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. Such a temperament is typically constructed of at least
one chain of tempered "fifth" intervals, the chains being anchored apart
by integer multiples of a certain anchor separation increment. Among
proportional beating temperaments already extant 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 an unjust 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 yesterday 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
their 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. They have, respectively, 10, 22 & 27 tones per 2/1 "octave." 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 out 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, & 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

This was cross-posted, so I am reposting my reply here where the topic belongs. I mention that so far as I know the only person doing equal beating tetrads was me, but I have to figure someone like Erv got there first. What do people know about equal being tetrads? As I've pointed out, these are easily concoted in rank three seven-limit temperaments.

--- In MakeMicroMusic@yahoogroups.com, "manuphonic" <manuphonic@ wrote:

>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.

The only person I know of who has done any work with equal beating tetrads is me, and I certainly haven't seen these before. They look very interesting; I'll certainly look at them. I'm a little mystified why you posted this here and not on tuning, as this is a tuning topic rather than a MMM topic.

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:

> Equal or proportional beating effects, played on instruments
> whose voices have harmonic timbres, including strings,
> woodwinds & free reeds, can sound impressive.

What makes you say so?

> Synchronized beating really makes an unjust chord
> cohere!

I have evidence to the contrary.

-Carl

After correcting for quickly detected errors, ambiguities & omissions in
the original version of this posting, here is a text closer to what I
should have said to introduce the 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. Such a temperament is typically constructed of at least
one chain of tempered "fifth" intervals, the chains being anchored apart
by integer multiples of a certain anchor separation increment. Among
proportional beating temperaments already extant 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 an unjust 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 yesterday 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
their 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. They have, respectively, 10, 22 & 27 tones per 2/1 "octave." 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 out 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)
4:6:7 equal beating in 3 keysigs (A, E-down, C-down)
8:10:13 equal beating in 3 keysigs (E-down, C-down, G-down)
both 4:6:7 equal beating & differently 8:10:13 equal beating in 2
of those keysigs (E-down, C-down)
4:5:6 equal beating in 3 keysigs (C, G, D)

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, & 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

--- In tuning@yahoogroups.com, "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. Such a temperament is typically constructed of at least
> one chain of tempered "fifth" intervals, the chains being anchored
apart
> by integer multiples of a certain anchor separation increment. Among
> proportional beating temperaments already extant 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 an unjust
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 yesterday 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
> their 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. They have, respectively, 10, 22 & 27 tones per 2/1 "octave." 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 out 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, & 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
>

Hi Gene. Thanks for looking into this. If you or Ervin Wilson already
pioneered this territory, it should be scenic enough for me! My original
posting fumbled the point but, as for the ovovo temperaments themselves,
besides their 7-limit equal beating tetrads, their 13-limit proportional
beating chords could perhaps offer enough to repay your interest.
I cross-posted to increase the likelihood that the identity of the ovovo
temperaments' real first discoverer would emerge, seeking a simple
readership broadening effect. In retrospect the cross-posting may seem a
breach of etiquette, so I apologize. I'd been supposing that any
temperament with 22 or 27 tones per 2/1 "octave" would automatically be
on-topic in MakeMicroMusic. Sorry.
==
Mark
--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...>
wrote:
>
> This was cross-posted, so I am reposting my reply here where the topic
belongs. I mention that so far as I know the only person doing equal
beating tetrads was me, but I have to figure someone like Erv got there
first. What do people know about equal being tetrads? As I've pointed
out, these are easily concoted in rank three seven-limit temperaments.
>
>
> --- In MakeMicroMusic@yahoogroups.com, "manuphonic" <manuphonic@
wrote:
>
>
> >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.
>
> The only person I know of who has done any work with equal beating
tetrads is me, and I certainly haven't seen these before. They look very
interesting; I'll certainly look at them. I'm a little mystified why you
posted this here and not on tuning, as this is a tuning topic rather
than a MMM topic.
>

Hi Carl. If a thing does sound impressive to me, I have to believe it
can. Not that it must sound impressive to all people everywhere, but
that it can sound impressive to some human ears. So I doubt that I
overstated my case on that point.
Perhaps I overstated my case about equal beating effects making unjust
chords cohere. I have the impression I've experienced that phenomenon,
but I can be wrong. You mentioned evidence to the contrary. I'm
interested.
Thanks!
==
Mark L. Vines
a/k/a Manu Phonic
--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "manuphonic" manuphonic@ wrote:
>
> > Equal or proportional beating effects, played on instruments
> > whose voices have harmonic timbres, including strings,
> > woodwinds & free reeds, can sound impressive.
>
> What makes you say so?
>
> > Synchronized beating really makes an unjust chord
> > cohere!
>
> I have evidence to the contrary.
>
> -Carl
>

--- "manuphonic" <manuphonic@...> wrote:
> Hi Carl. If a thing does sound impressive to me, I have to
> believe it can. Not that it must sound impressive to all
> people everywhere, but that it can sound impressive to some
> human ears. So I doubt that I overstated my case on
> that point. Perhaps I overstated my case about equal beating
> effects making unjust chords cohere. I have the impression
> I've experienced that phenomenon, but I can be wrong.
> You mentioned evidence to the contrary. I'm interested.
> Thanks!
> ==
> Mark L. Vines
> a/k/a Manu Phonic

Hi Mark,

There have been extensive investigations here on equal- and
proportional-beating triads in the past (though not tetrads).
I've synthesized many examples, posted them here and played
them for family members. I've designed proportional-beating
scales and tuned pianos to them. Kurt Bigler (list member
and my former neighbor) and I also worked with a professional
tuner who specializes in them -- with three pianos between us.
My observations are:

* The effect is subtle, even in synthesized examples. On a
piano, variations in the harmonicity of the strings and
precise tuning of the unisons make the effect very hard to
perceive. Real-world ensembles (you mentioned woodwinds) do
not have the intonation control necessary to perform in
particular temperaments like this.

* When auditioned with carefully synthesized sustained chords,
proportional-beating makes beats MORE noticeable because the
minima and maxima of individual beats often become aligned.
You may like this but most find beating undesirable.

* In one survey of major triad tunings, which included a
variety of simple-proportion beat rates as well as golden-
and silver-ratio beat rates (maximally irrational), what
predicted listener preference was good old error from just
intonation.

* The above goes for sustained chords. In polyphonic music
at typical tempi -- even when synthesized -- the effect is
inaudible.

* Proportional-beating tunings are very seductive (on paper)
to many tuning theorists (including myself).

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> * Proportional-beating tunings are very seductive (on paper)
> to many tuning theorists (including myself).

You might want to look again using synth tuning with equal beating tetrads. My impression, not based on any scientific tests, is that the effect is more pronounced with tetrads.

On Sun, Jan 2, 2011 at 9:50 PM, Carl Lumma <carl@...> wrote:
>
> Hi Mark,
>
> There have been extensive investigations here on equal- and
> proportional-beating triads in the past (though not tetrads).
> I've synthesized many examples, posted them here and played
> them for family members.

It might be that we're going about equal beating the wrong way. It's
my gut feeling that the concept of equal beating is related to the
concept of periodicity buzz, and perhaps there's a way to apply group
theory to the whole shebang and see how to transform periodicity buzz
(JI) into different equal beating forms (temperament). It might well
turn out that there are ways of dealing with equal beating that are
more pleasant because they start to resemble the "natural" equal
beating sound of JI, whereas the beating for the examples I've seen so
far is slower and doesn't sound much like JI.

I'll have to map this out some day once I finish all of the other
half-finished stuff that I have half-finished.

-Mike

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> Lastly, ovovo27. Has anyone else used or published this temperament or a
> close relative already?

I think you've got the field to yourself. Your scales are equal temperaments tweaked so as to give equal or proportional beating in various places. What I did was quite different--instead of tweaking an equal temperament into an irregular version, I started from a rank three temperament and kept it regular, solving an algebraic equation to get the tuning which produced the same proportional beating regularly. The tuning was already quite accurate and so ameliorating tuning errors was not a consideration, the desire was to achieve a special tuning effect.

The next step, of course, would be to compose something in one of these scales.

Gene wrote:
>You might want to look again using synth tuning with equal
>beating tetrads. My impression, not based on any scientific
>tests, is that the effect is more pronounced with tetrads.

Can you give some tetrads to try?

Mike wrote:
> It might be that we're going about equal beating the wrong way.
> It's my gut feeling that the concept of equal beating is related
> to the concept of periodicity buzz,

It may be, but there's no way to reproduce that effect with
non-JI intervals.

-Carl

On Mon, Jan 3, 2011 at 1:21 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
> > It might be that we're going about equal beating the wrong way.
> > It's my gut feeling that the concept of equal beating is related
> > to the concept of periodicity buzz,
>
> It may be, but there's no way to reproduce that effect with
> non-JI intervals.

Equal beating could be way to reproduce that effect with non-JI
intervals. Or rather, if there were a "beating space," like we have a
tuning space, then periodicity buzz would be at the origin.

It might be that all of the examples we're hearing so far are rather
hokey and don't really sound much like real periodicity buzz, and
hence the reaction to them has been rather poor.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
> >You might want to look again using synth tuning with equal
> >beating tetrads. My impression, not based on any scientific
> >tests, is that the effect is more pronounced with tetrads.
>
> Can you give some tetrads to try?

If "e" is a small quantity, either positive or negative, then
if we put 3+e, 5+e, 7+e into close root position, we get 5/4+e/4, 3/2+e/2, 7/4+e/4. If we start instead from 3+e, 5+2e, 7+2e we get instead 5/4+e/2, 3/2+e/2, 7/4+e/2. Also, the signs don't need to match, we could for instance have instead 5/4+e/2, 3/2+e/2, 7/4-e/2.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Equal beating could be way to reproduce that effect with non-JI
> intervals.

Nope.

> Or rather, if there were a "beating space," like we have a
> tuning space, then periodicity buzz would be at the origin.

?

> It might be that all of the examples we're hearing so far are
> rather hokey and don't really sound much like real periodicity
> buzz, and hence the reaction to them has been rather poor.

It's that they're fundamentally different phenomena.

