How to tune 7-tet by way of Just Intonation

Hello All,

I discovered 'rediscovered?' by accident the other day in some Just
Intonation
number crunching a curious fact-one can generate a near perfect 7-tet
tuning by
way of the 3rd, 5th, and 7th harmonics together. The octave arrived
at after 7
cycles of this generator interval will only be 1 cent off (good
enough for Swedish
polkas) !!!!
Here are 'les numbres':

3/2 * 5/4 * 7/4 = 105/64

multiplying 105/64 by itself.....including octave reductions-

0. 1.0000

1. 1.64062 (105/64)

2. 1.34583 (105/64)^2

3. 1.10400 etc.

4. 1.81125 etc.

5. 1.48579

6. 1.21881

7. 1.99961 (1 cent error!)

In this way, one could quite laboriously tune 7-tet by unaided
ear!!!!

Start with a tone, tune up a beatless fifth, from that a beatless
third, and from that
a beatless harmonic 7th, and you have a 5/7 of an octave. Continue
from there in
similar fashion to fill in the gaps.....

Is this published anywhere?

Best,
Aaron K. Johnson

hi aaron (there seem to be a lot of aarons around tuning lately,
aaron wolf, aaron hunt, and now you) . . .

this is interesting. we knew that one could tune 9-equal very
accurately and quickly through successive 7:6s, and (thanks to
kirnberger) 12-equal very accurately though laboriously through
successive schisma (32805:32768)-diminished 3:2s, etc., but this is
probably a new one.

the 8th note in your chain will differ from the 1st by the ratio
140737488355328:140710042265625, not even a cent as you say but only
1/3 of a cent. we might call this ratio the "johnston comma" . . .

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
> Hello All,
>
> I discovered 'rediscovered?' by accident the other day in some Just
> Intonation
> number crunching a curious fact-one can generate a near perfect 7-
tet
> tuning by
> way of the 3rd, 5th, and 7th harmonics together. The octave arrived
> at after 7
> cycles of this generator interval will only be 1 cent off (good
> enough for Swedish
> polkas) !!!!
> Here are 'les numbres':
>
> 3/2 * 5/4 * 7/4 = 105/64
>
> multiplying 105/64 by itself.....including octave reductions-
>
> 0. 1.0000
>
> 1. 1.64062 (105/64)
>
> 2. 1.34583 (105/64)^2
>
> 3. 1.10400 etc.
>
> 4. 1.81125 etc.
>
> 5. 1.48579
>
> 6. 1.21881
>
> 7. 1.99961 (1 cent error!)
>
> In this way, one could quite laboriously tune 7-tet by unaided
> ear!!!!
>
> Start with a tone, tune up a beatless fifth, from that a beatless
> third, and from that
> a beatless harmonic 7th, and you have a 5/7 of an octave. Continue
> from there in
> similar fashion to fill in the gaps.....
>
> Is this published anywhere?
>
> Best,
> Aaron K. Johnson

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> hi aaron (there seem to be a lot of aarons around tuning lately,
> aaron wolf, aaron hunt, and now you) . . .
>
> this is interesting. we knew that one could tune 9-equal very
> accurately and quickly through successive 7:6s, and (thanks to
> kirnberger) 12-equal very accurately though laboriously through
> successive schisma (32805:32768)-diminished 3:2s, etc., but this is
> probably a new one.
>
> the 8th note in your chain will differ from the 1st by the ratio
> 140737488355328:140710042265625, not even a cent as you say but only
> 1/3 of a cent. we might call this ratio the "johnston comma" . . .

Paul,

I'm excited by your reply!!! I'm glad to know that I discovered it! I feel like one of
those astronomers who discovers a comet.....

Just make sure you call it 'Johnson's comma' not 'Johnston's'...no 't' in my last
name....Ben Johnston already has made a name for himself!

