equal beating ovovo temperaments

In Common Era years, between 1600 & 1707, somebody like Abraham
Verheyen, Joseph Sauveur, Gottfried Keller or one of their colleagues,
tweaking what we now call 1/5 comma meantone temperament, discovered how
to tune for equal beating 4:5:6 major triads in 8 of 12 key signatures.
Since that time a variety of other temperaments with equal beating
chords & proportional beating chords have been discovered. Many people
on this forum are better acquainted than I with most of them; I'm not an
expert on any. I happen to like George Secor's Hi-Tol 29. Which do you
like?

Equal or proportional beating effects, played on instruments whose
voices have harmonic timbres, including strings, woodwinds & free reeds,
can sound impressive. Synchronized beating really makes a chord cohere!
For the improviser or composer using such a temperament, certain
keysig-specific moods & colors are set by the beat rates, which are
knowable in advance.

Mark L. Vines here, posting from near Austin, Texas, USA. Late in 2010
through today I either discovered, or else independently rediscovered, a
few temperaments with equal beating septimal tetrads like 4:5:6:7 or
8:9:14:15 in some of their keysigs. I stumbled across the ingredients
while seeking across the mathemusical landscape for an unrelated goal;
my find a lucky happenstance that now has me pitching uncomfortably high
above my actual competence level in the metatuning field.

Perhaps if I briefly describe three of these temperaments, you
mathemusical metatuning people would please graciously tell me whether
any or each of them has already been used or published &, if so, by
whom. I haven't found these temperaments in the Scala archive but, then
again, it's a big archive & my search techniques could be faulty. If
possible, please point us to published evidence of any prior use or
claim.

Kindly also point any mistake you may find in my work. Also, please
forgive my D-centric notation systems, which do little more than crudely
tag the notes according to the chain in which each belongs. Letter
naturals, flats & sharps are assigned only to notes from the origin or
zero-anchor chain. Ups, downs & their doubles, denoted by slash,
backslash & graphemic substitute characters, are assigned only to notes
from corresponding nonzero-anchor chains.

In case I discovered any of these temperaments first, I refer to them as
my ovovo temperaments, recalling their beating tetrads as I first heard
them, in reed organ voice, rather like ovovovovovo....

We'll start with ovovo10. Has anyone else used or published this
temperament or a close relative already?

nickname: ovovo10
full name: vines_ovovo10eb5w6w7
10 key signatures per 2/1 "octave"
chain link span 719.339096 cents
2 chains anchored at 0 & 1 times 375.798846 cents
3 step sizes, near 102, 105 & 137 cents
4:5:6:7 proportional beating in all 5 letter natural keysigs (C, G, D,
A, E)
4:5:6:7 equal beating in 1 of those keysigs (D)
4:5:6 equal beating in 3 keysigs (C, G, D)
4:6:7 equal beating in 4 keysigs (G, D, A, G-down)

ovovo10
keysig D4, edited Scala readouts:

0: 1/1 D
1: 137.121 cents E\ En
2: 238.678 cents E
3: 375.799 cents G\ Gn
4: 480.661 cents G
5: 617.782 cents A\ An
6: 719.339 cents A
7: 856.460 cents C\ Cn
8: 961.322 cents D
9: 1098.442 cents D\ Dn
10: 2/1 D 1 octave

|

Base frequency : 293.0000 Hertz
Beat frequencies of 5/4 3/2 13/8 7/4
0: 0.000: -8.8709 8.8709 35.2150 -8.8709
1: 137.121: -38.6990 9.6021 -38.6990 -9.6021
2: 238.678: -6.9889 12.1293 48.8510 -10.1822
3: 375.799: -44.4196 13.1291 -44.4196 -11.0215
4: 480.661: -11.7097 11.7097 46.4842 -16.8501
5: 617.782: -51.0832 12.6749 -61.3640 -18.2390
6: 719.339: -9.2254 13.4406 53.3556 -13.4406
7: 856.460: -58.6344 14.5485 -58.6344 -14.5485
8: 961.322: -15.4569 15.4569 61.3597 -22.2423
9: 1098.442: -72.5708 16.7310 -81.0012 -24.0757
10: 1200.000: -17.7418 17.7418 70.4300 -17.7418

Total abs. beats : 317.6588 128.2929 529.3838 149.0729
Average abs. beats: 31.7659 12.8293 52.9384 14.9073
Highest abs. beats: 72.5708 16.7310 81.0012 24.0757

Now for ovovo22. Has anyone else used or published this temperament or a
close relative already?

nickname: ovovo22
full name: vines_ovovo22eb9w14w15
22 key signatures per 2/1 "octave"
chain link span 707.922315 cents
3 chains anchored at 0, 1 & 2 times 387.517045 cents
3 step sizes, near 44, 60 & 67 cents
8:9:14:15 equal beating in 4 keysigs (C, G, D, E-down)
8:9:10:11:12:13:14:15 proportional beating in 3 of those keysigs (C, G,
D)
4:5:6:7 proportional beating in 6 keysigs
4:5:6 proportional beating in 10 keysigs
4:6:7 proportional beating in 12 keysigs

ovovo22
keysig D4, edited Scala readouts:

0: 1/1 D
1: 67.112 cents D/ Du
2: 127.500 cents D// Dw
3: 171.672 cents E\ En
4: 215.845 cents E
5: 276.233 cents F
6: 343.345 cents F/ Fu
7: 387.517 cents G\\ Gm
8: 431.689 cents G\ Gn
9: 492.078 cents G
10: 559.189 cents G/ Gu
11: 619.578 cents G// Gw
12: 663.750 cents A\ An
13: 707.922 cents A
14: 768.311 cents A/ Au
15: 835.423 cents A// Aw
16: 879.595 cents B\ Bn
17: 923.767 cents B
18: 984.155 cents C
19: 1051.267 cents C/ Cu
20: 1095.439 cents D\\ Dm
21: 1155.828 cents D\ Dn
22: 2/1 D 1 octave