-Carl

Hi Carl. Fascinating, persuasive evidence you've presented there. My
experience agrees with you on a key point: equal beating makes beating
more noticeable, not less. Seems highly probable now that your case
against equal beating consonance has merit. I was only claiming
coherence for eb, not full consonance, but the evidence you cite
supports you even there to some degree. I'd still commend the ovovo22
temperament to your attention, however. Its best 4:5:6 major triads
don't beat anywhere near as rapidly as they do in the other ovovo
temperaments; equal beating in ovovo22 occurs instead in its best
keysigs at 8:9:14:15, where the beating is quite audible in a
synthesized instrumental voice of harmonic timbre. Cheers!
==
Mark

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- "manuphonic" manuphonic@ wrote:
> > Hi Carl. If a thing does sound impressive to me, I have to
> > believe it can. Not that it must sound impressive to all
> > people everywhere, but that it can sound impressive to some
> > human ears. So I doubt that I overstated my case on
> > that point. Perhaps I overstated my case about equal beating
> > effects making unjust chords cohere. I have the impression
> > I've experienced that phenomenon, but I can be wrong.
> > You mentioned evidence to the contrary. I'm interested.
> > Thanks!
> > ==
> > Mark L. Vines
> > a/k/a Manu Phonic
>
> Hi Mark,
>
> There have been extensive investigations here on equal- and
> proportional-beating triads in the past (though not tetrads).
> I've synthesized many examples, posted them here and played
> them for family members. I've designed proportional-beating
> scales and tuned pianos to them. Kurt Bigler (list member
> and my former neighbor) and I also worked with a professional
> tuner who specializes in them -- with three pianos between us.
> My observations are:
>
> * The effect is subtle, even in synthesized examples. On a
> piano, variations in the harmonicity of the strings and
> precise tuning of the unisons make the effect very hard to
> perceive. Real-world ensembles (you mentioned woodwinds) do
> not have the intonation control necessary to perform in
> particular temperaments like this.
>
> * When auditioned with carefully synthesized sustained chords,
> proportional-beating makes beats MORE noticeable because the
> minima and maxima of individual beats often become aligned.
> You may like this but most find beating undesirable.
>
> * In one survey of major triad tunings, which included a
> variety of simple-proportion beat rates as well as golden-
> and silver-ratio beat rates (maximally irrational), what
> predicted listener preference was good old error from just
> intonation.
>
> * The above goes for sustained chords. In polyphonic music
> at typical tempi -- even when synthesized -- the effect is
> inaudible.
>
> * Proportional-beating tunings are very seductive (on paper)
> to many tuning theorists (including myself).
>
> -Carl
>

Here (in cents) are the main ovovo tetrads of interest:
8:9:14:15 equal beating from ovovo22:
215.844629
984.155371
1095.439360
4:5:6:7 proportional beating from ovovo22:
387.517045
707.922315
984.155371
4:5:6:7 equal beating from ovovo27:
393.056073
713.177722
973.644278
4:5:6:7 equal beating from ovovo10:
375.798846
719.339096
961.321809
Thanks for looking into this!
==
Mark L. Vines
a/k/a Manu Phonic

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
> >You might want to look again using synth tuning with equal
> >beating tetrads. My impression, not based on any scientific
> >tests, is that the effect is more pronounced with tetrads.
>
> Can you give some tetrads to try?
>
> Mike wrote:
> > It might be that we're going about equal beating the wrong way.
> > It's my gut feeling that the concept of equal beating is related
> > to the concept of periodicity buzz,
>
> It may be, but there's no way to reproduce that effect with
> non-JI intervals.
>
> -Carl
>

A scala file would be great - If I had one I'd be glad to try to
compose something.

On Sun, Jan 2, 2011 at 8:00 PM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
> >
> > Lastly, ovovo27. Has anyone else used or published this temperament or a
> > close relative already?

>
> The next step, of course, would be to compose something in one of these scales.

On Mon, Jan 3, 2011 at 2:39 AM, Carl Lumma <carl@...> wrote:
>
> > It might be that all of the examples we're hearing so far are
> > rather hokey and don't really sound much like real periodicity
> > buzz, and hence the reaction to them has been rather poor.
>
> It's that they're fundamentally different phenomena.

How so? What's the cause of periodicity buzz if not partials beating
against other partials?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > It's that they're fundamentally different phenomena.
>
> How so? What's the cause of periodicity buzz if not partials beating
> against other partials?

Yes, it's (high frequency) partials beating, not fundamentals.
Feel free to exhibit a proportional-beating chord that sounds
like it has periodicity buzz.

-Carl

Hi Gene,

> > Can you give some tetrads to try?
>
> If "e" is a small quantity, either positive or negative, then
> if we put 3+e, 5+e, 7+e into close root position, we get
> 5/4+e/4, 3/2+e/2, 7/4+e/4. If we start instead from 3+e, 5+2e,
> 7+2e we get instead 5/4+e/2, 3/2+e/2, 7/4+e/2. Also, the signs
> don't need to match, we could for instance have instead 5/4+e/2,
> 3/2+e/2, 7/4-e/2.

Can you give a tetrad and its beat ratios?

-Carl

Hi Mark,
Thanks for your posting.
First let me tell you I agree with you, and I am not alone, that both
equal, and proportional beating have strong and useful acoustical and
musical properties. They can be of many different kinds also, and
with various applications.
I have been myself investigating in several hundreds of such equal-
beating and proportional beating tunings, notably in very
concentrated areas where I could experience, and several times in
blind test situation, very different specific musical colors, even
between very close generators. It's clear to me.
Such extended fifths eq-b tunings you can find among those I included
in the African, Persian, and Indonesian sections of the Ethno 2
collection, and other eq-b tunings, available in my folder in the TL
files. I invite you for example to check the Soria tuning (Persian
folder), discussed in this list in its 17 and 19 notes versions, with
a 705.5685 c. generator, that has triple equal-beating of fourths,
septimal minor thirds with 7ths harmonics everywhere, and much more.
I have a preference myself for the rational versions issued from the
algorithm whenever possible, as the beatings of chords in different
keys will be also in proportional beating between them, but both
techniques have their utility.
If I understand well, what you do here is different, I call it
"ribbon" temperaments, that is the cumulation of 2 or 3 chains (of
extended fifths, in your case), transposed by a specific interval,
here a major third, tuned to be in equal beating with the other
intervals ? I have to hear how it sounds, but that's a good idea.
Of the temperaments you posted, two have known generators to me :

ovovo27 has a very famous slendro generator, that I used in the
triple eq-b natté.scl tuning (Ethno2 /Indonesian folder) ; it is also
well known by the mathematicians in its complementary fourth as the
first Pivot-Vijayaraghavan number, 1.3247179572447

ovovo10 is a declinaison of the same algorithm family, which I named
myself Eronyme.
Its fifth ratio is the solution of x^3 = 5x^2 - 8 and therefore has
in addition of its eq-b properties another property (even more
important to me), of differential coherence, as it is expressed by
the algorithm. In case it is of some interest to you, it expresses
the -c of a 23/16 in your temperament, which would be a major third
above the #2 note (238.678 c.) - it's present in the harmonics and
it's a super nice harmony. Because your tuning includes major third transpositions it could be pertinent with your temperament, however I
don't fully understand the logic of it and why for example between
notes 2 & 8 you have 723.644 c., and not 719.339 ?

ovovo22's generator is not known of me. If you don't mind I will
include it with your name in my (near to 2000 entries) fractal
generators database. If you can explain it to me either offlist or
not, you're welcome.
What means "ovovo" BTW ?
- - - - - - -
Jacques

Mark L. Vines wrote :
> 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 an
> unjust 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.
.../...

> 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)
> 4:6:7 equal beating in 3 keysigs (A, E-down, C-down)
> 8:10:13 equal beating in 3 keysigs (E-down, C-down, G-down)
> both 4:6:7 equal beating & differently 8:10:13 equal beating in 2
> of those keysigs (E-down, C-down)
> 4:5:6 equal beating in 3 keysigs (C, G, D)
>
> 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, & 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.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> I have been myself investigating in several hundreds of such equal-
> beating and proportional beating tunings, notably in very
> concentrated areas where I could experience, and several times in
> blind test situation, very different specific musical colors, even
> between very close generators. It's clear to me.

Hi Jacques,

Can you provide any audio files?

Thanks,
-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
>
> > I have been myself investigating in several hundreds of such equal-
> > beating and proportional beating tunings, notably in very
> > concentrated areas where I could experience, and several times in
> > blind test situation, very different specific musical colors, even
> > between very close generators. It's clear to me.
>
> Hi Jacques,
>
> Can you provide any audio files?
>
> Thanks,
> -Carl

Hi Carl,
No, these are not things I recorded, but I tested for example all the tunings I designed for the Ethno collection of course, and to give you an idea I choosed my meantones among a collection I have of about 350 different eq-b and/or -c meantones. This gives you an idea of how close some would be. Would the use of rational versions accentuate the differences ? I would think so also. Of course I know this is all subjective acoustics, except for the beats pulsations that can be heard, seen, and verified on recorded signals. But what can I say ? yes these differences of musicality were really strong and clear to me, and is this my personal experience...
- - - -
Jacques

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Gene,
>
> > > Can you give some tetrads to try?
> >
> > If "e" is a small quantity, either positive or negative, then
> > if we put 3+e, 5+e, 7+e into close root position, we get
> > 5/4+e/4, 3/2+e/2, 7/4+e/4. If we start instead from 3+e, 5+2e,
> > 7+2e we get instead 5/4+e/2, 3/2+e/2, 7/4+e/2. Also, the signs
> > don't need to match, we could for instance have instead 5/4+e/2,
> > 3/2+e/2, 7/4-e/2.
>
> Can you give a tetrad and its beat ratios?

You can concoct your own examples to try, but what about 1-501/400-301/200-701/400 as an example?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> You can concoct your own examples to try, but what about 1-501/400-301/200-701/400 as an example?

Or else 1-501/400-601/400-701/400, or 1-499/400-599/400-699/400, or 1-1001/800-601/400-1401/800, etc etc etc.

Gene wrote:

> > Can you give a tetrad and its beat ratios?
>
> You can concoct your own examples to try, but what
> about 1-501/400-301/200-701/400 as an example?

I'm curious what you think the beat ratios are, how
you would recommend writing them, etc.