Best to all,
Aaron Krister Johnson

>
> --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
> > Hello All,
> >
> > I discovered 'rediscovered?' by accident the other day in some Just
> > Intonation
> > number crunching a curious fact-one can generate a near perfect 7-
> tet
> > tuning by
> > way of the 3rd, 5th, and 7th harmonics together. The octave arrived
> > at after 7
> > cycles of this generator interval will only be 1 cent off (good
> > enough for Swedish
> > polkas) !!!!
> > Here are 'les numbres':
> >
> > 3/2 * 5/4 * 7/4 = 105/64
> >
> > multiplying 105/64 by itself.....including octave reductions-
> >
> > 0. 1.0000
> >
> > 1. 1.64062 (105/64)
> >
> > 2. 1.34583 (105/64)^2
> >
> > 3. 1.10400 etc.
> >
> > 4. 1.81125 etc.
> >
> > 5. 1.48579
> >
> > 6. 1.21881
> >
> > 7. 1.99961 (1 cent error!)
> >
> > In this way, one could quite laboriously tune 7-tet by unaided
> > ear!!!!
> >
> > Start with a tone, tune up a beatless fifth, from that a beatless
> > third, and from that
> > a beatless harmonic 7th, and you have a 5/7 of an octave. Continue
> > from there in
> > similar fashion to fill in the gaps.....
> >
> > Is this published anywhere?
> >
> > Best,
> > Aaron K. Johnson

I did a search for other examples of this phenomenon, and found the
following 53 cases where 2^(p/q), with p,q < 100, is within 1/20th of
a cent of a 7-limit interval which is a semiconvergent for the
continued fraction for 2^(p/q). I list first p/q, then the 7-limit
approximation A to 2^(p/q), and then A 2^(-p/q) in cents.

Pondering this list, I note that 2^(1/87) is almost exactly the small
septimal comma of size 126/125, and 2^(1/44) is almost exactly the
large septimal comma of size 64/63. Other intervals of note making an
appearance are 35/24, 32/25, 25/16, 27/20, and 21/20. All of these
lead to corresponding 7-limit commas; for instance (126/125)^87 / 2 is
a comma of size about 1/7 of a cent.

34/83 3645/2744 -.000265
1/87 126/125 .001664
86/87 125/63 -.001663
4/97 1029/1000 .007043
31/54 1715/1152 -.007458
23/54 2304/1715 .007456
1/44 64/63 -.008635
43/44 63/32 .008635
45/82 256/175 .010081
37/82 175/128 -.010081
4/85 405/392 .011315
33/94 125/98 .012733
61/94 196/125 -.012733
55/84 540/343 -.013289
29/84 343/270 .013289
45/68 405/256 .016071
37/62 245/162 .016492
25/62 324/245 -.016491
36/79 48/35 -.020063
43/79 35/24 .020061
45/49 189/100 .022665
4/49 200/189 -.022665
51/65 441/256 .023354
14/65 512/441 -.023353
37/51 625/378 -.024289
26/73 32/25 -.024688
47/73 25/16 .024687
40/43 343/180 -.025066
31/69 512/375 -.026578
38/69 375/256 .026578
17/33 343/240 .027187
16/33 480/343 -.027186
58/67 625/343 -.028835
9/67 686/625 .028833
32/91 245/192 .032504
36/77 112/81 -.033058
41/77 81/56 .033058
42/97 27/20 -.036340
55/97 40/27 .036340
34/57 189/125 -.039706
23/57 250/189 .039705
66/71 40/21 .039849
5/71 21/20 -.039849
11/90 160/147 .040233
79/90 147/80 -.040233
47/82 729/490 -.040400
23/32 288/175 -.043332
9/32 175/144 .043332
59/97 343/225 .043382
2/7 128/105 .048237
5/7 105/64 -.048236
4/19 81/70 .048805
15/19 140/81 -.048805

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I did a search for other examples of this phenomenon, and found the
> following 53 cases where 2^(p/q), with p,q < 100,

q being the et. the last ones on your list, the first two of course
being what aaron found, are most interesting.

> 2/7 128/105 .048237
> 5/7 105/64 -.048236
> 4/19 81/70 .048805
> 15/19 140/81 -.048805

so you can tune 19-equal "by ear" too! awesome . . .

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > I did a search for other examples of this phenomenon, and found
the
> > following 53 cases where 2^(p/q), with p,q < 100,
>
> q being the et. the last ones on your list, the first two of course
> being what aaron found, are most interesting.
>
> > 2/7 128/105 .048237
> > 5/7 105/64 -.048236
> > 4/19 81/70 .048805
> > 15/19 140/81 -.048805
>
> so you can tune 19-equal "by ear" too! awesome . . .