|

Base frequency : 293.0000 Hertz
Beat frequencies of 9/8 5/4 11/8 3/2
13/8 7/4 15/8
0: 0.000: 18.2415 1.0186 14.6876 3.0350
-11.2156 18.2415 18.2415
1: 67.112: 8.2640 -19.0011 2.2228 -0.3989
37.0580 18.9626 1.1806
2: 127.500: 19.6357 -19.6756 -30.0647 3.2670
38.3735 -1.1305 -42.8956
3: 171.672: 20.1431 1.1248 -30.8417 3.3514
-67.5863 20.1431 20.1431
4: 215.845: 20.6637 16.7823 2.4221 3.4380
-12.7048 20.6637 67.7108
5: 276.233: 21.3972 1.1948 17.2284 3.5600
-54.6882 30.8417 21.3972
6: 343.345: 22.2429 -22.2881 -34.0568 3.7008
43.4688 22.2429 -48.5914
7: 387.517: 54.0746 -5.8337 -34.9369 3.7964
-76.5605 22.8178 1.4206
8: 431.689: -31.1701 19.0107 2.7438 14.5459
-14.3918 23.4074 76.7016
9: 492.078: 24.2383 1.3535 19.5161 4.0328
-14.9027 24.2383 24.2383
10: 559.189: 10.9807 -25.2476 -38.5789 4.1922
49.2407 25.1964 -55.0435
11: 619.578: 26.0908 -26.1438 -39.9484 4.3410
50.9886 -1.5022 -56.9974
12: 663.750: 26.7651 1.4946 -40.9808 4.4532
-89.8050 -1.5410 26.7651
13: 707.922: 27.4567 1.5332 22.1074 4.5682
-16.8815 27.4568 89.9704
14: 768.311: 28.4314 1.5877 22.8922 4.7304
-72.6667 40.9808 28.4314
15: 835.423: 29.5552 -29.6152 -45.2528 4.9174
57.7590 29.5552 -64.5656
16: 879.595: 30.3190 1.6931 -46.4223 5.0445
-101.7295 30.3190 1.8876
17: 923.767: 73.7083 25.2604 3.6458 5.1748
-19.1231 31.1025 101.9170
18: 984.155: 32.2066 1.7985 25.9319 5.3585
-19.8019 32.2066 32.2066
19: 1051.267: 33.4796 -33.5476 -51.2616 5.5703
65.4284 33.4796 -73.1388
20: 1095.439: 81.3920 -8.7807 -52.5863 21.3427
-115.2373 34.3448 2.1382
21: 1155.828: 35.5640 1.9859 -54.4530 5.9171
-119.3280 -2.0476 35.5640
22: 1200.000: 36.4830 2.0373 29.3752 6.0700
-22.4312 36.4830 36.4830

Total abs. beats : 676.0203 265.9715 632.7823 118.7365
1148.9401 492.4220 891.1462
Average abs. beats: 30.7282 12.0896 28.7628 5.3971
52.2246 22.3828 40.5066
Highest abs. beats: 81.3920 33.5476 54.4530 21.3427
119.3280 40.9808 101.9170

Lastly, ovovo27. Has anyone else used or published this temperament or a
close relative already?

nickname: ovovo27
full name: vines_ovovo27eb5w6w7
27 key signatures per 2/1 "octave"
chain link span 713.177722 cents
3 chains anchored at -1, 0 & 1 times 393.056073 cents
3 step sizes, near 34, 39 & 60 cents
4:5:7 equal beating in 13 keysigs
4:5:6:7 equal beating in 12 of those keysigs, including all 7 letter
natural keysigs
(F, C, G, D, A, E, B, G-double-up, E-flat-up, B-flat-up, F-up, C-up)
8:10:11:12:14 proportional beating in 4 of those keysigs (F, C, G, D)
8:10:11:12:13:14 proportional beating in 1 of those keysigs (D)
4:5:6 equal beating in 15 keysigs
4:6:7 equal beating in 18 keysigs

ovovo27
keysig D, edited Scala readouts:

0: 1/1 D
1: 34.111 cents Eb
2: 93.766 cents Eb/ Ebu
3: 132.589 cents E\\ Em
4: 166.701 cents E\ En
5: 226.355 cents E
6: 260.467 cents F
7: 320.122 cents F/ Fu
8: 354.233 cents F// Fw
9: 393.056 cents F#\ F#n
10: 452.711 cents F#
11: 486.822 cents G
12: 546.477 cents G/ Gu
13: 580.588 cents G// Gw
14: 619.412 cents A\\ Am
15: 653.523 cents A\ An
16: 713.178 cents A
17: 747.289 cents Bb
18: 806.944 cents Bb/ Bbu
19: 841.055 cents Bb// Bbw
20: 879.878 cents B\ Bn
21: 939.533 cents B
22: 973.645 cents C
23: 1033.299 cents C/ Cu
24: 1067.411 cents C// Cw
25: 1106.234 cents C#\ C#n
26: 1165.889 cents C#
27: 2/1 D 1 octave

|

Base frequency : 293.0000 Hertz
Beat frequencies of 5/4 11/8 3/2 13/8
7/4
0: 0.000: 5.7166 -8.9995 5.7166 1.1611
5.7166
1: 34.111: -23.4353 -9.1786 5.8304 11.7745
-35.0960
2: 93.766: 6.0348 16.6251 6.0348 12.1873
6.0348
3: 132.589: 25.3927 59.3298 3.5757 1.2535
6.1716
4: 166.701: -6.0796 -9.9092 6.2945 63.6365
-37.8895
5: 226.355: 6.5151 62.6318 6.5151 1.3233
6.5151
6: 260.467: 6.6448 -10.4607 6.6448 13.4192
6.6448
7: 320.122: 6.8777 18.9473 6.8777 13.8896
6.8777
8: 354.233: 7.0146 -58.2132 7.0146 14.1660
13.8923
9: 393.056: 29.5154 -11.2933 7.1737 1.4571
7.1737
10: 452.711: 2.2333 71.3803 7.4252 1.5081
7.4252
11: 486.822: 7.5729 -11.9218 7.5729 15.2936
7.5729
12: 546.477: 7.8384 21.5939 -9.9287 -60.8953
7.8384
13: 580.588: 7.9944 -66.3444 7.9944 16.1447
7.9944
14: 619.412: 33.6381 -12.8708 8.1757 1.6605
8.1757
15: 653.523: -8.0538 -13.1269 8.3384 84.3005
-50.1929
16: 713.178: 8.6307 82.9695 8.6307 1.7530
8.6307
17: 747.289: -35.3815 -13.8574 8.8024 17.7766
-52.9864
18: 806.944: 9.1110 25.0998 9.1110 18.3998
9.1110
19: 841.055: 9.2924 -77.1160 9.2924 18.7660
18.4034
20: 879.878: 39.0996 -14.9604 9.5031 96.0756
9.5031
21: 939.533: 9.8362 94.5587 9.8362 1.9978
9.8362
22: 973.645: 10.0320 -15.7931 10.0320 20.2597
10.0320
23: 1033.299: 10.3837 28.6058 10.3837 -80.6690
10.3837
24: 1067.411: 10.5903 -87.8876 10.5903 21.3872
10.5903
25: 1106.234: 44.5610 -17.0501 10.8305 2.1998
10.8305
26: 1165.889: 3.3718 107.7667 -14.1996 2.2769
11.2102
27: 1200.000: 11.4332 -17.9990 11.4332 2.3222
11.4333