-Carl

I wrote :

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> >
> > > I have been myself investigating in several hundreds of such > equal-
> > > beating and proportional beating tunings, notably in very
> > > concentrated areas where I could experience, and several times in
> > > blind test situation, very different specific musical colors, even
> > > between very close generators. It's clear to me.
> >
> > Hi Jacques,
> >
> > Can you provide any audio files?
> >
> > Thanks,
> > -Carl
>
> Hi Carl,
> No, these are not things I recorded, but I tested for example all > the tunings I designed for the Ethno collection of course, and to > give you an idea I choosed my meantones among a collection I have > of about 350 different eq-b and/or -c meantones. This gives you an > idea of how close some would be. Would the use of rational versions > accentuate the differences ? I would think so also. Of course I > know this is all subjective acoustics, except for the beats > pulsations that can be heard, seen, and verified on recorded > signals. But what can I say ? yes these differences of musicality > were really strong and clear to me, and is this my personal > experience...
> - - - -
> Jacques

...Remembering now I have recorded samples of equal-beating :
One is during one minute at the end of my Ethno demo "Xiuhtecatl" where you can hear a succession of eq-b chords, except that these are not harmonic beatings, but small commas beats inside different chords in a pseudo-sinus sound. Don't know if it is of interest to you. May be one thing it shows is that the beats add a much more pleasant dimension than if they were absent, as we can easily imagine how dull this part would be without them !
https://www.dropbox.com/s/9grkajrdg7uo6ds

Another one is something I remember now I sent to Johnny Reinhard as it demonstrates the equal-beating property of Werckmeister III, as expressed by the "Comptine" algorithm, and how it naturally helps its tuning by ear.
The sound is quite horrible, because rendered with low-resolution sawtooth waves on an amateur sound application ; it is just meant to let you hear the eq-b of the concerned dyads, separetely, and together. Do you want that ?
Or, was your question more about audible spectral difference between very close meantones of different eq-b algorithms ? then I can think about such scala files, if some are interested to experiment.
- - - - - - - -
Jacques

Mike wrote :

> > > It might be that we're going about equal beating the wrong way.
> > > It's my gut feeling that the concept of equal beating is related
> > > to the concept of periodicity buzz,
> >(Carl) :
> > It may be, but there's no way to reproduce that effect with
> > non-JI intervals.
>
> Equal beating could be way to reproduce that effect with non-JI
> intervals. Or rather, if there were a "beating space," like we have a
> tuning space, then periodicity buzz would be at the origin.
>
> It might be that all of the examples we're hearing so far are rather
> hokey and don't really sound much like real periodicity buzz, and
> hence the reaction to them has been rather poor.
>
> -Mike

Carl wrote :

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > It's that they're fundamentally different phenomena.
> >(Mike) :
> > How so? What's the cause of periodicity buzz if not partials beating
> > against other partials?
>
> Yes, it's (high frequency) partials beating, not fundamentals.
> Feel free to exhibit a proportional-beating chord that sounds
> like it has periodicity buzz.
>
> -Carl

Since I could not find any information about what you guys call "periodicity buzz", here I come with my request again :
What do you understand by "periodicity buzz" ?
It's important to have clear definitions of the words we use.
In my sense, a buzz is a more or less low frequency sound, not a beat. I can hear such "buzz", or even several types of them, in accurately tuned JI chords.
And a beat is something else, let's say an amplitude modulation, but under 20 hz maximum.
Tempered chords, of whether harmonic or inharmonic sounds, generate beats.
Then I know JI chords can also have harmonic (or inharmonic) beats, in addition.
But at the moment, because english is not my native langage, and also because I probably missed some of the numerous exchanges on the subject, I am mainly confused because I have no clue to what is exactly meant on this list by "periodicity buzz".
- - - - - - - -
Jacques

In ovovo10 there are two key signatures, E\ & C\, featuring pentads that
simulate periodicity buzz. In these pentads, two notes depart from 3/2 &
7/4 at the same beat rate, while two other notes depart from 5/4 & 13/8
at proportional, much higher, but also mutually matching rates.
Sensitive to slight differences in the timing of chordal note onsets,
periodicity-buzzlike effects are most audible in these pentads at low
tonic frequencies. Try it.

In the Opts dialog box, set Scala's notation for system E10, where E\
(E-down) is called Eb (E-flat). Choose an offset of Eb & a frequency of
79.2876 Hz. OK the box, then reopen it & repeat until the offset is
correct. Use the Input box to enter the following pitches in cents:

343.540
719.339
824.201
961.322

That's the pentad. Don't enter a 1200.000 cent pitch; just OK the box.
Using the Shift button & the Play keyboard set to a single "octave,"
swipe all the notes as nearly simultaneously as you can manage. Hold the
chord. Hear the buzz?
==
Mark L. Vines

--- In tuning@...m, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia battaglia01@ wrote:
>
> > > It's that they're fundamentally different phenomena.
> >
> > How so? What's the cause of periodicity buzz if not partials beating
> > against other partials?
>
> Yes, it's (high frequency) partials beating, not fundamentals.
> Feel free to exhibit a proportional-beating chord that sounds
> like it has periodicity buzz.
>
> -Carl
>

Mike> How so? What's the cause of periodicity buzz if not partials beating against other partials?

Carl> Yes, it's (high frequency) partials beating, not fundamentals. Feel free to exhibit a proportional-beating chord that sounds like it has periodicity buzz.

I could be confused, but Carl's answer comes across ambiguously or even "back to front" to me; I think a correct statement is:
* beating is caused by high frequency partials beating
* periodicity buzz seems to appear when several notes are sounded together that are harmonics of a low VF - too low to be assigned a pitch, but high enough to be heard as a buzz (?)

I got this impression from searching this forum for "periodicity buzz" and looking at the oldest results first. e.g. messages #1804, #6124, #8171.

I hope this will help Jacques, if only by prompting someone to correct me!
Steve M.

Hi Gene! The path by which I arrived, at the ovovo equal beating
temperaments, from equal temperaments meaning logarithmically equal
divisions of the 2/1 "octave," seemed a random & arduous one, so I'm
surprised you spotted it.

I'd evaluated how well certain prime harmonics & subharmonics are
approximated in 19ed2 or 2^(n/19) equal temperament & in 49ed2 or
2^(n/49) equal temperament, then tried to predict how much closer their
approximations would get in 19*49= 931ed2 or 2^(n/931) equal
temperament. While studying my predictive mix of success & failure, I
noticed a "fifth" which tempers out the difference between 16/9 & 7/4 in
such a way that 3/2 & 7/4 have proportional beating. Previously I had
noticed proportional beating effects only from variant "fifths"
tempering out the difference between 81/64 & 5/4, or from the use of
just intervals as chain anchor separation increments, meaning that
either "fifths" were selected to give 3/2 & 5/4 proportional beating or
else 5/4 was kept beatless. Keller did it the one way & Secor the other
& I hadn't expected a third approach.

So, this 931ed2 linkage of 3/2 beats with 7/4 beats by the width of its
"fifth" was novel enough to me that I started playing around with
variant "fifths" tempered by variant slices of that 64/63 interval,
looking & sometimes listening for equal beating effects. Playing around
led to temperaments with 4:5:6:7 equal beating tetrads that, when I
first heard them in a reed organ voice, sported an effect that sounded
to my ears like "ovovovovovo..." Hence I call them, & their 8:9:14:15
equal beating cousins, ovovo temperaments. But I never dreamed that
their connection with 931 equal temperament could be spotted on sight!
==
Mark L. Vines
a/k/a Manu Phonic

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...>
wrote:
>
>
>
> --- In tuning@yahoogroups.com, "manuphonic" manuphonic@ wrote:
> >
> > Lastly, ovovo27. Has anyone else used or published this temperament
or a
> > close relative already?
>
> I think you've got the field to yourself. Your scales are equal
temperaments tweaked so as to give equal or proportional beating in
various places. What I did was quite different--instead of tweaking an
equal temperament into an irregular version, I started from a rank three
temperament and kept it regular, solving an algebraic equation to get
the tuning which produced the same proportional beating regularly. The
tuning was already quite accurate and so ameliorating tuning errors was
not a consideration, the desire was to achieve a special tuning effect.
>
> The next step, of course, would be to compose something in one of
these scales.
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
>
> > > Can you give a tetrad and its beat ratios?
> >
> > You can concoct your own examples to try, but what
> > about 1-501/400-301/200-701/400 as an example?
>
> I'm curious what you think the beat ratios are, how
> you would recommend writing them, etc.

I have a method I've been using. If f is the fifth, t is the third, and s is the seventh, then one can form the triple [(4t-5)/(2f-3), (4s-7)/(2f-3), (4s-7)/(4t-5)], which for the above example gives [1,1,1]. The last ratio is redundant, so that you can just use the first two. I'd call it a [1,1] septimal brat.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> https://www.dropbox.com/s/9grkajrdg7uo6ds

Dropbox never works for me.

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
While studying my predictive mix of success & failure, I
> noticed a "fifth" which tempers out the difference between 16/9 & 7/4 in
> such a way that 3/2 & 7/4 have proportional beating.

In other words, it tempers out the Archytas comma, 64/63, which a synch beating scale I just posted here does also. That's a good choice to achieve synch beating, and 81/80, which you mention below, also works, though the scale I presented for that wasn't as successful.

Steve wrote :

> I could be confused, but Carl's answer comes across ambiguously or > even "back to
> front" to me; I think a correct statement is:
> * beating is caused by high frequency partials beating
> * periodicity buzz seems to appear when several notes are sounded > together that
> are harmonics of a low VF - too low to be assigned a pitch, but > high enough to
> be heard as a buzz (?)
>
> I got this impression from searching this forum for "periodicity > buzz" and
> looking at the oldest results first. e.g. messages #1804, #6124, > #8171.
>
> I hope this will help Jacques, if only by prompting someone to > correct me!
> Steve M.

Thanks Steve,
Simple definitions and quite exactly what I thought "periodicity buzz", in my sense was !
I would only add two complementary exceptions :
* beating can also be generated of course by close fundamental chords such as small commas, double strings or reeds etc. ;
* I would not exclude a high enough VF to be a pitch, this would only make it more "buzzing"...

Very nice to read these old posts from Paul Erlich ! Thank you very much, it's all clear now.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

On Tue, Jan 4, 2011 at 10:02 AM, martinsj013 <martinsj@...> wrote:
>
> Mike> How so? What's the cause of periodicity buzz if not partials beating against other partials?
>
> Carl> Yes, it's (high frequency) partials beating, not fundamentals. Feel free to exhibit a proportional-beating chord that sounds like it has periodicity buzz.
>
> I could be confused, but Carl's answer comes across ambiguously or even "back to front" to me; I think a correct statement is:
> * beating is caused by high frequency partials beating
> * periodicity buzz seems to appear when several notes are sounded together that are harmonics of a low VF - too low to be assigned a pitch, but high enough to be heard as a buzz (?)