Since 64/63 is one step of 44-et, you can tune 22-equal also.

>> I did a search for other examples of this phenomenon, and found the
>> following 53 cases where 2^(p/q), with p,q < 100,
>
>q being the et. the last ones on your list, the first two of course
>being what aaron found, are most interesting.
>
>> 2/7 128/105 .048237
>> 5/7 105/64 -.048236
>> 4/19 81/70 .048805
>> 15/19 140/81 -.048805
>
>so you can tune 19-equal "by ear" too! awesome . . .

Aren't these too laborious to really be practical? Meanwhile,
can't 19-et be tuned as a chain of minor thirds?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I did a search for other examples of this phenomenon, and found
the
> >> following 53 cases where 2^(p/q), with p,q < 100,
> >
> >q being the et. the last ones on your list, the first two of
course
> >being what aaron found, are most interesting.
> >
> >> 2/7 128/105 .048237
> >> 5/7 105/64 -.048236
> >> 4/19 81/70 .048805
> >> 15/19 140/81 -.048805
> >
> >so you can tune 19-equal "by ear" too! awesome . . .
>
> Aren't these too laborious to really be practical?

the 7-equal ones involve tuning 21 just intervals to get the whole
et, not too bad. the 19-equal ones require 76 . . . still less than
kirnberger's 96 for 12-equal . . .

> Meanwhile,
> can't 19-et be tuned as a chain of minor thirds?

apparently not to this degree of accuracy . . .

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> the 8th note in your chain will differ from the 1st by the ratio
> 140737488355328:140710042265625, not even a cent as you say but only
> 1/3 of a cent. we might call this ratio the "johnson comma" . . .

Or more specifically "Johnson's schismina".

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > the 8th note in your chain will differ from the 1st by the ratio
> > 140737488355328:140710042265625, not even a cent as you say but
only
> > 1/3 of a cent. we might call this ratio the "johnson comma" . . .
>
> Or more specifically "Johnson's schismina".

Johnschisma?

In a message dated 8/20/03 11:59:43 PM Eastern Daylight Time,
perlich@aya.yale.edu writes:

> the 7-equal ones involve tuning 21 just intervals to get the whole
> et, not too bad. the 19-equal ones require 76 . . . still less than
> kirnberger's 96 for 12-equal . . .
>
>

Of course, Kirnberger was always against 12-equal. He declared a monochord
was necessary. Even so, he came up with a technique though laborious that did
give rather accurate ET. The fact that 96 notes would be used fits well with
Kirnberger's aims after all.

Johnny

Hello All,

I know Gene Ward Smith did an excellent inquiry into further examples of the
phenomena I stumbled across, and he inspired me to do a little python script which
would do the same thing, albeit layed out differently. I wanted to know "What are
the n-tets that can be tuned by ear to within 3 cent accuracy, within the 7-limit, and
each prime being used no more than 4 times, limiting the worst case target interval
to being arrived at by 12 just intervals (which is bordering on really impractical)..."

Below is my chart, which is sorted by cents error (last column), least to greatest.
Note my beloved discovery (the very elegant and accurate 7-tet approximation) is
near the very top, but I found an somewhat impractical 21-tet and 3-tet by ear
which ousted it for accuracy, if not sheer elegance. I'm sure that if you allow the
parameters to grow, you can find very arbitrarily close approximations to any n-tet,
which would make 21-tet run for its money. But the beauty of the 7-tet discovery
was its accuracy AND simplicity combined! (It sort of makes me want to write in
7-tet for the rest of my life....I have a 'causa natura' justification for it now---hehe!!!)

Note the interesting 1-tet (octave) approximation using 7-limit overtones!

BTW, I like "Johnson's comma" the best ;) The others started sounding like rare
African diseases......

And yes, Carl, 19-tet is fairly well approximated by a chain of 6:5's.....still the most
expediant way.

Anyone who would like this script for further parameter tinkering, let me know by
emailing me at <akj@rcn.com>.....

cheers,
Aaron.