Total abs. beats : 380.8476 1028.4914 222.3249 595.6325
382.7292
Average abs. beats: 14.1055 38.0923 8.2343 22.0605
14.1752
Highest abs. beats: 44.5610 107.7667 14.1996 96.0756
52.9864

Having discovered these three equal beating ovovo temperaments only by
chance, if indeed I did more than rediscover them at all, & lacking much
of the talent & knowledge that many people on this forum have in
abundance, I am inordinately proud of these metatunings, & entertain
egotistical hopes that they will bear my name on into the mathemusical
future. Puncturing such vanity should be fun so, if any of these
temperaments is extant already, please don't scruple to say so.

Regardless of their first discoverer's identity, you might enjoy playing
with these ovovo temperaments.

Meanwhile, by the Common Era calendar, Happy New 2011!
==
Mark L. Vines
a/k/a Manu Phonic

[Non-text portions of this message have been removed]

Hi Mark,

Why not post this to the regular tuning list? You will no doubt find
more help there. Jacques Dudon will have a lot of good info for you
about this.

-Mike

On Sat, Jan 1, 2011 at 11:48 PM, manuphonic <manuphonic@...> wrote:
>
>
>
> In Common Era years, between 1600 & 1707, somebody like Abraham
> Verheyen, Joseph Sauveur, Gottfried Keller or one of their colleagues,
> tweaking what we now call 1/5 comma meantone temperament, discovered how
> to tune for equal beating 4:5:6 major triads in 8 of 12 key signatures.
> Since that time a variety of other temperaments with equal beating
> chords & proportional beating chords have been discovered. Many people
> on this forum are better acquainted than I with most of them; I'm not an
> expert on any. I happen to like George Secor's Hi-Tol 29. Which do you
> like?
>
> Equal or proportional beating effects, played on instruments whose
> voices have harmonic timbres, including strings, woodwinds & free reeds,
> can sound impressive. Synchronized beating really makes a chord cohere!
> For the improviser or composer using such a temperament, certain
> keysig-specific moods & colors are set by the beat rates, which are
> knowable in advance.
>
> Mark L. Vines here, posting from near Austin, Texas, USA. Late in 2010
> through today I either discovered, or else independently rediscovered, a
> few temperaments with equal beating septimal tetrads like 4:5:6:7 or
> 8:9:14:15 in some of their keysigs. I stumbled across the ingredients
> while seeking across the mathemusical landscape for an unrelated goal;
> my find a lucky happenstance that now has me pitching uncomfortably high
> above my actual competence level in the metatuning field.
>
> Perhaps if I briefly describe three of these temperaments, you
> mathemusical metatuning people would please graciously tell me whether
> any or each of them has already been used or published &, if so, by
> whom. I haven't found these temperaments in the Scala archive but, then
> again, it's a big archive & my search techniques could be faulty. If
> possible, please point us to published evidence of any prior use or
> claim.
>
> Kindly also point any mistake you may find in my work. Also, please
> forgive my D-centric notation systems, which do little more than crudely
> tag the notes according to the chain in which each belongs. Letter
> naturals, flats & sharps are assigned only to notes from the origin or
> zero-anchor chain. Ups, downs & their doubles, denoted by slash,
> backslash & graphemic substitute characters, are assigned only to notes
> from corresponding nonzero-anchor chains.
>
> In case I discovered any of these temperaments first, I refer to them as
> my ovovo temperaments, recalling their beating tetrads as I first heard
> them, in reed organ voice, rather like ovovovovovo....
>
> We'll start with ovovo10. Has anyone else used or published this
> temperament or a close relative already?
>
> nickname: ovovo10
> full name: vines_ovovo10eb5w6w7
> 10 key signatures per 2/1 "octave"
> chain link span 719.339096 cents
> 2 chains anchored at 0 & 1 times 375.798846 cents
> 3 step sizes, near 102, 105 & 137 cents
> 4:5:6:7 proportional beating in all 5 letter natural keysigs (C, G, D,
> A, E)
> 4:5:6:7 equal beating in 1 of those keysigs (D)
> 4:5:6 equal beating in 3 keysigs (C, G, D)
> 4:6:7 equal beating in 4 keysigs (G, D, A, G-down)
>
> ovovo10
> keysig D4, edited Scala readouts:
>
> 0: 1/1 D
> 1: 137.121 cents E\ En
> 2: 238.678 cents E
> 3: 375.799 cents G\ Gn
> 4: 480.661 cents G
> 5: 617.782 cents A\ An
> 6: 719.339 cents A
> 7: 856.460 cents C\ Cn
> 8: 961.322 cents D
> 9: 1098.442 cents D\ Dn
> 10: 2/1 D 1 octave
>
> |
>
> Base frequency : 293.0000 Hertz
> Beat frequencies of 5/4 3/2 13/8 7/4
> 0: 0.000: -8.8709 8.8709 35.2150 -8.8709
> 1: 137.121: -38.6990 9.6021 -38.6990 -9.6021
> 2: 238.678: -6.9889 12.1293 48.8510 -10.1822
> 3: 375.799: -44.4196 13.1291 -44.4196 -11.0215
> 4: 480.661: -11.7097 11.7097 46.4842 -16.8501
> 5: 617.782: -51.0832 12.6749 -61.3640 -18.2390
> 6: 719.339: -9.2254 13.4406 53.3556 -13.4406
> 7: 856.460: -58.6344 14.5485 -58.6344 -14.5485
> 8: 961.322: -15.4569 15.4569 61.3597 -22.2423
> 9: 1098.442: -72.5708 16.7310 -81.0012 -24.0757
> 10: 1200.000: -17.7418 17.7418 70.4300 -17.7418
>
> Total abs. beats : 317.6588 128.2929 529.3838 149.0729
> Average abs. beats: 31.7659 12.8293 52.9384 14.9073
> Highest abs. beats: 72.5708 16.7310 81.0012 24.0757
>
> Now for ovovo22. Has anyone else used or published this temperament or a
> close relative already?
>
> nickname: ovovo22
> full name: vines_ovovo22eb9w14w15
> 22 key signatures per 2/1 "octave"
> chain link span 707.922315 cents
> 3 chains anchored at 0, 1 & 2 times 387.517045 cents
> 3 step sizes, near 44, 60 & 67 cents
> 8:9:14:15 equal beating in 4 keysigs (C, G, D, E-down)
> 8:9:10:11:12:13:14:15 proportional beating in 3 of those keysigs (C, G,
> D)
> 4:5:6:7 proportional beating in 6 keysigs
> 4:5:6 proportional beating in 10 keysigs
> 4:6:7 proportional beating in 12 keysigs
>
> ovovo22
> keysig D4, edited Scala readouts:
>
> 0: 1/1 D
> 1: 67.112 cents D/ Du
> 2: 127.500 cents D// Dw
> 3: 171.672 cents E\ En
> 4: 215.845 cents E
> 5: 276.233 cents F
> 6: 343.345 cents F/ Fu
> 7: 387.517 cents G\\ Gm
> 8: 431.689 cents G\ Gn
> 9: 492.078 cents G
> 10: 559.189 cents G/ Gu
> 11: 619.578 cents G// Gw
