Scales with two kinds of fifth

Here are some examples of MOS scales with both meantone and nearly
pure fifths. The presence of a lot of small steps in these scales is
not an accident, it results from the fact that the difference between
a pure fifth and a meantone fifth is a fraction of a comma, which is a
small step. I hope Ozan will take a look at them.

! twofifths1.scl
152&159[75] in 159-et tuning
75
!
7.547170
15.094340
22.641509
30.188679
37.735849
135.849057
143.396226
150.943396
158.490566
166.037736
173.584906
181.132075
188.679245
196.226415
203.773585
211.320755
309.433962
316.981132
324.528302
332.075472
339.622642
347.169811
354.716981
362.264151
369.811321
377.358491
475.471698
483.018868
490.566038
498.113208
505.660377
513.207547
520.754717
528.301887
535.849057
543.396226
550.943396
649.056604
656.603774
664.150943
671.698113
679.245283
686.792453
694.339623
701.886792
709.433962
716.981132
724.528302
822.641509
830.188679
837.735849
845.283019
852.830189
860.377358
867.924528
875.471698
883.018868
890.566038
988.679245
996.226415
1003.773585
1011.320755
1018.867925
1026.415094
1033.962264
1041.509434
1049.056604
1056.603774
1064.150943
1162.264151
1169.811321
1177.358491
1184.905660
1192.452830
1200.000000

! twofifths2.scl
19&159[64] in 159-et tuning
64
!
7.547170
15.094340
22.641509
67.924528
75.471698
83.018868
128.301887
135.849057
143.396226
150.943396
196.226415
203.773585
211.320755
256.603774
264.150943
271.698113
316.981132
324.528302
332.075472
339.622642
384.905660
392.452830
400.000000
445.283019
452.830189
460.377358
505.660377
513.207547
520.754717
528.301887
573.584906
581.132075
588.679245
633.962264
641.509434
649.056604
656.603774
701.886792
709.433962
716.981132
762.264151
769.811321
777.358491
822.641509
830.188679
837.735849
845.283019
890.566038
898.113208
905.660377
950.943396
958.490566
966.037736
1011.320755
1018.867925
1026.415094
1033.962264
1079.245283
1086.792453
1094.339623
1139.622642
1147.169811
1154.716981
1200.000000

! toof1.scl
12&224[80] in 224-et tuning
80
!
5.357143
10.714286
16.071429
21.428571
26.785714
32.142857
101.785714
107.142857
112.500000
117.857143
123.214286
128.571429
133.928571
203.571429
208.928571
214.285714
219.642857
225.000000
230.357143
300.000000
305.357143
310.714286
316.071429
321.428571
326.785714
332.142857
401.785714
407.142857
412.500000
417.857143
423.214286
428.571429
433.928571
503.571429
508.928571
514.285714
519.642857
525.000000
530.357143
600.000000
605.357143
610.714286
616.071429
621.428571
626.785714
632.142857
701.785714
707.142857
712.500000
717.857143
723.214286
728.571429
733.928571
803.571429
808.928571
814.285714
819.642857
825.000000
830.357143
900.000000
905.357143
910.714286
916.071429
921.428571
926.785714
932.142857
1001.785714
1007.142857
1012.500000
1017.857143
1023.214286
1028.571429
1033.928571
1103.571429
1108.928571
1114.285714
1119.642857
1125.000000
1130.357143
1200.000000

! toof2.scl
31&224[69] in 224-et tuning
69
!
5.357143
37.500000
42.857143
75.000000
80.357143
112.500000
117.857143
150.000000
155.357143
160.714286
192.857143
198.214286
230.357143
235.714286
267.857143
273.214286
305.357143
310.714286
342.857143
348.214286
353.571429
385.714286
391.071429
423.214286
428.571429
460.714286
466.071429
498.214286
503.571429
508.928571
541.071429
546.428571
578.571429
583.928571
616.071429
621.428571
653.571429
658.928571
691.071429
696.428571
701.785714
733.928571
739.285714
771.428571
776.785714
808.928571
814.285714
846.428571
851.785714
857.142857
889.285714
894.642857
926.785714
932.142857
964.285714
969.642857
1001.785714
1007.142857
1039.285714
1044.642857
1050.000000
1082.142857
1087.500000
1119.642857
1125.000000
1157.142857
1162.500000
1194.642857
1200.000000