(in cents)
21-tet: 3^4 5^0 7^3 error: 0.25218
3-tet: 3^2 5^-2 7^1 error: 0.32544
7-tet: 3^1 5^1 7^1 error: 0.33765 (my claim to fame in some obsure tome)
22-tet: 3^4 5^0 7^2 error: 0.37996
25-tet: 3^4 5^-1 7^3 error: 0.39977
3-tet: 3^4 5^-4 7^2 error: 0.65088
6-tet: 3^2 5^-2 7^1 error: 0.65088
7-tet: 3^2 5^2 7^2 error: 0.67530
14-tet: 3^1 5^1 7^1 error: 0.67530
1-tet: 3^1 5^2 7^-4 error: 0.72120 (interesting octave approximation!!)
9-tet: 3^3 5^-2 7^0 error: 0.86183
25-tet: 3^1 5^4 7^1 error: 0.89407
11-tet: 3^3 5^-1 7^-2 error: 0.89423
33-tet: 3^1 5^1 7^-3 error: 0.89715
19-tet: 3^4 5^-1 7^-1 error: 0.92728
9-tet: 3^2 5^-2 7^1 error: 0.97632
17-tet: 3^3 5^4 7^-1 error: 1.00282
7-tet: 3^3 5^3 7^3 error: 1.01295
21-tet: 3^1 5^1 7^1 error: 1.01295
20-tet: 3^3 5^4 7^1 error: 1.08471
21-tet: 3^3 5^-1 7^2 error: 1.26514
6-tet: 3^4 5^-4 7^2 error: 1.30177
12-tet: 3^2 5^-2 7^1 error: 1.30177
7-tet: 3^4 5^4 7^4 error: 1.35061
14-tet: 3^2 5^2 7^2 error: 1.35061
28-tet: 3^1 5^1 7^1 error: 1.35061
16-tet: 3^4 5^-4 7^-2 error: 1.38664
32-tet: 3^2 5^-2 7^-1 error: 1.38664
22-tet: 3^1 5^1 7^4 error: 1.40851
17-tet: 3^1 5^-2 7^0 error: 1.43126
2-tet: 3^1 5^2 7^-4 error: 1.44239
32-tet: 3^2 5^3 7^-4 error: 1.52056
30-tet: 3^4 5^-3 7^-1 error: 1.58866
15-tet: 3^2 5^-2 7^1 error: 1.62721
10-tet: 3^0 5^-1 7^3 error: 1.64006
10-tet: 3^0 5^1 7^-3 error: 1.64006
15-tet: 3^4 5^-2 7^-4 error: 1.66575
30-tet: 3^2 5^-1 7^-2 error: 1.66575
13-tet: 3^3 5^1 7^0 error: 1.67669
35-tet: 3^1 5^1 7^1 error: 1.68826
18-tet: 3^3 5^-2 7^0 error: 1.72365
8-tet: 3^1 5^-4 7^-3 error: 1.77941
22-tet: 3^3 5^-1 7^-2 error: 1.78847
3-tet: 3^3 5^0 7^-3 error: 1.83815
9-tet: 3^1 5^0 7^-1 error: 1.83815
38-tet: 3^4 5^-1 7^-1 error: 1.85456
31-tet: 3^1 5^1 7^-4 error: 1.91775
23-tet: 3^2 5^-4 7^3 error: 1.94409
9-tet: 3^4 5^-4 7^2 error: 1.95265
18-tet: 3^2 5^-2 7^1 error: 1.95265
5-tet: 3^1 5^0 7^2 error: 1.96593
34-tet: 3^3 5^4 7^-1 error: 2.00565
4-tet: 3^2 5^3 7^2 error: 2.01183
14-tet: 3^3 5^3 7^3 error: 2.02591
21-tet: 3^2 5^2 7^2 error: 2.02591
37-tet: 3^1 5^-4 7^0 error: 2.09462
3-tet: 3^1 5^2 7^-4 error: 2.16359
20-tet: 3^3 5^3 7^4 error: 2.19540
21-tet: 3^2 5^-2 7^1 error: 2.27809
17-tet: 3^4 5^2 7^-1 error: 2.43408
27-tet: 3^3 5^-2 7^0 error: 2.58548
12-tet: 3^4 5^-4 7^2 error: 2.60353
24-tet: 3^2 5^-2 7^1 error: 2.60353
33-tet: 3^3 5^-1 7^-2 error: 2.68270
2-tet: 3^3 5^1 7^3 error: 2.68713
9-tet: 3^4 5^-2 7^-1 error: 2.69998
14-tet: 3^4 5^4 7^4 error: 2.70121
28-tet: 3^2 5^2 7^2 error: 2.70121
24-tet: 3^1 5^2 7^4 error: 2.73469
32-tet: 3^4 5^-4 7^-2 error: 2.77328
9-tet: 3^1 5^-2 7^2 error: 2.81447
19-tet: 3^1 5^-1 7^0 error: 2.81555
17-tet: 3^2 5^-4 7^0 error: 2.86251
34-tet: 3^1 5^-2 7^0 error: 2.86251
4-tet: 3^1 5^2 7^-4 error: 2.88479
27-tet: 3^2 5^-2 7^1 error: 2.92897