> 12: 663.750 cents A\ An
> 13: 707.922 cents A
> 14: 768.311 cents A/ Au
> 15: 835.423 cents A// Aw
> 16: 879.595 cents B\ Bn
> 17: 923.767 cents B
> 18: 984.155 cents C
> 19: 1051.267 cents C/ Cu
> 20: 1095.439 cents D\\ Dm
> 21: 1155.828 cents D\ Dn
> 22: 2/1 D 1 octave
>
> |
>
> Base frequency : 293.0000 Hertz
> Beat frequencies of 9/8 5/4 11/8 3/2
> 13/8 7/4 15/8
> 0: 0.000: 18.2415 1.0186 14.6876 3.0350
> -11.2156 18.2415 18.2415
> 1: 67.112: 8.2640 -19.0011 2.2228 -0.3989
> 37.0580 18.9626 1.1806
> 2: 127.500: 19.6357 -19.6756 -30.0647 3.2670
> 38.3735 -1.1305 -42.8956
> 3: 171.672: 20.1431 1.1248 -30.8417 3.3514
> -67.5863 20.1431 20.1431
> 4: 215.845: 20.6637 16.7823 2.4221 3.4380
> -12.7048 20.6637 67.7108
> 5: 276.233: 21.3972 1.1948 17.2284 3.5600
> -54.6882 30.8417 21.3972
> 6: 343.345: 22.2429 -22.2881 -34.0568 3.7008
> 43.4688 22.2429 -48.5914
> 7: 387.517: 54.0746 -5.8337 -34.9369 3.7964
> -76.5605 22.8178 1.4206
> 8: 431.689: -31.1701 19.0107 2.7438 14.5459
> -14.3918 23.4074 76.7016
> 9: 492.078: 24.2383 1.3535 19.5161 4.0328
> -14.9027 24.2383 24.2383
> 10: 559.189: 10.9807 -25.2476 -38.5789 4.1922
> 49.2407 25.1964 -55.0435
> 11: 619.578: 26.0908 -26.1438 -39.9484 4.3410
> 50.9886 -1.5022 -56.9974
> 12: 663.750: 26.7651 1.4946 -40.9808 4.4532
> -89.8050 -1.5410 26.7651
> 13: 707.922: 27.4567 1.5332 22.1074 4.5682
> -16.8815 27.4568 89.9704
> 14: 768.311: 28.4314 1.5877 22.8922 4.7304
> -72.6667 40.9808 28.4314
> 15: 835.423: 29.5552 -29.6152 -45.2528 4.9174
> 57.7590 29.5552 -64.5656
> 16: 879.595: 30.3190 1.6931 -46.4223 5.0445
> -101.7295 30.3190 1.8876
> 17: 923.767: 73.7083 25.2604 3.6458 5.1748
> -19.1231 31.1025 101.9170
> 18: 984.155: 32.2066 1.7985 25.9319 5.3585
> -19.8019 32.2066 32.2066
> 19: 1051.267: 33.4796 -33.5476 -51.2616 5.5703
> 65.4284 33.4796 -73.1388
> 20: 1095.439: 81.3920 -8.7807 -52.5863 21.3427
> -115.2373 34.3448 2.1382
> 21: 1155.828: 35.5640 1.9859 -54.4530 5.9171
> -119.3280 -2.0476 35.5640
> 22: 1200.000: 36.4830 2.0373 29.3752 6.0700
> -22.4312 36.4830 36.4830
>
> Total abs. beats : 676.0203 265.9715 632.7823 118.7365
> 1148.9401 492.4220 891.1462
> Average abs. beats: 30.7282 12.0896 28.7628 5.3971
> 52.2246 22.3828 40.5066
> Highest abs. beats: 81.3920 33.5476 54.4530 21.3427
> 119.3280 40.9808 101.9170
>
> Lastly, ovovo27. Has anyone else used or published this temperament or a
> close relative already?
>
> nickname: ovovo27
> full name: vines_ovovo27eb5w6w7
> 27 key signatures per 2/1 "octave"
> chain link span 713.177722 cents
> 3 chains anchored at -1, 0 & 1 times 393.056073 cents
> 3 step sizes, near 34, 39 & 60 cents
> 4:5:7 equal beating in 13 keysigs
> 4:5:6:7 equal beating in 12 of those keysigs, including all 7 letter
> natural keysigs
> (F, C, G, D, A, E, B, G-double-up, E-flat-up, B-flat-up, F-up, C-up)
> 8:10:11:12:14 proportional beating in 4 of those keysigs (F, C, G, D)
> 8:10:11:12:13:14 proportional beating in 1 of those keysigs (D)
> 4:5:6 equal beating in 15 keysigs
> 4:6:7 equal beating in 18 keysigs
>
> ovovo27
> keysig D, edited Scala readouts:
>
> 0: 1/1 D
> 1: 34.111 cents Eb
> 2: 93.766 cents Eb/ Ebu
> 3: 132.589 cents E\\ Em
> 4: 166.701 cents E\ En
> 5: 226.355 cents E
> 6: 260.467 cents F
> 7: 320.122 cents F/ Fu
> 8: 354.233 cents F// Fw
> 9: 393.056 cents F#\ F#n
> 10: 452.711 cents F#
> 11: 486.822 cents G
> 12: 546.477 cents G/ Gu
> 13: 580.588 cents G// Gw
> 14: 619.412 cents A\\ Am
> 15: 653.523 cents A\ An
> 16: 713.178 cents A
> 17: 747.289 cents Bb
> 18: 806.944 cents Bb/ Bbu
> 19: 841.055 cents Bb// Bbw
> 20: 879.878 cents B\ Bn
> 21: 939.533 cents B
> 22: 973.645 cents C
> 23: 1033.299 cents C/ Cu
> 24: 1067.411 cents C// Cw
> 25: 1106.234 cents C#\ C#n
> 26: 1165.889 cents C#
> 27: 2/1 D 1 octave
>
> |
>
> Base frequency : 293.0000 Hertz
> Beat frequencies of 5/4 11/8 3/2 13/8
> 7/4
> 0: 0.000: 5.7166 -8.9995 5.7166 1.1611
> 5.7166
> 1: 34.111: -23.4353 -9.1786 5.8304 11.7745
> -35.0960
> 2: 93.766: 6.0348 16.6251 6.0348 12.1873
> 6.0348
> 3: 132.589: 25.3927 59.3298 3.5757 1.2535
> 6.1716
> 4: 166.701: -6.0796 -9.9092 6.2945 63.6365
> -37.8895
> 5: 226.355: 6.5151 62.6318 6.5151 1.3233
> 6.5151
> 6: 260.467: 6.6448 -10.4607 6.6448 13.4192
> 6.6448
> 7: 320.122: 6.8777 18.9473 6.8777 13.8896
> 6.8777
> 8: 354.233: 7.0146 -58.2132 7.0146 14.1660
> 13.8923
> 9: 393.056: 29.5154 -11.2933 7.1737 1.4571
> 7.1737
> 10: 452.711: 2.2333 71.3803 7.4252 1.5081
> 7.4252
> 11: 486.822: 7.5729 -11.9218 7.5729 15.2936
> 7.5729
> 12: 546.477: 7.8384 21.5939 -9.9287 -60.8953
> 7.8384
> 13: 580.588: 7.9944 -66.3444 7.9944 16.1447
> 7.9944
> 14: 619.412: 33.6381 -12.8708 8.1757 1.6605
> 8.1757
> 15: 653.523: -8.0538 -13.1269 8.3384 84.3005
> -50.1929
> 16: 713.178: 8.6307 82.9695 8.6307 1.7530
> 8.6307
> 17: 747.289: -35.3815 -13.8574 8.8024 17.7766
> -52.9864
> 18: 806.944: 9.1110 25.0998 9.1110 18.3998
> 9.1110
> 19: 841.055: 9.2924 -77.1160 9.2924 18.7660
> 18.4034
> 20: 879.878: 39.0996 -14.9604 9.5031 96.0756
> 9.5031
> 21: 939.533: 9.8362 94.5587 9.8362 1.9978
> 9.8362
> 22: 973.645: 10.0320 -15.7931 10.0320 20.2597
> 10.0320
> 23: 1033.299: 10.3837 28.6058 10.3837 -80.6690
> 10.3837
> 24: 1067.411: 10.5903 -87.8876 10.5903 21.3872
> 10.5903
> 25: 1106.234: 44.5610 -17.0501 10.8305 2.1998
> 10.8305
> 26: 1165.889: 3.3718 107.7667 -14.1996 2.2769
> 11.2102
> 27: 1200.000: 11.4332 -17.9990 11.4332 2.3222
> 11.4333
>
> Total abs. beats : 380.8476 1028.4914 222.3249 595.6325
> 382.7292
> Average abs. beats: 14.1055 38.0923 8.2343 22.0605
> 14.1752
> Highest abs. beats: 44.5610 107.7667 14.1996 96.0756
> 52.9864
>
> Having discovered these three equal beating ovovo temperaments only by
> chance, if indeed I did more than rediscover them at all, & lacking much
> of the talent & knowledge that many people on this forum have in
> abundance, I am inordinately proud of these metatunings, & entertain
> egotistical hopes that they will bear my name on into the mathemusical
> future. Puncturing such vanity should be fun so, if any of these
> temperaments is extant already, please don't scruple to say so.
>
> Regardless of their first discoverer's identity, you might enjoy playing
> with these ovovo temperaments.
>
> Meanwhile, by the Common Era calendar, Happy New 2011!
> ==
> Mark L. Vines
> a/k/a Manu Phonic