No, these won't work Gene. 4 fifths up the chain should give a ~390 cent E
for segah, as well as a ~410 cent E for buselik.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 23 �ubat 2006 Per�embe 1:05
Subject: [tuning] Scales with two kinds of fifth

> Here are some examples of MOS scales with both meantone and nearly
> pure fifths. The presence of a lot of small steps in these scales is
> not an accident, it results from the fact that the difference between
> a pure fifth and a meantone fifth is a fraction of a comma, which is a
> small step. I hope Ozan will take a look at them.
>
> ! twofifths1.scl
> 152&159[75] in 159-et tuning
> 75
> !
> 7.547170
> 15.094340
> 22.641509
> 30.188679
> 37.735849
> 135.849057
> 143.396226
> 150.943396
> 158.490566
> 166.037736
> 173.584906
> 181.132075
> 188.679245
> 196.226415
> 203.773585
> 211.320755
> 309.433962
> 316.981132
> 324.528302
> 332.075472
> 339.622642
> 347.169811
> 354.716981
> 362.264151
> 369.811321
> 377.358491
> 475.471698
> 483.018868
> 490.566038
> 498.113208
> 505.660377
> 513.207547
> 520.754717
> 528.301887
> 535.849057
> 543.396226
> 550.943396
> 649.056604
> 656.603774
> 664.150943
> 671.698113
> 679.245283
> 686.792453
> 694.339623
> 701.886792
> 709.433962
> 716.981132
> 724.528302
> 822.641509
> 830.188679
> 837.735849
> 845.283019
> 852.830189
> 860.377358
> 867.924528
> 875.471698
> 883.018868
> 890.566038
> 988.679245
> 996.226415
> 1003.773585
> 1011.320755
> 1018.867925
> 1026.415094
> 1033.962264
> 1041.509434
> 1049.056604
> 1056.603774
> 1064.150943
> 1162.264151
> 1169.811321
> 1177.358491
> 1184.905660
> 1192.452830
> 1200.000000
>
> ! twofifths2.scl
> 19&159[64] in 159-et tuning
> 64
> !
> 7.547170
> 15.094340
> 22.641509
> 67.924528
> 75.471698
> 83.018868
> 128.301887
> 135.849057
> 143.396226
> 150.943396
> 196.226415
> 203.773585
> 211.320755
> 256.603774
> 264.150943
> 271.698113
> 316.981132
> 324.528302
> 332.075472
> 339.622642
> 384.905660
> 392.452830
> 400.000000
> 445.283019
> 452.830189
> 460.377358
> 505.660377
> 513.207547
> 520.754717
> 528.301887
> 573.584906
> 581.132075
> 588.679245
> 633.962264
> 641.509434
> 649.056604
> 656.603774
> 701.886792
> 709.433962
> 716.981132
> 762.264151
> 769.811321
> 777.358491
> 822.641509
> 830.188679
> 837.735849
> 845.283019
> 890.566038
> 898.113208
> 905.660377
> 950.943396
> 958.490566
> 966.037736
> 1011.320755
> 1018.867925
> 1026.415094
> 1033.962264
> 1079.245283
> 1086.792453
> 1094.339623
> 1139.622642
> 1147.169811
> 1154.716981
> 1200.000000
>
> ! toof1.scl
> 12&224[80] in 224-et tuning
> 80
> !
> 5.357143
> 10.714286
> 16.071429
> 21.428571
> 26.785714
> 32.142857
> 101.785714
> 107.142857
> 112.500000
> 117.857143
> 123.214286
> 128.571429
> 133.928571
> 203.571429
> 208.928571
> 214.285714
> 219.642857
> 225.000000
> 230.357143
> 300.000000
> 305.357143
> 310.714286
> 316.071429
> 321.428571
> 326.785714
> 332.142857
> 401.785714
> 407.142857
> 412.500000
> 417.857143
> 423.214286
> 428.571429
> 433.928571
> 503.571429
> 508.928571
> 514.285714
> 519.642857
> 525.000000
> 530.357143
> 600.000000
> 605.357143
> 610.714286
> 616.071429
> 621.428571
> 626.785714
> 632.142857
> 701.785714
> 707.142857
> 712.500000
> 717.857143
> 723.214286
> 728.571429
> 733.928571
> 803.571429
> 808.928571
> 814.285714
> 819.642857
> 825.000000
> 830.357143
> 900.000000
> 905.357143
> 910.714286
> 916.071429
> 921.428571
> 926.785714
> 932.142857
> 1001.785714
> 1007.142857
> 1012.500000
> 1017.857143
> 1023.214286
> 1028.571429
> 1033.928571
> 1103.571429
> 1108.928571
> 1114.285714
> 1119.642857
> 1125.000000
> 1130.357143
> 1200.000000
>
> ! toof2.scl
> 31&224[69] in 224-et tuning
> 69
> !
> 5.357143
> 37.500000
> 42.857143
> 75.000000
> 80.357143
> 112.500000
> 117.857143
> 150.000000
> 155.357143
> 160.714286
> 192.857143
> 198.214286
> 230.357143
> 235.714286
> 267.857143
> 273.214286
> 305.357143
> 310.714286
> 342.857143
> 348.214286
> 353.571429
> 385.714286
> 391.071429
> 423.214286
> 428.571429
> 460.714286
> 466.071429
> 498.214286
> 503.571429
> 508.928571
> 541.071429
> 546.428571
> 578.571429
> 583.928571
> 616.071429
> 621.428571
> 653.571429
> 658.928571
> 691.071429
> 696.428571
> 701.785714
> 733.928571
> 739.285714
> 771.428571
> 776.785714
> 808.928571
> 814.285714
> 846.428571
> 851.785714
> 857.142857
> 889.285714
> 894.642857
> 926.785714
> 932.142857
> 964.285714
> 969.642857
> 1001.785714
> 1007.142857
> 1039.285714
> 1044.642857
> 1050.000000
> 1082.142857
> 1087.500000
> 1119.642857
> 1125.000000
> 1157.142857
> 1162.500000
> 1194.642857
> 1200.000000
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> No, these won't work Gene. 4 fifths up the chain should give a ~390
cent E
> for segah, as well as a ~410 cent E for buselik.