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> 1-tet: 3^1 5^2 7^-4 error: 0.72120 (interesting octave
approximation!!) 2401/2400 is a well-regarded comma in these parts.

If we sort out the 5-limit ones, we get

> 9-tet: 3^3 5^-2 7^0 error: 0.86183

Ennealimma

> 17-tet: 3^1 5^-2 7^0 error: 1.43126
> 13-tet: 3^3 5^1 7^0 error: 1.67669
> 18-tet: 3^3 5^-2 7^0 error: 1.72365
> 37-tet: 3^1 5^-4 7^0 error: 2.09462
> 27-tet: 3^3 5^-2 7^0 error: 2.58548
> 19-tet: 3^1 5^-1 7^0 error: 2.81555
> 17-tet: 3^2 5^-4 7^0 error: 2.86251

This is a square

> 34-tet: 3^1 5^-2 7^0 error: 2.86251

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> Note the interesting 1-tet (octave) approximation using 7-limit
overtones!

> 1-tet: 3^1 5^2 7^-4 error: 0.72120 (interesting octave
>approximation!!)

this is also known as 2400:2401 or the "breedsma", an essential
interval in defining many of the more powerful 7-limit temperaments.
it only requires tuning four 7-limit consonant intervals to tune it
up: 6:7 * 5:7 * 10:7 * 8:7 = 2400:2401, and it's easy (as graham
breed did) to create a short chord cycle in JI that drifts by this
tiny interval every time it goes around. in temperaments that
distribute out this comma, the drift is eliminated, and each of the
consonant intervals suffers a mistuning that need be no larger than a
quarter breedsma, or 0.18 cents.

for even more accuracy, use the "ragisma", 4374:4375 or 3^7 5^-4 7^-
1, important for some temperaments of high complexity and stunning
accuracy. it requires a chain of seven 7-limit consonances to tune
this one, for example 3:1 * 3:1 * 3:5 * 3:5 * 3:5 * 6:5 * 3:7 =
4374:4375.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
>
> > Note the interesting 1-tet (octave) approximation using 7-limit
> overtones!
>
> > 1-tet: 3^1 5^2 7^-4 error: 0.72120 (interesting octave
> >approximation!!)
>
> this is also known as 2400:2401 or the "breedsma", an essential
> interval in defining many of the more powerful 7-limit temperaments.
> it only requires tuning four 7-limit consonant intervals to tune it
> up: 6:7 * 5:7 * 10:7 * 8:7 = 2400:2401, and it's easy (as graham
> breed did) to create a short chord cycle in JI that drifts by this
> tiny interval every time it goes around. in temperaments that
> distribute out this comma, the drift is eliminated, and each of the
> consonant intervals suffers a mistuning that need be no larger than a
> quarter breedsma, or 0.18 cents.
>
> for even more accuracy, use the "ragisma", 4374:4375 or 3^7 5^-4 7^-
> 1, important for some temperaments of high complexity and stunning
> accuracy. it requires a chain of seven 7-limit consonances to tune
> this one, for example 3:1 * 3:1 * 3:5 * 3:5 * 3:5 * 6:5 * 3:7 =
> 4374:4375.

This is interesting stuff, Paul, thanks for pointing it out to me...

BTW...I revised my little Python script so as to only show temperaments which are
accesible by a combination of no more than 4 JI ratios per note, to within 3 cents
accuracy....results below. BTW, do you know of much written in 34-tet or 44-tet?
I've heard of Nel Haverstick, etc. using 34-tet, but haven't seen much about 44-tet.
(I know 22-tet is popular). I'd also like to discuss some musical and structural
possibilities inherent in these temperaments!!!