Correction! My earlier description of the ovovo10 equal beating
temperament should be amended as follows:

4:6:7 equal beating in 3 keysigs (A, E-down, C-down)

8:10:13 equal beating in 3 keysigs (E-down, C-down, G-down)
simultaneous 4:6:7 & 8:10:13 equal beating in 2 of those keysigs
(E-down, C-down)

Alas, my descriptions of the ovovo temperaments may contain other such
errors. For accurate rate information about these phenomenal equal
beaters, rely on the computed Scala readouts rather than my fallible
descriptions!
==
Mark L. Vines
a/k/a Manu Phonic

--- In metatuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> nickname: ovovo10
> full name: vines_ovovo10eb5w6w7
> 10 key signatures per 2/1 "octave"
> chain link span 719.339096 cents
> 2 chains anchored at 0 & 1 times 375.798846 cents
> 3 step sizes, near 102, 105 & 137 cents
> 4:5:6:7 proportional beating in all 5 letter natural keysigs (C, G, D,
> A, E)
> 4:5:6:7 equal beating in 1 of those keysigs (D)
> 4:5:6 equal beating in 3 keysigs (C, G, D)
> 4:6:7 equal beating in 4 keysigs (G, D, A, G-down)
Snip.
> Meanwhile, by the Common Era calendar, Happy New 2011!
> ==
> Mark L. Vines
> a/k/a Manu Phonic
>
> [Non-text portions of this message have been removed]
>

[Non-text portions of this message have been removed]