twofifths1 has 36 major thirds of size 385 cents, 29 somewhat sharp
390.5 cent fifths, 15 Pythagorean major thirds of size 407.5, and
eight 14/11 major thirds of size 415 cents.

twofifths2 has 49 385 cent major thirds, 30 390.5 cent major thirds,
no Pythagorean major thirds, and no 14/11 major thirds.

toof1 has 48 385.7 cent major thirds, 72 391 cent major thirds, 64
Pythagorean major thirds, and 40 14/11 major thirds.

toof2 has 61 385.7 cent major thirds, 15 391 cent major thirds, no
Pythagorean major thirds, and 23 14/11 major thirds.

Your scale has 25 385 cent major thirds, 53 390.5 cent major thirds,
52 Pythagorean thirds, and 27 14/11 thirds.

toof1 seems to be the winner here. It clobbers your scale in terms of
all four of the above kinds of thirds, as well as in quantities of
pure and meantone fifths, and purity of intonation. I suspect if you
looked more carefully, you would find much more than you are now finding.

On the other hand, my scale clobbers toof with its notational integrity by
default. For gosh sakes, E is 402 cents with nothing down till 332 cents. A
is a horrible 900 cents with nothing below till 830 cents. This scale
certainly does not fulfil my expectations in the least. Many Maqams are
mutilated on several degrees, distorted out of recognition and made a
mockery of.