I don't think that's what causes periodicity buzz. My unscientific
hunch is that periodicity buzz is caused by two things:

1) Overtones beating against other overtones (e.g. for 4:5:6:7, the
third harmonic of 5 will beat against the second harmonic of 7, and so
on). Or if we're dealing with a cluster chord like 8:9:10:11:12, the
notes beating against each other.
2) Nonlinear effects (perhaps DPOAE's, check them out on google) which
cause additional harmonics to be generated so that the above can
happen even when this is happening with only sines.

I think that what ends up happening in the case of "periodicity buzz"
is that all of this beating ends up being in a regular pattern - e.g.
a polyrhythm. Since the pattern is regular, your brain hears it as
being highly autocorrelated, and hence it sounds like "buzz" rather
than like nonsense.

A good way to study this phenomenon more might be to play around with
just triads in really low registers; critical band effects down there
will be amplified.

However, I'm not entirely sure that this is correct; I think that
something in the way that the brain performs its time-frequency
analysis might be involved. There is a rule when it comes to doing a
mixed time-frequency analysis on a signal, which is that the higher
frequency resolution you have, the lower time resolution and vice
versa. I think that the brain is performing the analysis in such a way
that there is more frequency resolution at lower frequencies and more
time resolution at high frequencies. This could be related to critical
bandwidths in the ear as they change across the spectrum.

All in all this is a very interesting phenomenon, and I'd like to
mathematically tie it into equal beating if possible (they seem to
share a common origin). But if I get engaged in any more research at
the moment, I'm going to disappear off of the face of the earth...

-Mike

Hi Jacques,

On Tue, Jan 4, 2011 at 6:37 AM, Jacques Dudon <fotosonix@wanadoo.fr> wrote:
>
> Since I could not find any information about what you guys call "periodicity buzz", here I come with my request again :
> What do you understand by "periodicity buzz" ?

I responded in your other thread, but I'll repost here. I guess the
meaning of "buzz" is difficult to communicate because it's an English
onomatopoeia; it "sounds like" what it is.

Say "zzzzzzzzzzzzzzzzz." Periodicity buzz is that sound that you hear,
like a bee "buzzing" around, when a just chord is played, especially
if it's played with a really harsh and complex timbre.

> In my sense, a buzz is a more or less low frequency sound, not a beat. I can hear such "buzz", or even several types of them, in accurately tuned JI chords.

This low frequency sound that you're calling a "buzz" is probably what
we're referring to as the "VF" or the virtual fundamental.

> And a beat is something else, let's say an amplitude modulation, but under 20 hz maximum.
> Tempered chords, of whether harmonic or inharmonic sounds, generate beats.
> Then I know JI chords can also have harmonic (or inharmonic) beats, in addition.

JI chords end up having harmonic beats with one another, and this is
what I think "periodicity buzz" is. The two of them are related.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
> >You might want to look again using synth tuning with equal
> >beating tetrads. My impression, not based on any scientific
> >tests, is that the effect is more pronounced with tetrads.
>
> Can you give some tetrads to try?
>
> Mike wrote:
> > It might be that we're going about equal beating the wrong way.
> > It's my gut feeling that the concept of equal beating is related
> > to the concept of periodicity buzz,
>
> It may be, but there's no way to reproduce that effect with
> non-JI intervals.

I have a file using a 15-odd-limit proportional-beating temperament -- a live performance, in fact (given 35 years ago) -- that demonstrates that the foregoing statement is incorrect:
http://xenharmony.wikispaces.com/space/showimage/improv29.mp3
The periodicity buzz is particularly evident toward the very end, when an 11-limit chord progression is played an octave lower than in the rest of the piece.

As the following message indicates, you even had it on your website at one time:
/tuning/topicId_53615.html#53615
The temperament (which Mark Vines happened to mention in the message that began this thread) can be identified by following the above link to other messages. For a quick summary of how the tempering was done, see:
/tuning-math/message/10586

Some of the intervals are just, and some (including the 5ths) are tempered, so that every tetrad in the recording is tempered. What makes it proportional-beating is that the error of the tempered intervals is either ~1.62 cents or twice that amount.

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> > It may be, but there's no way to reproduce that effect with
> > non-JI intervals.
>
> I have a file using a 15-odd-limit proportional-beating
> temperament -- a live performance, in fact (given 35 years ago)
> -- that demonstrates that the foregoing statement is incorrect:
> http://xenharmony.wikispaces.com/space/showimage/improv29.mp3
> The periodicity buzz is particularly evident toward the very end,
> when an 11-limit chord progression is played an octave lower
> than in the rest of the piece.

Can you give the chord or chords in question?

-Carl

Hi Steve, sorry for the late response, Yahoo was giving errors
when I tried to reply earlier.

> I could be confused, but Carl's answer comes across ambiguously
> or even "back to front" to me; I think a correct statement is:
> * beating is caused by high frequency partials beating
> * periodicity buzz seems to appear when several notes are sounded
> together that are harmonics of a low VF - too low to be assigned a
> pitch, but high enough to be heard as a buzz (?)

No, "beating" usually refers to beating between fundamentals
or lower partials. Nobody knows what causes periodicity buzz
but our best guess is that it is rapid proportional beating
between upper partials.

-Carl

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Since I could not find any information about what you guys call
> "periodicity buzz", here I come with my request again :
> What do you understand by "periodicity buzz" ?

Hi Jacques, sorry for the confusion. It is hard to explain.
Like Mike said, it is something akin to soft, rapid clipping
that sometimes is heard in very accurately-tuned 5-limit
chords, and much more frequently heard in extended just
intonation. Intervals like 7:4 and 7:6 usually exhibit it.

-Carl

Hi Jacques,

> ...Remembering now I have recorded samples of equal-beating :
> One is during one minute at the end of my Ethno demo "Xiuhtecatl"
> where you can hear a succession of eq-b chords, except that these
> are not harmonic beatings, but small commas beats inside different
> chords in a pseudo-sinus sound. Don't know if it is of interest
> to you. May be one thing it shows is that the beats add a much
> more pleasant dimension than if they were absent, as we can easily
> imagine how dull this part would be without them !
> https://www.dropbox.com/s/9grkajrdg7uo6ds

I remember this demo!

Starting at ~ 2:35 there are some high-pitched tones
and beating. Very briefly at 2:47-2:48 I hear something
that sounds vaguely like periodicity buzz. I don't hear
it anywhere else in the piece.

I asked for audio demos because I originally thought you
were referring to your work with photonic disks, which
have a unique sound I cannot easily reproduce. Now that
I see you were referring to other work I can say audio
files are less necessary. What is necessary is to compare
chords of like error (deviation from JI) but with different
beat ratios, where only those with simpler beat ratios
exhibit some desirable quality (periodicity buzz or other).
That is the only test that matters for claims pertaining
to beat ratios and I am hoping to have time to put one
together for tetrads (previous tests on triads failed to
show benefit of proportional beating). If you have suggestions
for chords to try I am collecting them now (some from Gene
and some from Mark already). If you would like to submit
the chord at 2:47 or any others feel free!

-Carl

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Hi Mark,
> Thanks for your posting.
Thank you, & greetings!
> First let me tell you I agree with you, and I am not alone, that both
> equal, and proportional beating have strong and useful acoustical and
> musical properties. They can be of many different kinds also, and
> with various applications.>
> I have been myself investigating in several hundreds of such equal-
> beating and proportional beating tunings, notably in very
> concentrated areas where I could experience, and several times in
> blind test situation, very different specific musical colors, even
> between very close generators. It's clear to me.>
> Such extended fifths eq-b tunings you can find among those I included
> in the African, Persian, and Indonesian sections of the Ethno 2
> collection, and other eq-b tunings, available in my folder in the TL
> files. I invite you for example to check the Soria tuning (Persian
> folder), discussed in this list in its 17 and 19 notes versions, with
> a 705.5685 c. generator, that has triple equal-beating of fourths,
> septimal minor thirds with 7ths harmonics everywhere, and much more.
> I have a preference myself for the rational versions issued from the
> algorithm whenever possible, as the beatings of chords in different
> keys will be also in proportional beating between them, but both
> techniques have their utility.
What a fascinating collection!
> If I understand well, what you do here is different, I call it
> "ribbon" temperaments, that is the cumulation of 2 or 3 chains (of
> extended fifths, in your case), transposed by a specific interval,
> here a major third, tuned to be in equal beating with the other
> intervals ? I have to hear how it sounds, but that's a good idea.
> Of the temperaments you posted, two have known generators to me :
>
> ovovo27 has a very famous slendro generator, that I used in the
> triple eq-b natté.scl tuning (Ethno2 /Indonesian folder) ; it is
also
> well known by the mathematicians in its complementary fourth as the
> first Pivot-Vijayaraghavan number, 1.3247179572447

Took me awhile to catch up with you here. I'd never heard of a Pisot
number <http://mathworld.wolfram.com/PisotNumber.html> before. Now that
I've read a bit about them, I can say that, not only are the chains in
ovovo27 made from links that invert the smallest Pisot number, but that
there are also two other Pisot numbers with, respectively, 11/8 & 13/8
beating rates that are closely approximated in both ovovo27 & ovovo22.
> ovovo10 is a declinaison of the same algorithm family, which I named
> myself Eronyme.
> Its fifth ratio is the solution of x^3 = 5x^2 - 8 and therefore has
> in addition of its eq-b properties another property (even more
> important to me), of differential coherence, as it is expressed by
> the algorithm. In case it is of some interest to you, it expresses
> the -c of a 23/16 in your temperament, which would be a major third
> above the #2 note (238.678 c.) - it's present in the harmonics and
> it's a super nice harmony. Because your tuning includes major third
> transpositions it could be pertinent with your temperament, however I
> don't fully understand the logic of it and why for example between
> notes 2 & 8 you have 723.644 c., and not 719.339 ?
In ovovo10, reckoning the chain of tempered "fifth" intervals from D at
0.000, the "fifth" is so insanely wide that E (2 links after D) & F (3
links before D) converge. They're less than 3.305 cents apart, which
means that a practical temperament can only use one of those two
pitches. I chose to keep E at 238.678 cents because keeping F at 241.983
cents would have left me with four different step sizes instead of
three.