Hi Mike! == Thanks, after reading your comment I submitted postings on
these ovovo temperaments to the tuning, tuning-math and MakeMicroMusic
lists. Then, of course, I found errors & omissions in my description of
ovovo10 that I had to correct, & I cannot submit corrections to any
moderated list until my original post appears there. No doubt I've made
other descriptive errors as well, but the temperaments themselves are
just fabulous. One has 10, another 22 & another 27 tones per 2/1
"octave." Check 'em out! == Mark
--- In metatuning@yahoogroups.com, Mike Battaglia <battaglia01@...>
wrote:
>
> Hi Mark,
>
> Why not post this to the regular tuning list? You will no doubt find
> more help there. Jacques Dudon will have a lot of good info for you
> about this.
>
> -Mike
>
>
> On Sat, Jan 1, 2011 at 11:48 PM, manuphonic manuphonic@... wrote:
> >
> >
> >
> > In Common Era years, between 1600 & 1707, somebody like Abraham
> > Verheyen, Joseph Sauveur, Gottfried Keller or one of their
colleagues,
> > tweaking what we now call 1/5 comma meantone temperament, discovered
how
> > to tune for equal beating 4:5:6 major triads in 8 of 12 key
signatures.
> > Since that time a variety of other temperaments with equal beating
> > chords & proportional beating chords have been discovered. Many
people
> > on this forum are better acquainted than I with most of them; I'm
not an
> > expert on any. I happen to like George Secor's Hi-Tol 29. Which do
you
> > like?
> >
> > Equal or proportional beating effects, played on instruments whose
> > voices have harmonic timbres, including strings, woodwinds & free
reeds,
> > can sound impressive. Synchronized beating really makes a chord
cohere!
> > For the improviser or composer using such a temperament, certain
> > keysig-specific moods & colors are set by the beat rates, which are
> > knowable in advance.
> >
> > Mark L. Vines here, posting from near Austin, Texas, USA. Late in
2010
> > through today I either discovered, or else independently
rediscovered, a
> > few temperaments with equal beating septimal tetrads like 4:5:6:7 or
> > 8:9:14:15 in some of their keysigs. I stumbled across the
ingredients
> > while seeking across the mathemusical landscape for an unrelated
goal;
> > my find a lucky happenstance that now has me pitching uncomfortably
high
> > above my actual competence level in the metatuning field.
> >
> > Perhaps if I briefly describe three of these temperaments, you
> > mathemusical metatuning people would please graciously tell me
whether
> > any or each of them has already been used or published &, if so, by
> > whom. I haven't found these temperaments in the Scala archive but,
then
> > again, it's a big archive & my search techniques could be faulty. If
> > possible, please point us to published evidence of any prior use or
> > claim.
> >
> > Kindly also point any mistake you may find in my work. Also, please
> > forgive my D-centric notation systems, which do little more than
crudely
> > tag the notes according to the chain in which each belongs. Letter
> > naturals, flats & sharps are assigned only to notes from the origin
or
> > zero-anchor chain. Ups, downs & their doubles, denoted by slash,
> > backslash & graphemic substitute characters, are assigned only to
notes
> > from corresponding nonzero-anchor chains.
> >
> > In case I discovered any of these temperaments first, I refer to
them as
> > my ovovo temperaments, recalling their beating tetrads as I first
heard
> > them, in reed organ voice, rather like ovovovovovo....
> >
> > We'll start with ovovo10. Has anyone else used or published this
> > temperament or a close relative already?
> >
> > nickname: ovovo10
> > full name: vines_ovovo10eb5w6w7
> > 10 key signatures per 2/1 "octave"
> > chain link span 719.339096 cents
> > 2 chains anchored at 0 & 1 times 375.798846 cents
> > 3 step sizes, near 102, 105 & 137 cents
> > 4:5:6:7 proportional beating in all 5 letter natural keysigs (C, G,
D,
> > A, E)
> > 4:5:6:7 equal beating in 1 of those keysigs (D)
> > 4:5:6 equal beating in 3 keysigs (C, G, D)
> > 4:6:7 equal beating in 4 keysigs (G, D, A, G-down)
> >
> > ovovo10
> > keysig D4, edited Scala readouts:
> >
> > 0: 1/1 D
> > 1: 137.121 cents E\ En
> > 2: 238.678 cents E
> > 3: 375.799 cents G\ Gn
> > 4: 480.661 cents G
> > 5: 617.782 cents A\ An
> > 6: 719.339 cents A