Do you have anything better to suggest Gene? Sheer quantity of consonances
do not impress me at all. What I'm looking for is functionality.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 24 �ubat 2006 Cuma 1:05
Subject: [tuning] Re: Scales with two kinds of fifth

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > No, these won't work Gene. 4 fifths up the chain should give a ~390
> cent E
> > for segah, as well as a ~410 cent E for buselik.
>
> twofifths1 has 36 major thirds of size 385 cents, 29 somewhat sharp
> 390.5 cent fifths, 15 Pythagorean major thirds of size 407.5, and
> eight 14/11 major thirds of size 415 cents.
>
> twofifths2 has 49 385 cent major thirds, 30 390.5 cent major thirds,
> no Pythagorean major thirds, and no 14/11 major thirds.
>
> toof1 has 48 385.7 cent major thirds, 72 391 cent major thirds, 64
> Pythagorean major thirds, and 40 14/11 major thirds.
>
> toof2 has 61 385.7 cent major thirds, 15 391 cent major thirds, no
> Pythagorean major thirds, and 23 14/11 major thirds.
>
> Your scale has 25 385 cent major thirds, 53 390.5 cent major thirds,
> 52 Pythagorean thirds, and 27 14/11 thirds.
>
> toof1 seems to be the winner here. It clobbers your scale in terms of
> all four of the above kinds of thirds, as well as in quantities of
> pure and meantone fifths, and purity of intonation. I suspect if you
> looked more carefully, you would find much more than you are now finding.
>
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> On the other hand, my scale clobbers toof with its notational
integrity by
> default. For gosh sakes, E is 402 cents with nothing down till 332
cents. A
> is a horrible 900 cents with nothing below till 830 cents.

If your C is the zero scale degree, then A is 900 cents. If it is
degrees 1 through 6, then it is a whole range of possible A values
flatter than this. One of them is probably close to the A you want.
The same comment applies to E.

This scale
> certainly does not fulfil my expectations in the least. Many Maqams are
> mutilated on several degrees, distorted out of recognition and made a
> mockery of.

Give an example of one, and I'll check to see if this is true. It
would help to know what I'm aiming for here.

> Do you have anything better to suggest Gene? Sheer quantity of
consonances
> do not impress me at all. What I'm looking for is functionality.

I don't think you've really looked at this scale yet.

I have rotated the scale to the third degree. The gaps in the structure of
80 MOS 224-tET are still too wide for convenient approximations of 13/12,
12/11 and 11/10.

I appreciate your help, but this must be improved further before I am
totally satisfied with it.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 24 �ubat 2006 Cuma 1:46
Subject: [tuning] Re: Scales with two kinds of fifth

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > On the other hand, my scale clobbers toof with its notational
> integrity by
> > default. For gosh sakes, E is 402 cents with nothing down till 332
> cents. A
> > is a horrible 900 cents with nothing below till 830 cents.
>
> If your C is the zero scale degree, then A is 900 cents. If it is
> degrees 1 through 6, then it is a whole range of possible A values
> flatter than this. One of them is probably close to the A you want.
> The same comment applies to E.
>
> This scale
> > certainly does not fulfil my expectations in the least. Many Maqams are
> > mutilated on several degrees, distorted out of recognition and made a
> > mockery of.
>
> Give an example of one, and I'll check to see if this is true. It
> would help to know what I'm aiming for here.
>
> > Do you have anything better to suggest Gene? Sheer quantity of
> consonances
> > do not impress me at all. What I'm looking for is functionality.
>
> I don't think you've really looked at this scale yet.
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I have rotated the scale to the third degree. The gaps in the
structure of
> 80 MOS 224-tET are still too wide for convenient approximations of
13/12,
> 12/11 and 11/10.

What is the set of intervals which you want approximated? So far we
have octaves, fifths, fourths, major and minor thirds, 13/12, 12/11,
11/10, and probably 14/11, as well as a meantone fifth and a superpyth
fifth. What else?

Include the 7 and 17 limit consonances, and we are done.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 24 �ubat 2006 Cuma 2:29
Subject: [tuning] Re: Scales with two kinds of fifth

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I have rotated the scale to the third degree. The gaps in the
> structure of
> > 80 MOS 224-tET are still too wide for convenient approximations of
> 13/12,
> > 12/11 and 11/10.
>
> What is the set of intervals which you want approximated? So far we
> have octaves, fifths, fourths, major and minor thirds, 13/12, 12/11,
> 11/10, and probably 14/11, as well as a meantone fifth and a superpyth
> fifth. What else?
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Include the 7 and 17 limit consonances, and we are done.

If you want all consonances up to the 17 limit plus three kinds of
fifth, you'll end up with something impractical, I think. Why not give
the specific intervals you truly need?