> ovovo22's generator is not known of me. If you don't mind I will
> include it with your name in my (near to 2000 entries) fractal
> generators database. If you can explain it to me either offlist or
> not, you're welcome.
That would be a cool thrill. I made ovovo22 by tempering out the 64/63
interval in such a fashion that 9/8 & 7/4 would beat at the same rate,
then approximating 5/4 so that 15/8 would have the same beating rate,
choosing between sharp & flat 5/4 approximations in order to have
similar or slower beating rates neat 11/8 & 13/8. I didn't realize it,
but those 11/8 & 13/8 beaters, found along the +2 anchor chain, are also
close approximations of two Pisot numbers.
> What means "ovovo" BTW ?
> - - - - - - -
> Jacques

That's my truncated impression of the way 4:5:6:7 beating, from a subset
of the temperament I now call ovovo27, sounded when I first heard it, in
reed organ voice: ovovovovovo....
==
Mark L. Vines
a/k/a Manu Phonic

Hi Chris. You make fine music & I would love to hear what you do with any of the ovovo temperaments. I've posted their Scala files (with corrections; always the careless errors!) both under Messages in the "ovovo Scala files" topic & under Files in the "manuphonic" folder. Note that the pathname line in each file gives only the filename, whereas I guess the full pathname that the file will have in your system needs to be there upon opening.
==
Mark L. Vines
a/k/a Manu Phonic

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> A scala file would be great - If I had one I'd be glad to try to
> compose something.
>
> On Sun, Jan 2, 2011 at 8:00 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> > >
> > > Lastly, ovovo27. Has anyone else used or published this temperament or a
> > > close relative already?
>
> >
> > The next step, of course, would be to compose something in one of these scales.
>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> Simple definitions and quite exactly what I thought "periodicity
> buzz", in my sense was !
> I would only add two complementary exceptions :
> * beating can also be generated of course by close fundamental chords
> such as small commas, double strings or reeds etc. ;
> * I would not exclude a high enough VF to be a pitch, this would only
> make it more "buzzing"...

Jacques,
I am sure you have seen that Mike B and Carl have said that my explanation is incorrect. Sorry to all, I got the wrong impression from those old messages.

I am sure I should really listen for myself - but I am intrigued that Carl says it can be heard with e.g. 7:6 and 7:4 - do you mean the purely tuned dyad *on its own*?

Steve M.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > > It may be, but there's no way to reproduce that effect with
> > > non-JI intervals.
> >
> > I have a file using a 15-odd-limit proportional-beating
> > temperament -- a live performance, in fact (given 35 years ago)
> > -- that demonstrates that the foregoing statement is incorrect:
> > http://xenharmony.wikispaces.com/space/showimage/improv29.mp3
> > The periodicity buzz is particularly evident toward the very end,
> > when an 11-limit chord progression is played an octave lower
> > than in the rest of the piece.
>
> Can you give the chord or chords in question?

At 3:45-3:46 there is a tempered 6:7:9:11 chord followed by a tempered 4:5:6:7 chord: 2:3 is 1.62 cents wide, 7:9 is 3.25 cents wide, 7:11 is 3.19 cents wide; the other intervals are exact (except for 9:11, which is almost exact, about 0.05 cents narrow).

Perhaps my reply was a bit hasty: since these are irrationally tempered chords, there can't be any periodicity buzz. Are we instead hearing a virtual fundamental?

--George

Thanks for your very accurate listening, Carl. Now I remember that in this passage I noticed a very strange phenomena, most of which is destroyed in the mp3 unfortunately : while the played sounds are absolutely steady (but with changing volumes), at times you can have the illusion (?) that some of the combinatory tones glide between notes, in a crying effect (I would even say some kind of a "cavernous stomach noise" ! !)
These sounds being produced by digital synthesis here what those combinatory tones are exactly can be discussed, but the effect does not seem different from rubbed cristal glasses played together, which typically produce difference tones.
This scale being based on differential coherence I just verified here that most of them are in tune with the scale, which means they can well be difference tones.
You may not agree with that, but in the spot you mention between 2:47" and 2:48", I can hear indeed a groaning effect that is not radically different from either low difference tones, or subharmonics (which can also be produced by difference tones between difference tones), things that I hear regularly with my photosonic disks.
Whatever it is, and now that I have a better idea of what you mean by periodicity buzz, it could very well be this phenomena I experienced with the meta-version of Miracle temperament, that I commented to Cameron Bobro in these words (TL, 5th of Feb.2010) :

"On my first test what striked me were the 7/6, I found them terrific ! Nothing new, I found these are quasi-perfectly just."
"Then one scale I was happy with that came out is this kind of weird "septimal Shur" :

1/1
150.59055
315.81365
466.40420
699.60630
816.20735
966.79790
2/1

The sound was an oboe imitation and I forgot what was the chord exactly, I think a 7:6:4 but what I experimented as unusual with it was a strong surbrillance of the partials + a groaning effect (at first I thought I was having some electric distorsion).

Not that I believe that a "meta-temperament" has special acoustic properties in itself, but you may have a candidate here to test in the chord [0 - 699.6063 - 966.7979] cents, and a scale context to compare with other chords too.

- - - - - - - -
Jacques

Carl wrote :

> Starting at ~ 2:35 there are some high-pitched tones
> and beating. Very briefly at 2:47-2:48 I hear something
> that sounds vaguely like periodicity buzz. I don't hear
> it anywhere else in the piece.
>
> I asked for audio demos because I originally thought you
> were referring to your work with photosonic disks, which
> have a unique sound I cannot easily reproduce. Now that
> I see you were referring to other work I can say audio
> files are less necessary. What is necessary is to compare
> chords of like error (deviation from JI) but with different
> beat ratios, where only those with simpler beat ratios
> exhibit some desirable quality (periodicity buzz or other).
> That is the only test that matters for claims pertaining
> to beat ratios and I am hoping to have time to put one
> together for tetrads (previous tests on triads failed to
> show benefit of proportional beating). If you have suggestions
> for chords to try I am collecting them now (some from Gene
> and some from Mark already). If you would like to submit
> the chord at 2:47 or any others feel free!
>
> -Carl

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> I am sure I should really listen for myself - but I am intrigued
> that Carl says it can be heard with e.g. 7:6 and 7:4 - do you
> mean the purely tuned dyad *on its own*?

Yep. It's also audible in a 5:6:7 composed of sines...
http://lumma.org/temp/567.wav
so much for what I just told you about beating in
upper partials. . . -Carl

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> > > http://xenharmony.wikispaces.com/space/showimage/improv29.mp3

> > Can you give the chord or chords in question?
>
> At 3:45-3:46 there is a tempered 6:7:9:11 chord followed by a
> tempered 4:5:6:7 chord: 2:3 is 1.62 cents wide, 7:9 is 3.25 cents
> wide, 7:11 is 3.19 cents wide; the other intervals are exact
> (except for 9:11, which is almost exact, about 0.05 cents narrow).
>
> Perhaps my reply was a bit hasty: since these are irrationally
> tempered chords, there can't be any periodicity buzz. Are we
> instead hearing a virtual fundamental?

I don't know what periodicity buzz is, but I hear it all
over your piece. And I suspect that is because the tempering
is very slight. There is no problem with 72-ET (Prent Rogers
music) and probably not 41 or 46 either (I should try it).

-Carl

Hi Jacques,

It looks like I am going to have to go back to "square one"
with the periodicity buzz phenomenon (idiom for having to throw
out one's ideas and start over at the beginning). It may be
that we are hearing the zero-crossings of the waveform itself??

In the meantime I will proceed with my test of proportional-
beating tetrads that *controls for error from JI* (in the
sense of a scientific study).

Harumph. (onomatopoeia expressing frustration :)

-Carl

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Thanks for your very accurate listening, Carl. Now I remember
> that in this passage I noticed a very strange phenomena, most of
> which is ...

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> > Simple definitions and quite exactly what I thought "periodicity
> > buzz", in my sense was !
> > I would only add two complementary exceptions :
> > * beating can also be generated of course by close fundamental chords
> > such as small commas, double strings or reeds etc. ;
> > * I would not exclude a high enough VF to be a pitch, this would only
> > make it more "buzzing"...
>
> Jacques,
> I am sure you have seen that Mike B and Carl have said that my explanation is incorrect. Sorry to all, I got the wrong impression from those old messages.

But your explanation was correct in the sense Paul Erlich was using this term. What would be useful now would be to have some recordings on which everyone would agree to say "this is a periodicity buzz !" - then we could probably do a more concrete analysis of the phenomena in question, by trying to reproduce it while changing the conditions. In the meantime, we are only making hypothesis, about perhaps even different phenomenas.

> I am sure I should really listen for myself - but I am intrigued that Carl says it can be heard with e.g. 7:6 and 7:4 - do you mean the purely tuned dyad *on its own*?
>
> Steve M.
>

OK, so there are a few options here:

1) The fundamentals in 5:6:7 are close enough within the critical band
to beat slightly, and since they're harmonically related the beating
will be periodic and polyrhythmic. There's also some buzz on the upper
8:9 in this example: http://lumma.org/temp/689.wav
2) there could be some nonlinearity in your ear (DPOAE's) or in the
speakers intensifying this, although this is for the moment
unfalsifiable
3) some kind of nonuniformity in the time-frequency response of the
ear. I hope this isn't really the cause because things will get really
complicated.

My initial analysis, from a time-domain standpoint, of the cause of
periodicity buzz is related to this: as you add more harmonics
together, with -NO- rolloff, the waveform changes from a single sine
wave to an impulse train. That is, the waveform starts to look like
this:

_|_|_|_|_|_|_|_|_|_|_|_

Where f(t)=0 at all values of t except for integer multiples of the
period, where it is infinite.

I was actually taught that this waveform is called "buzz." It's pretty
harsh. Here are some examples:

http://www.mikebattagliamusic.com/music/441HzBuzz.wav
http://www.mikebattagliamusic.com/music/220.5HzBuzz.wav
http://www.mikebattagliamusic.com/music/110.25HzBuzz.wav
http://www.mikebattagliamusic.com/music/55.125HzBuzz.wav
http://www.mikebattagliamusic.com/music/27.5625HzBuzz.wav

Here's the last one again, but only with sines being added up to 2756.25 Hz:

http://www.mikebattagliamusic.com/music/27.5625HzIncompleteBuzz.wav

Note the harmonic ringing out on top. Believe it or not, this sine
wave isn't louder than any of the others - I simply just made the
algorithm stop adding cosines after 2756.25 Hz. There is no actual
resonance at that frequency. It's simply that since I'm not adding
other harmonics above it, there's nothing to interfere with it at the
"sharp" end of the critical band at that frequency, and you actually
hear it.