> > 7: 856.460 cents C\ Cn
> > 8: 961.322 cents D
> > 9: 1098.442 cents D\ Dn
> > 10: 2/1 D 1 octave
> >
> > |
> >
> > Base frequency : 293.0000 Hertz
> > Beat frequencies of 5/4 3/2 13/8 7/4
> > 0: 0.000: -8.8709 8.8709 35.2150 -8.8709
> > 1: 137.121: -38.6990 9.6021 -38.6990 -9.6021
> > 2: 238.678: -6.9889 12.1293 48.8510 -10.1822
> > 3: 375.799: -44.4196 13.1291 -44.4196 -11.0215
> > 4: 480.661: -11.7097 11.7097 46.4842 -16.8501
> > 5: 617.782: -51.0832 12.6749 -61.3640 -18.2390
> > 6: 719.339: -9.2254 13.4406 53.3556 -13.4406
> > 7: 856.460: -58.6344 14.5485 -58.6344 -14.5485
> > 8: 961.322: -15.4569 15.4569 61.3597 -22.2423
> > 9: 1098.442: -72.5708 16.7310 -81.0012 -24.0757
> > 10: 1200.000: -17.7418 17.7418 70.4300 -17.7418
> >
> > Total abs. beats : 317.6588 128.2929 529.3838 149.0729
> > Average abs. beats: 31.7659 12.8293 52.9384 14.9073
> > Highest abs. beats: 72.5708 16.7310 81.0012 24.0757
> >
> > Now for ovovo22. Has anyone else used or published this temperament
or a
> > close relative already?
> >
> > nickname: ovovo22
> > full name: vines_ovovo22eb9w14w15
> > 22 key signatures per 2/1 "octave"
> > chain link span 707.922315 cents
> > 3 chains anchored at 0, 1 & 2 times 387.517045 cents
> > 3 step sizes, near 44, 60 & 67 cents
> > 8:9:14:15 equal beating in 4 keysigs (C, G, D, E-down)
> > 8:9:10:11:12:13:14:15 proportional beating in 3 of those keysigs (C,
G,
> > D)
> > 4:5:6:7 proportional beating in 6 keysigs
> > 4:5:6 proportional beating in 10 keysigs
> > 4:6:7 proportional beating in 12 keysigs
> >
> > ovovo22
> > keysig D4, edited Scala readouts:
> >
> > 0: 1/1 D
> > 1: 67.112 cents D/ Du
> > 2: 127.500 cents D// Dw
> > 3: 171.672 cents E\ En
> > 4: 215.845 cents E
> > 5: 276.233 cents F
> > 6: 343.345 cents F/ Fu
> > 7: 387.517 cents G\\ Gm
> > 8: 431.689 cents G\ Gn
> > 9: 492.078 cents G
> > 10: 559.189 cents G/ Gu
> > 11: 619.578 cents G// Gw
> > 12: 663.750 cents A\ An
> > 13: 707.922 cents A
> > 14: 768.311 cents A/ Au
> > 15: 835.423 cents A// Aw
> > 16: 879.595 cents B\ Bn
> > 17: 923.767 cents B
> > 18: 984.155 cents C
> > 19: 1051.267 cents C/ Cu
> > 20: 1095.439 cents D\\ Dm
> > 21: 1155.828 cents D\ Dn
> > 22: 2/1 D 1 octave
> >
> > |
> >
> > Base frequency : 293.0000 Hertz
> > Beat frequencies of 9/8 5/4 11/8 3/2
> > 13/8 7/4 15/8
> > 0: 0.000: 18.2415 1.0186 14.6876 3.0350
> > -11.2156 18.2415 18.2415
> > 1: 67.112: 8.2640 -19.0011 2.2228 -0.3989
> > 37.0580 18.9626 1.1806
> > 2: 127.500: 19.6357 -19.6756 -30.0647 3.2670
> > 38.3735 -1.1305 -42.8956
> > 3: 171.672: 20.1431 1.1248 -30.8417 3.3514
> > -67.5863 20.1431 20.1431
> > 4: 215.845: 20.6637 16.7823 2.4221 3.4380
> > -12.7048 20.6637 67.7108
> > 5: 276.233: 21.3972 1.1948 17.2284 3.5600
> > -54.6882 30.8417 21.3972
> > 6: 343.345: 22.2429 -22.2881 -34.0568 3.7008
> > 43.4688 22.2429 -48.5914
> > 7: 387.517: 54.0746 -5.8337 -34.9369 3.7964
> > -76.5605 22.8178 1.4206
> > 8: 431.689: -31.1701 19.0107 2.7438 14.5459
> > -14.3918 23.4074 76.7016
> > 9: 492.078: 24.2383 1.3535 19.5161 4.0328
> > -14.9027 24.2383 24.2383
> > 10: 559.189: 10.9807 -25.2476 -38.5789 4.1922
> > 49.2407 25.1964 -55.0435
> > 11: 619.578: 26.0908 -26.1438 -39.9484 4.3410
> > 50.9886 -1.5022 -56.9974
> > 12: 663.750: 26.7651 1.4946 -40.9808 4.4532
> > -89.8050 -1.5410 26.7651
> > 13: 707.922: 27.4567 1.5332 22.1074 4.5682
> > -16.8815 27.4568 89.9704
> > 14: 768.311: 28.4314 1.5877 22.8922 4.7304
> > -72.6667 40.9808 28.4314
> > 15: 835.423: 29.5552 -29.6152 -45.2528 4.9174
> > 57.7590 29.5552 -64.5656
> > 16: 879.595: 30.3190 1.6931 -46.4223 5.0445
> > -101.7295 30.3190 1.8876
> > 17: 923.767: 73.7083 25.2604 3.6458 5.1748
> > -19.1231 31.1025 101.9170
> > 18: 984.155: 32.2066 1.7985 25.9319 5.3585
> > -19.8019 32.2066 32.2066
> > 19: 1051.267: 33.4796 -33.5476 -51.2616 5.5703
> > 65.4284 33.4796 -73.1388
> > 20: 1095.439: 81.3920 -8.7807 -52.5863 21.3427
> > -115.2373 34.3448 2.1382
> > 21: 1155.828: 35.5640 1.9859 -54.4530 5.9171
> > -119.3280 -2.0476 35.5640
> > 22: 1200.000: 36.4830 2.0373 29.3752 6.0700
> > -22.4312 36.4830 36.4830
> >
> > Total abs. beats : 676.0203 265.9715 632.7823 118.7365
> > 1148.9401 492.4220 891.1462
> > Average abs. beats: 30.7282 12.0896 28.7628 5.3971
> > 52.2246 22.3828 40.5066
> > Highest abs. beats: 81.3920 33.5476 54.4530 21.3427
> > 119.3280 40.9808 101.9170
