/3275.41274

Weighted LST37 696.9982482088972
TF(14*F+ 60*S+ 300*OCT)/614
E34.956753 E3224.5694
E51.679279 E522.819978
E5/36.636032 E5/3244.03691
E356.636032 E35227.38938
E355/313.27206 E355/3271.42629

Weighted LST57 696.8269269798556
TF(56*T+ 100*S+ 612*OCT)/1224
E35.128074 E3226.29714
E5.9939941 E52.9880242
E5/36.122068 E5/3237.47972
E356.122068 E35227.28517
E355/312.24414 E355/3264.76488

Weighted LST357 696.9229
TF(70*F+ 168*T+ 300*S+1836*OCT)/3742
E35.032145 E3225.32248
E51.377709 E521.898082
E5/36.409854 E5/3241.08623
E356.409854 E35227.22057
E355/312.81971 E355/3268.30679

LucyTuning695.4929658426684
TF(OCT/PI +2*OCT)/4
E36.462035 E3241.7579
E54.34185 E5218.85167
E5/32.120184 E5/324.495183
E3510.80389 E35260.60956
E355/312.92407 E355/3265.10474

Wilson's Meta-Meantone695.6304372402378
TFOCT*LOG(1.49453018048#)/LOG(2#)
E36.324564 E3240.00011
E53.791965 E5214.379
E5/32.532599 E5/326.414056
E3510.11653 E35254.3791
E355/312.64913 E355/3260.79316

Wilson's Meta-1/5-Comma Meantone697.0675841384062
TFOCT*LOG(1.49577134782#)/LOG(2#)
E34.887417 E3223.88684
E51.956623 E523.828372
E5/36.844039 E5/3246.84087
E356.844039 E35227.71521
E355/313.68808 E355/3274.55609

3 1/3-Comma, 6/5 Just694.7862376664825
TF(T-F+2 *OCT)/3
E37.168763 E3251.39117
E57.168763 E5251.39117
E5/30 E5/320
E3514.33753 E352102.7823
E355/314.33753 E355/32102.7823

3op5, 1/5-Comma, 15/8 Just697.6537429460444
TF(T+F+2 *OCT)/5
E34.301258 E3218.50082
E54.301258 E5218.50082
E5/38.602516 E5/3274.00328
E358.602516 E35237.00164
E355/317.20503 E355/32111.0049


10-tet, F720, TMajor 360
E318.045 E32325.622
E526.31371 E52692.4116
E5/344.35871 E5/321967.695
E3544.35871 E3521018.034
E355/388.71742 E355/322985.729

15-tet, Fr0, TMajor400, tminor 320
E318.045 E32325.622
E513.68629 E52187.3144
E5/34.358713 E5/3218.99838
E3531.73129 E352512.9364
E355/336.09 E355/32531.9348


The harmonic 7 in the these tunings is generated as 10 fifths up.
Fp1.955+ T86.3137+ S�8.8259+ OCT00.
Wigthed, LSTLeast Square, TFmpered fifth
The error analysis for septimal intervals is not shown. The septimal
tunings were included as they are also good for 3 and 5 limit intervals.

The weightings used in the "variant" tunings are various combinations of
3, 5, 5/3, and their reciprocals.

--John











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