A more rigorous formalization of periodicity buzz would have to spring
out of this. I think that this can be explained entirely with critical
band effects, but if not we're going to have to delve into this:

http://en.wikipedia.org/wiki/Continuous_wavelet_transform

And I'd still like to see a smooth transformation between perfectly
polyrhythmic periodicity buzz and perfectly polyrhythmic equal
beating, although I'm not sure how exactly it would work.

-Mike

PS: for the mathy folks, note that I said that the frequency response
of an impulse train is all harmonics being present with no rolloff.
That is, the Fourier transform of an impulse train is another impulse
train, but the period of one becomes the frequency of the other. If
f=1/f, you end up with an eigenfunction of the Fourier transform,
which might be useful for some to understand it more deeply.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > > > http://xenharmony.wikispaces.com/space/showimage/improv29.mp3
>
> > > Can you give the chord or chords in question?
> >
> > At 3:45-3:46 there is a tempered 6:7:9:11 chord followed by a
> > tempered 4:5:6:7 chord: 2:3 is 1.62 cents wide, 7:9 is 3.25 cents
> > wide, 7:11 is 3.19 cents wide; the other intervals are exact
> > (except for 9:11, which is almost exact, about 0.05 cents narrow).
> >
> > Perhaps my reply was a bit hasty: since these are irrationally
> > tempered chords, there can't be any periodicity buzz. Are we
> > instead hearing a virtual fundamental?
>
> I don't know what periodicity buzz is, but I hear it all
> over your piece. And I suspect that is because the tempering
> is very slight. There is no problem with 72-ET (Prent Rogers
> music) and probably not 41 or 46 either (I should try it).
>
> -Carl

I do hear "it" too almost all the way and it sounds to me as virtual fundamentals, from close to JI chords, reinforced by the various drones - or periodicity buzz in the Paul Erlich sense, whatever...
Anyway, interesting harmonies you were doing 35 years ago, George !
- - - -
Jacques

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> My initial analysis, from a time-domain standpoint, of the cause of
> periodicity buzz is related to this: as you add more harmonics
> together, with -NO- rolloff, the waveform changes from a single sine
> wave to an impulse train.

Shouldn't the first order of business be a definition of periodicity buzz? Might be a good article for you to write on the Xenwiki.

Mike wrote:

> 1) The fundamentals in 5:6:7 are close enough within the critical
> band to beat slightly,

Not in the example I posted.

> 2) there could be some nonlinearity in your ear (DPOAE's) or in
> the speakers intensifying this, although this is for the moment
> unfalsifiable

It could be beating sum tones, but it doesn't sound that way
to me and I listened at low volume levels through very good
equipment.

>My initial analysis, from a time-domain standpoint, of the cause
>of periodicity buzz is related to this: as you add more harmonics
>together, with -NO- rolloff, the waveform changes from a single
>sine wave to an impulse train. [snip]
> Here are some examples:
> http://www.mikebattagliamusic.com/music/441HzBuzz.wav
> http://www.mikebattagliamusic.com/music/220.5HzBuzz.wav
> http://www.mikebattagliamusic.com/music/110.25HzBuzz.wav
> http://www.mikebattagliamusic.com/music/55.125HzBuzz.wav
> http://www.mikebattagliamusic.com/music/27.5625HzBuzz.wav
> Here's the last one again, but only with sines being added
> up to 2756.25 Hz:
> http://www.mikebattagliamusic.com/music/27.5625HzIncompleteBuzz.wav

Hm, interesting!

-Carl

On Wed, Jan 5, 2011 at 4:40 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > My initial analysis, from a time-domain standpoint, of the cause of
> > periodicity buzz is related to this: as you add more harmonics
> > together, with -NO- rolloff, the waveform changes from a single sine
> > wave to an impulse train.
>
> Shouldn't the first order of business be a definition of periodicity buzz? Might be a good article for you to write on the Xenwiki.

OK, I'll get on that at some point, although I'm extremely busy this
next week...

-Mike

On Thu, Jan 6, 2011 at 2:04 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > 1) The fundamentals in 5:6:7 are close enough within the critical
> > band to beat slightly,
>
> Not in the example I posted.

Isn't 6:7 within the critical bandwidth at Eb above middle C? But
either way, the effect is subtle with 5:6:7, and if there's "subtle"
beating at 5:6 and "subtle" beating at 6:7, and they reinforce one
another in some kind of periodic pattern, that should be more apparent
than the beating at 5:6 or 6:7 alone.

I bet if you move this up an octave, there will be less periodicity
buzz, and if you move it down an octave, more... I also wonder what
would happen if you tried 5:5.9:7.8 and 5:6.1:7.2.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Isn't 6:7 within the critical bandwidth at Eb above middle C?

Nope, and even if it were, it wouldn't produce beating until
it was narrower than CB/2.

> I bet if you move this up an octave, there will be less periodicity
> buzz, and if you move it down an octave, more... I also wonder what
> would happen if you tried 5:5.9:7.8 and 5:6.1:7.2.

I just tried it at 200 and 400 Hz and the buzz changed rate
but not intensity or quality.

I must say, it appears it is the shape of the waveform itself...

-Carl

I wrote:

> I must say, it appears it is the shape of the waveform itself...

One way to test this would be to arrange the phases so that
the waveform amplitude is as flat as possible. That's beyond
my synthesis capabilities at the moment. So I decided to
test it dichotically.

stereo, L = R = 4:6:7
http://lumma.org/temp/467.wav

stereo, L = 6, R = 4:7
http://lumma.org/temp/6and47.wav

stereo, L = 7, R = 4:6
http://lumma.org/temp/7and46.wav

Through good in-ears, I hear:

1st file - clear period buzz

2nd file - very weak/rapid buzz equal to that when
listening to the R channel alone

3rd file - very weak/rapid buzz equal to that when
listening to the R channel alone

You can test the distortion in your equipment by listening
to the L channel by itself (in the 2nd and 3rd files) and
verifying the purity of the tone.

In short: for me it doesn't occur dichotically.

Comments?

-Carl

On Thu, Jan 6, 2011 at 3:24 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Isn't 6:7 within the critical bandwidth at Eb above middle C?
>
> Nope, and even if it were, it wouldn't produce beating until
> it was narrower than CB/2.

I don't understand, why do you say that? Roughness and beating are the
same thing - the critical band isn't a perfect rectangular bandpass
filter, it's a smooth slope. Specifically it's that a sine wave
simultaneously excites all of the hairs nearby a certain peak
frequency in the basilar membrane, so why wouldn't 6:7 be close enough
to produce some kind of small perturbation?

> > I bet if you move this up an octave, there will be less periodicity
> > buzz, and if you move it down an octave, more... I also wonder what
> > would happen if you tried 5:5.9:7.8 and 5:6.1:7.2.
>
> I just tried it at 200 and 400 Hz and the buzz changed rate
> but not intensity or quality.

This is true for me too, assuming low volumes, but volume does change
it; e.g. louder volume - more buzz. This might be because of
nonlinearities in my listening equipment though. But I'm really
stumped here.

> I must say, it appears it is the shape of the waveform itself...

Yes, but in the same regard, the time-domain "shape" of 440 Hz + 441
Hz is going to be a 440.5 Hz sine wave with a clear amplitude
envelope... but there's more to why we hear beating than that. We're
not hearing the time-domain waveform, but a time frequency
representation of the waveform.

I'm stumped here - might be that there's a second critical band in the
auditory midbrain. Any time-frequency analysis generates a "critical
band" per se, and there's obviously some kind of time-frequency
analysis going on in the brain. I've noticed people making reference
to some "dual critical band hypothesis" in the literature as well.

-Mike

Mike wrote:
> I don't understand, why do you say that? Roughness and beating
> are the same thing

They're not, that's why they have separate names.

> Specifically it's that a sine wave
> simultaneously excites all of the hairs nearby a certain peak
> frequency in the basilar membrane, so why wouldn't 6:7 be close
> enough to produce some kind of small perturbation?

7/6 is larger than a CB in this range and anything larger
than CB/2 is well-resolved enough not to create anything like
periodicity buzz. That's easy to test with irrational
intervals.

-Carl

On Thu, Jan 6, 2011 at 4:27 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
> > I don't understand, why do you say that? Roughness and beating
> > are the same thing
>
> They're not, that's why they have separate names.

But they're generated by the same mechanism. When the beating is of
low amplitude and high frequency, we say it's roughness, when the
beating is of high amplitude and low frequency, we say it's beating.

> > Specifically it's that a sine wave
> > simultaneously excites all of the hairs nearby a certain peak
> > frequency in the basilar membrane, so why wouldn't 6:7 be close
> > enough to produce some kind of small perturbation?
>
> 7/6 is larger than a CB in this range and anything larger
> than CB/2 is well-resolved enough not to create anything like
> periodicity buzz. That's easy to test with irrational
> intervals.

What do you mean by "well-resolved?"

-Mike

On Thu, Jan 6, 2011 at 3:54 AM, Carl Lumma <carl@...> wrote:
>
> I wrote:
>
> > I must say, it appears it is the shape of the waveform itself...
>
> One way to test this would be to arrange the phases so that
> the waveform amplitude is as flat as possible. That's beyond
> my synthesis capabilities at the moment. So I decided to
> test it dichotically.

If there really is a polyrhythm going on, then that would, I guess,
alter the phases of the beats in the polyrhythm. So instead of 3
against 4 sounding like "pass the #$@#$ ketchup," it would sound like
a bunch of random stuff that happens to be periodic. Perhaps that
would make the buzz less significant.

I just ran some quick tests in MATLAB and what happens when the 5:6:7
sine waves are all in different phases is that you get crazy
spatialization effects going on. Or at least I do. Periodicity buzz
actually does noticeably vary here, at least to me. The more
interesting and relevant thing was when I generated the impulse train
again with the sines at random phases, which means we're now dealing
with periodic noise, not an impulse train. I'll post the examples
tomorrow because it's 4 AM now.

-Mike

Mike wrote:

> > 7/6 is larger than a CB in this range and anything larger
> > than CB/2 is well-resolved enough not to create anything like
> > periodicity buzz. That's easy to test with irrational
> > intervals.
>
> What do you mean by "well-resolved?"

There is no roughness in the 7:6 here. And it's looking more
and more to me like periodicity buzz is an audible time-domain
thing... -Carl

Mike Battaglia <battaglia01@...> wrote:

> But they're generated by the same mechanism. When the
> beating is of low amplitude and high frequency, we say
> it's roughness, when the beating is of high amplitude and
> low frequency, we say it's beating.