> >
> > Lastly, ovovo27. Has anyone else used or published this temperament
or a
> > close relative already?
> >
> > nickname: ovovo27
> > full name: vines_ovovo27eb5w6w7
> > 27 key signatures per 2/1 "octave"
> > chain link span 713.177722 cents
> > 3 chains anchored at -1, 0 & 1 times 393.056073 cents
> > 3 step sizes, near 34, 39 & 60 cents
> > 4:5:7 equal beating in 13 keysigs
> > 4:5:6:7 equal beating in 12 of those keysigs, including all 7 letter
> > natural keysigs
> > (F, C, G, D, A, E, B, G-double-up, E-flat-up, B-flat-up, F-up, C-up)
> > 8:10:11:12:14 proportional beating in 4 of those keysigs (F, C, G,
D)
> > 8:10:11:12:13:14 proportional beating in 1 of those keysigs (D)
> > 4:5:6 equal beating in 15 keysigs
> > 4:6:7 equal beating in 18 keysigs
> >
> > ovovo27
> > keysig D, edited Scala readouts:
> >
> > 0: 1/1 D
> > 1: 34.111 cents Eb
> > 2: 93.766 cents Eb/ Ebu
> > 3: 132.589 cents E\\ Em
> > 4: 166.701 cents E\ En
> > 5: 226.355 cents E
> > 6: 260.467 cents F
> > 7: 320.122 cents F/ Fu
> > 8: 354.233 cents F// Fw
> > 9: 393.056 cents F#\ F#n
> > 10: 452.711 cents F#
> > 11: 486.822 cents G
> > 12: 546.477 cents G/ Gu
> > 13: 580.588 cents G// Gw
> > 14: 619.412 cents A\\ Am
> > 15: 653.523 cents A\ An
> > 16: 713.178 cents A
> > 17: 747.289 cents Bb
> > 18: 806.944 cents Bb/ Bbu
> > 19: 841.055 cents Bb// Bbw
> > 20: 879.878 cents B\ Bn
> > 21: 939.533 cents B
> > 22: 973.645 cents C
> > 23: 1033.299 cents C/ Cu
> > 24: 1067.411 cents C// Cw
> > 25: 1106.234 cents C#\ C#n
> > 26: 1165.889 cents C#
> > 27: 2/1 D 1 octave
> >
> > |
> >
> > Base frequency : 293.0000 Hertz
> > Beat frequencies of 5/4 11/8 3/2 13/8
> > 7/4
> > 0: 0.000: 5.7166 -8.9995 5.7166 1.1611
> > 5.7166
> > 1: 34.111: -23.4353 -9.1786 5.8304 11.7745
> > -35.0960
> > 2: 93.766: 6.0348 16.6251 6.0348 12.1873
> > 6.0348
> > 3: 132.589: 25.3927 59.3298 3.5757 1.2535
> > 6.1716
> > 4: 166.701: -6.0796 -9.9092 6.2945 63.6365
> > -37.8895
> > 5: 226.355: 6.5151 62.6318 6.5151 1.3233
> > 6.5151
> > 6: 260.467: 6.6448 -10.4607 6.6448 13.4192
> > 6.6448
> > 7: 320.122: 6.8777 18.9473 6.8777 13.8896
> > 6.8777
> > 8: 354.233: 7.0146 -58.2132 7.0146 14.1660
> > 13.8923
> > 9: 393.056: 29.5154 -11.2933 7.1737 1.4571
> > 7.1737
> > 10: 452.711: 2.2333 71.3803 7.4252 1.5081
> > 7.4252
> > 11: 486.822: 7.5729 -11.9218 7.5729 15.2936
> > 7.5729
> > 12: 546.477: 7.8384 21.5939 -9.9287 -60.8953
> > 7.8384
> > 13: 580.588: 7.9944 -66.3444 7.9944 16.1447
> > 7.9944
> > 14: 619.412: 33.6381 -12.8708 8.1757 1.6605
> > 8.1757
> > 15: 653.523: -8.0538 -13.1269 8.3384 84.3005
> > -50.1929
> > 16: 713.178: 8.6307 82.9695 8.6307 1.7530
> > 8.6307
> > 17: 747.289: -35.3815 -13.8574 8.8024 17.7766
> > -52.9864
> > 18: 806.944: 9.1110 25.0998 9.1110 18.3998
> > 9.1110
> > 19: 841.055: 9.2924 -77.1160 9.2924 18.7660
> > 18.4034
> > 20: 879.878: 39.0996 -14.9604 9.5031 96.0756
> > 9.5031
> > 21: 939.533: 9.8362 94.5587 9.8362 1.9978
> > 9.8362
> > 22: 973.645: 10.0320 -15.7931 10.0320 20.2597
> > 10.0320
> > 23: 1033.299: 10.3837 28.6058 10.3837 -80.6690
> > 10.3837
> > 24: 1067.411: 10.5903 -87.8876 10.5903 21.3872
> > 10.5903
> > 25: 1106.234: 44.5610 -17.0501 10.8305 2.1998
> > 10.8305
> > 26: 1165.889: 3.3718 107.7667 -14.1996 2.2769
> > 11.2102
> > 27: 1200.000: 11.4332 -17.9990 11.4332 2.3222
> > 11.4333
> >
> > Total abs. beats : 380.8476 1028.4914 222.3249 595.6325
> > 382.7292
> > Average abs. beats: 14.1055 38.0923 8.2343 22.0605
> > 14.1752
> > Highest abs. beats: 44.5610 107.7667 14.1996 96.0756
> > 52.9864
> >
> > Having discovered these three equal beating ovovo temperaments only
by
> > chance, if indeed I did more than rediscover them at all, & lacking
much
> > of the talent & knowledge that many people on this forum have in
> > abundance, I am inordinately proud of these metatunings, & entertain
> > egotistical hopes that they will bear my name on into the
mathemusical
> > future. Puncturing such vanity should be fun so, if any of these
> > temperaments is extant already, please don't scruple to say so.
> >
> > Regardless of their first discoverer's identity, you might enjoy
playing
> > with these ovovo temperaments.
> >
> > Meanwhile, by the Common Era calendar, Happy New 2011!
> > ==
> > Mark L. Vines
> > a/k/a Manu Phonic
>

[Non-text portions of this message have been removed]