Bread and beer are generated by the same mechanism.
They're still different things that have different names.

Graham

Thanks for these examples, Mike.

They definitively sound like buzzes, and the "incomplete" one is very interesting in the way it boosts the last harmonic, this is also a super musical effect I think. Why not creating a virtual synthesizer with that technique ? I buy it !
Such "phase alignment of harmonics" are certainly good examples of periodicity buzz.

Note that there can be many other ways to synthesize similar buzzes.
For example if you create periodically any small accident in any waveform, such as a blank space in every "0" here :

^^^^^^^^^^0^^^^^^^^^^0^^^^^^^^^^0...

You will also generate a "periodicity buzz", with various spectra on top. I call that omissions. There can be bi-omissions, multi-omissions etc.
These buzzes are subharmonics, and that's also what the tibetan monks use in the Gyuto or Gyume styles of chanting, as well as the Tuva shepards in their "Kaarjira" singing style (have you heard that CD where a very young boy seems to have the voice of a huge geant), etc..
You were also talking about harmonic polyrhythms generating periodicity buzz, that works also (that what I've been experimenting since my very first photosonic disks) : any accerated rhythm has its own timbre, and not only periodic rhythms but also semi-periodic, composite waveforms, MOS, fractal rhythms, etc.
You also made the hypothesis that equal or proportional beating wether in triads or tetrads would be a possible cause, I also experimented they can have periodicity buzz, but could it be a cause I don't know, it has to be verified.
Then we heard different examples that were pointed out as "periodicity buzz", but could be simply explained by virtual fundamental or by difference tones generation.
My suggestion is that "periodicity buzz", if we strictly stick to what the words mean, covers all these phenomenas.
Unless this term has been used before in musical acoustics in a more special way, but I didn't find it - by googling it the only pertinent pages it refers to (or trying to be pertinent !) are from the Tuning List.
Let's hope this "buzz" arrives to a satisfying ketchup... :)
- - - - - - - -
Jacques

Mike wrote :

> My initial analysis, from a time-domain standpoint, of the cause of
> periodicity buzz is related to this: as you add more harmonics
> together, with -NO- rolloff, the waveform changes from a single sine
> wave to an impulse train. That is, the waveform starts to look like
> this:
>
> _|_|_|_|_|_|_|_|_|_|_|_
>
> Where f(t)=0 at all values of t except for integer multiples of the
> period, where it is infinite.
>
> I was actually taught that this waveform is called "buzz." It's pretty
> harsh. Here are some examples:
>
> http://www.mikebattagliamusic.com/music/441HzBuzz.wav
> http://www.mikebattagliamusic.com/music/220.5HzBuzz.wav
> http://www.mikebattagliamusic.com/music/110.25HzBuzz.wav
> http://www.mikebattagliamusic.com/music/55.125HzBuzz.wav
> http://www.mikebattagliamusic.com/music/27.5625HzBuzz.wav
>
> Here's the last one again, but only with sines being added up to > 2756.25 Hz:
>
> http://www.mikebattagliamusic.com/music/27.5625HzIncompleteBuzz.wav
>
> Note the harmonic ringing out on top. Believe it or not, this sine
> wave isn't louder than any of the others - I simply just made the
> algorithm stop adding cosines after 2756.25 Hz. There is no actual
> resonance at that frequency. It's simply that since I'm not adding
> other harmonics above it, there's nothing to interfere with it at the
> "sharp" end of the critical band at that frequency, and you actually
> hear it.
>
> A more rigorous formalization of periodicity buzz would have to spring
> out of this. I think that this can be explained entirely with critical
> band effects, but if not we're going to have to delve into this:
>
> http://en.wikipedia.org/wiki/Continuous_wavelet_transform
>
> And I'd still like to see a smooth transformation between perfectly
> polyrhythmic periodicity buzz and perfectly polyrhythmic equal
> beating, although I'm not sure how exactly it would work.
>
> -Mike

Mark wrote :

(Jacques) :
> > ovovo27 has a very famous slendro generator, that I used in the
> > triple eq-b natté.scl tuning (Ethno2 /Indonesian folder) ; it is
> also
> > well known by the mathematicians in its complementary fourth as the
> > first Pisot-Vijayaraghavan number, 1.3247179572447
>
> Took me awhile to catch up with you here. I'd never heard of a Pisot
> number <http://mathworld.wolfram.com/PisotNumber.html> before. Now
> that
> I've read a bit about them, I can say that, not only are the chains in
> ovovo27 made from links that invert the smallest Pisot number,

It does not matter that it inverts it or not, both this generator and
its complement arrive to the same tuning.

> > ovovo22's generator is not known of me. If you don't mind I will
> > include it with your name in my (near to 2000 entries) fractal
> > generators database. If you can explain it to me either offlist or
> > not, you're welcome.
> That would be a cool thrill. I made ovovo22 by tempering out the 64/63
> interval in such a fashion that 9/8 & 7/4 would beat at the same rate,
> then approximating 5/4 so that 15/8 would have the same beating rate,
> choosing between sharp & flat 5/4 approximations in order to have
> similar or slower beating rates neat 11/8 & 13/8.

Could you express it (just the fifth or fourth ratio) in algebraic
form ? that would be lots simpler.
(also because different sets of intervals can have same beat rates,
but in opposite directions, at different octaves, etc.)

> I didn't realize it,
> but those 11/8 & 13/8 beaters, found along the +2 anchor chain, are
> also
> close approximations of two Pisot numbers

That's more anecdotic, unless it confers any special acoustic
quality, does it ?
- - - - - -
Jacques

On Thu, Jan 6, 2011 at 5:22 AM, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > But they're generated by the same mechanism. When the
> > beating is of low amplitude and high frequency, we say
> > it's roughness, when the beating is of high amplitude and
> > low frequency, we say it's beating.
>
> Bread and beer are generated by the same mechanism.
> They're still different things that have different names.
>
> Graham

Your witty analogy notwithstanding, they are not two different things.
I used the example to make a point about how the critical band works.
They have two different names for the same reason that a tropical
storm and a hurricane have two different names. And if I am mistaken,
then I still don't see what facet of the picture I'm missing.

-Mike

On Thu, Jan 6, 2011 at 5:15 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > 7/6 is larger than a CB in this range and anything larger
> > > than CB/2 is well-resolved enough not to create anything like
> > > periodicity buzz. That's easy to test with irrational
> > > intervals.
> >
> > What do you mean by "well-resolved?"
>
> There is no roughness in the 7:6 here. And it's looking more
> and more to me like periodicity buzz is an audible time-domain
> thing... -Carl

It could be an audible time-domain thing. Or perhaps specifically, it
could be that the brain partitions some of the signal into a "noise"
component, which is what I think is a likely possibility, and I have
no idea how the brain treats the noise vs pitched problem at all.

My original point is that the critical band is not an abrupt switch,
and the cochlea doesn't really resolve anything. The critical band is
telling you what hairs nearby resonate at the same frequency, and how
much. So if you play a single sine wave, the entire critical band
resonates - this doesn't cause chaos, however. I would assume because
of this (which you turned me onto... not sure why we've flipped sides
now):

http://en.wikipedia.org/wiki/Temporal_theory_%28hearing%29

Anyway, I'm wondering what a buzz waveform would look like if we ran
it through this:

http://iem.at/projekte/publications/iem_report/report02_98/

-Mike

I'll have to get to this later, more computer trouble for now.

On Thu, Jan 6, 2011 at 4:55 AM, Mike Battaglia <battaglia01@...> wrote:
> On Thu, Jan 6, 2011 at 3:54 AM, Carl Lumma <carl@...> wrote:
>>
>> I wrote:
>>
>> > I must say, it appears it is the shape of the waveform itself...
>>
>> One way to test this would be to arrange the phases so that
>> the waveform amplitude is as flat as possible. That's beyond
>> my synthesis capabilities at the moment. So I decided to
>> test it dichotically.
>
> If there really is a polyrhythm going on, then that would, I guess,
> alter the phases of the beats in the polyrhythm. So instead of 3
> against 4 sounding like "pass the #$@#$ ketchup," it would sound like
> a bunch of random stuff that happens to be periodic. Perhaps that
> would make the buzz less significant.
>
> I just ran some quick tests in MATLAB and what happens when the 5:6:7
> sine waves are all in different phases is that you get crazy
> spatialization effects going on. Or at least I do. Periodicity buzz
> actually does noticeably vary here, at least to me. The more
> interesting and relevant thing was when I generated the impulse train
> again with the sines at random phases, which means we're now dealing
> with periodic noise, not an impulse train. I'll post the examples
> tomorrow because it's 4 AM now.
>
> -Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
>
> > I must say, it appears it is the shape of the waveform itself...
>
> One way to test this would be to arrange the phases so that
> the waveform amplitude is as flat as possible. That's beyond
> my synthesis capabilities at the moment. So I decided to
> test it dichotically.
>
> stereo, L = R = 4:6:7
> http://lumma.org/temp/467.wav
>
> stereo, L = 6, R = 4:7
> http://lumma.org/temp/6and47.wav
>
> stereo, L = 7, R = 4:6
> http://lumma.org/temp/7and46.wav

None of them buzzed all that much. The third sample sounded clean; I thought the second was more like the first than it was like the third.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > stereo, L = R = 4:6:7
> > http://lumma.org/temp/467.wav
> >
> > stereo, L = 6, R = 4:7
> > http://lumma.org/temp/6and47.wav
> >
> > stereo, L = 7, R = 4:6
> > http://lumma.org/temp/7and46.wav
>
> None of them buzzed all that much. The third sample sounded
> clean; I thought the second was more like the first than it
> was like the third.

How are you listening? The first version should buzz,
in any case. -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> How are you listening? The first version should buzz,
> in any case. -Carl

Computer speakers. Would earphones be better?

Gene wrote:

> > stereo, L = R = 4:6:7
> > http://lumma.org/temp/467.wav
> > stereo, L = 6, R = 4:7
> > http://lumma.org/temp/6and47.wav
> > stereo, L = 7, R = 4:6
> > http://lumma.org/temp/7and46.wav

> > How are you listening? The first version should buzz,
> > in any case. -Carl
>
> Computer speakers. Would earphones be better?

Speakers should make all three files sound more alike.
Earphones, especially the in-ear kind, should make the
first one buzz more than the other two. At least that's
what I heard.

-Carl

> > > stereo, L = R = 4:6:7
> > > http://lumma.org/temp/467.wav
> > > stereo, L = 6, R = 4:7
> > > http://lumma.org/temp/6and47.wav
> > > stereo, L = 7, R = 4:6
> > > http://lumma.org/temp/7and46.wav

> Speakers should make all three files sound more alike.
> Earphones, especially the in-ear kind, should make the
> first one buzz more than the other two.

Still hoping someone can confirm or deny this effect!

-C.

On Sun, Jan 9, 2011 at 4:43 PM, Carl Lumma <carl@...> wrote:
>
> > > > stereo, L = R = 4:6:7
> > > > http://lumma.org/temp/467.wav
> > > > stereo, L = 6, R = 4:7
> > > > http://lumma.org/temp/6and47.wav
> > > > stereo, L = 7, R = 4:6
> > > > http://lumma.org/temp/7and46.wav
>
> > Speakers should make all three files sound more alike.
> > Earphones, especially the in-ear kind, should make the
> > first one buzz more than the other two.
>
> Still hoping someone can confirm or deny this effect!

Same applies here. I'm working on synthesizing the samples now.

-Mike

I also notice that the buzz happens on the phantom fundamental - that
is the note that is buzzing, specifically. There seems to be AM on the
phantom fundamental, not on the individual notes. Well, maybe slightly
on the individual notes, but mostly the fundamental. Does anyone else
notice something similar?

-Mike

On Sun, Jan 9, 2011 at 4:47 PM, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Jan 9, 2011 at 4:43 PM, Carl Lumma <carl@...> wrote:
>>
>> > > > stereo, L = R = 4:6:7
>> > > > http://lumma.org/temp/467.wav
>> > > > stereo, L = 6, R = 4:7
>> > > > http://lumma.org/temp/6and47.wav
>> > > > stereo, L = 7, R = 4:6
>> > > > http://lumma.org/temp/7and46.wav
>>
>> > Speakers should make all three files sound more alike.
>> > Earphones, especially the in-ear kind, should make the
>> > first one buzz more than the other two.
>>
>> Still hoping someone can confirm or deny this effect!
>
> Same applies here. I'm working on synthesizing the samples now.
>
> -Mike
>

I am not sure if that is the VF or a difference tone, but
yes, I hear it buzzing. I also hear the tones themselves
buzzing. It seems the amplitude variation in the original
waveform is projected onto the pitches we hear. -C.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I also notice that the buzz happens on the phantom
> fundamental - that is the note that is buzzing, specifically.
> There seems to be AM on the phantom fundamental, not on the
> individual notes. Well, maybe slightly on the individual notes,
> but mostly the fundamental. Does anyone else notice something
> similar?
>
> -Mike
>
>
>
> On Sun, Jan 9, 2011 at 4:47 PM, Mike Battaglia <battaglia01@...> wrote:
> > On Sun, Jan 9, 2011 at 4:43 PM, Carl Lumma <carl@...> wrote:
> >>
> >> > > > stereo, L = R = 4:6:7
> >> > > > http://lumma.org/temp/467.wav
> >> > > > stereo, L = 6, R = 4:7
> >> > > > http://lumma.org/temp/6and47.wav
> >> > > > stereo, L = 7, R = 4:6
> >> > > > http://lumma.org/temp/7and46.wav
> >>
> >> > Speakers should make all three files sound more alike.
> >> > Earphones, especially the in-ear kind, should make the
> >> > first one buzz more than the other two.
> >>
> >> Still hoping someone can confirm or deny this effect!
> >
> > Same applies here. I'm working on synthesizing the samples now.
> >
> > -Mike
> >
>

On Sun, Jan 9, 2011 at 4:54 PM, Carl Lumma <carl@...> wrote:
>
> I am not sure if that is the VF or a difference tone, but
> yes, I hear it buzzing. I also hear the tones themselves
> buzzing. It seems the amplitude variation in the original
> waveform is projected onto the pitches we hear. -C.

I think we might be hearing, literally, the impulse response of the VF
"filterbank" in the brain. Imagine taking a trumpet and slapping the
valve with your hand - you'll hear a pitched, decaying "thunk," which
is the impulse response of the trumpet. Now imagine instead of
slapping it with your hand you slapped it with an infinitely hard,
infinitely dense object, and you'd have the perfect impulse response
(your hand is soft and compliant, so the high frequencies will be
filtered out more with that).

A periodic impulse train is, in a sense, "slapping" this neurological
filterbank with an impulse x times per second, and we're hearing not
just a smooth VF, but a quick series of rapidly decaying VF's, which
is periodicity buzz.

I'm starting to realize that both this and the HE-filterbank "harmonic
laplace" transform I keep talking about are related mathematically by
something I've seen in the psychoacoustics literature called the
"double critical band hypothesis," but I'm not sure how to explain how
without blasting the list with more obscure signal processing
calculus. The general idea is that the fact that the VF's decay over
time would lead to spreading in the frequency domain, which leads to
another 'critical band' (just like the damping of the hair cells in
the ear leads to the concept of the first critical band).

-Mike

OK, I'm not sure how to make it flat, but here are some sines at
different phases:

http://www.mikebattagliamusic.com/music/567sines.wav - all sines at phase 0
http://www.mikebattagliamusic.com/music/567sines0-45-90.wav - sines at
phases 0, 45, and 90 degrees, in order from lowest to highest
http://www.mikebattagliamusic.com/music/567sines0-90-180.wav - sines
at phases 0, 90, and 180 degrees, in order from lowest to highest

A graph of the three:

http://www.mikebattagliamusic.com/music/sinesphase.png

I hear some interesting spatialization effects between the three sine
examples, but nothing changing with the buzz. Next I'll repeat this by
convolving an impulse train and a chirp and things will get more
interesting.

-Mike

On Sun, Jan 9, 2011 at 4:47 PM, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Jan 9, 2011 at 4:43 PM, Carl Lumma <carl@...> wrote:
>>
>> > > > stereo, L = R = 4:6:7
>> > > > http://lumma.org/temp/467.wav
>> > > > stereo, L = 6, R = 4:7
>> > > > http://lumma.org/temp/6and47.wav
>> > > > stereo, L = 7, R = 4:6
>> > > > http://lumma.org/temp/7and46.wav
>>
>> > Speakers should make all three files sound more alike.
>> > Earphones, especially the in-ear kind, should make the
>> > first one buzz more than the other two.
>>
>> Still hoping someone can confirm or deny this effect!
>
> Same applies here. I'm working on synthesizing the samples now.
>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> OK, I'm not sure how to make it flat, but here are some sines at
> different phases:

90 degrees rattles the furniture. 0 sounds similar, but with less tendency to cause earthquake-type damage. 180 degrees is pretty mild by comparison.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> OK, I'm not sure how to make it flat, but here are some sines at
> different phases:
>
> http://www.mikebattagliamusic.com/music/567sines.wav

???

On Sun, Jan 9, 2011 at 6:10 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > OK, I'm not sure how to make it flat, but here are some sines at
> > different phases:
> >
> > http://www.mikebattagliamusic.com/music/567sines.wav
>
> ???

Hahaha, oh man, I'm laughing so hard right now. I didn't realize what
Gene had meant before. Damn, I screwed this one up bad, lol.

I'm actually going to keep these up though, because there is something
interesting going on anyway, if you listen to the examples. But I'll
repost new ones at the right sample rate.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > > http://www.mikebattagliamusic.com/music/567sines.wav
> >
> > ???
>
> Hahaha, oh man, I'm laughing so hard right now. I didn't realize
> what Gene had meant before. Damn, I screwed this one up bad, lol.
>
> I'm actually going to keep these up though, because there is
> something interesting going on anyway, if you listen to the
> examples. But I'll repost new ones at the right sample rate.

Whew. -Carl

> Whew. -Carl

Also, can you try some random phases? -Carl

http://www.mikebattagliamusic.com/music/567sinesright.wav
http://www.mikebattagliamusic.com/music/567sinesright0-45-90.wav
http://www.mikebattagliamusic.com/music/567sinesright0-90-180.wav

Random phases will be coming up next

-Mike

On Sun, Jan 9, 2011 at 6:12 PM, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Jan 9, 2011 at 6:10 PM, Carl Lumma <carl@...> wrote:
>>
>> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>> >
>> > OK, I'm not sure how to make it flat, but here are some sines at
>> > different phases:
>> >
>> > http://www.mikebattagliamusic.com/music/567sines.wav
>>
>> ???
>
> Hahaha, oh man, I'm laughing so hard right now. I didn't realize what
> Gene had meant before. Damn, I screwed this one up bad, lol.
>
> I'm actually going to keep these up though, because there is something
> interesting going on anyway, if you listen to the examples. But I'll
> repost new ones at the right sample rate.
>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > I am not sure if that is the VF or a difference tone, but
> > yes, I hear it buzzing. I also hear the tones themselves
> > buzzing. It seems the amplitude variation in the original
> > waveform is projected onto the pitches we hear. -C.
>
> I think we might be hearing, literally, the impulse response
> of the VF "filterbank" in the brain.

Neat idea, but the VF filterbank (as you call it) is thought to
be located above the point where signals from the two ears are
combined. See here:
http://www.lloydwatts.com/images/Watts-AuditoryModels_final.pdf

Since periodicity buzz is not heard under dichotic conditions,
it seems likely that it occurs prior to VF processing.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> http://www.mikebattagliamusic.com/music/567sinesright.wav
> http://www.mikebattagliamusic.com/music/567sinesright0-45-90.wav
> http://www.mikebattagliamusic.com/music/567sinesright0-90-180.wav
>
> Random phases will be coming up next
>
> -Mike

They all seem to buzz about the same, though I do not hear
excessive amounts of buzz. Can you try 4:6:7 and/or a lower
fundamental frequency? -Carl

On Sun, Jan 9, 2011 at 8:49 PM, Carl Lumma <carl@...> wrote:
>
> > I think we might be hearing, literally, the impulse response
> > of the VF "filterbank" in the brain.
>
> Neat idea, but the VF filterbank (as you call it) is thought to
> be located above the point where signals from the two ears are
> combined. See here:
> http://www.lloydwatts.com/images/Watts-AuditoryModels_final.pdf
>
> Since periodicity buzz is not heard under dichotic conditions,
> it seems likely that it occurs prior to VF processing.

Very good point. The multiresolution analysis model will likely handle
this better than the idea I mentioned above.

-Mike