7-tet: 3^1 5^1 7^1 error: 0.33765
44-tet: 3^2 5^0 7^1 error: 0.37996
14-tet: 3^1 5^1 7^1 error: 0.67530
21-tet: 3^1 5^1 7^1 error: 1.01295
28-tet: 3^1 5^1 7^1 error: 1.35061
17-tet: 3^1 5^-2 7^0 error: 1.43126
10-tet: 3^0 5^-1 7^3 error: 1.64006
10-tet: 3^0 5^1 7^-3 error: 1.64006
13-tet: 3^3 5^1 7^0 error: 1.67669
35-tet: 3^1 5^1 7^1 error: 1.68826
9-tet: 3^1 5^0 7^-1 error: 1.83815
5-tet: 3^1 5^0 7^2 error: 1.96593
42-tet: 3^1 5^1 7^1 error: 2.02591
49-tet: 3^1 5^1 7^1 error: 2.36356
19-tet: 3^1 5^-1 7^0 error: 2.81555
34-tet: 3^1 5^-2 7^0 error: 2.86251

-Aaron.

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> I'd also like to discuss some musical and structural
> possibilities inherent in these temperaments!!!

well, i guess you've come to the right place! here's a
"bingo card" for 28-equal, showing the 5-limit consonances that can
be formed with each pitch as its immediate neighbors:

/tuning/files/perlich/28.gif

28-equal is one i've discussed early and often -- it's got nice
octatonic (diminished) scales, and you can modulate them around a
circle of 7-equal fifths if you want. 28-equal is not only a member
of the "diminished" family, it's also a member of the "negri" family
(which involves 9- and 10-note scales, all of whose steps except one
are equal), as you can see on the first graph here:

http://sonic-arts.org/dict/eqtemp.htm

the data for this graph is here:

/tuning/database?
method=reportRows&tbl=10&sortBy=4

the ninth row of this table, which begins "major diesis", corresponds
to the diminished, 'octatonic' system of constructing scales and
chords. as you can see, the optimal period and generator for this
scale are [300, 94.1] -- meaning the scale repeats itself every 300
cents, 4 times per octave, and that the generator for new pitches is
about 94.1 cents. in 28-equal this would be approximated by 85.7
cents, while in 12-equal it's 100 cents.

as you can also see in that row, the the first three primes 2, 3, and
5 are approximated in this system as [4,0] [6,1] and [9,1] in terms
of the generator. this means that the prime 2 is approximated by

4*300 + 0*94.1 cents = 1200 cents, or in 28-equal,
4*300 + 0*85.7 cents = 1200 cents;

the prime 3 is approximated by

6*300 + 1*94.1 cents = 1894.1 cents, or in 28-equal,
6*300 + 1*85.7 cents = 1885.7 cents;

the prime 5 is approximated by

9*300 + 1*94.1 cents = 2794.1 cents, or in 28-equal,
9*300 + 1*85.7 cents = 2785.7 cents.

the approximations other consonant intervals like 6:5 can be found by
vector addition: the represenation of 6:5 is the representation of 2
plus the represenation of 3 minus the representation of 5, since 6/5
= 2*3/5. the represenation of 6:5 is therefore

[4,0] + [6,1] - [9,1] = [1,0], so the approximation to 6:5 is always
300 cents in the diminished, 'octatonic' system.

the 8-tone-per octave scale created by applying the period an
infinite number of times and the generator once, is called the
octatonic or diminished scale and is probably familiar to you from
its 12-equal incarnation. trace it out on the 28-equal bingo card to
see that it contains 4 major triads and 4 minor triads per octave.
notice that these chords wouldn't all be possible if the bingo card
didn't repeat itself at the vector shown as "major diesis" on the
second graph back at

http://sonic-arts.org/dict/eqtemp.htm --

this is the reason the relevant row in the table is designated "major
diesis" (which is the name of the interval formed by the vector in
question on the just intonation bingo card, ratio 648:625).

the thirteenth row of the table corresponds to the "negri" system.
negri actually used the system within a 19-equal tuning, but it's
perfectly applicable (though with not as much harmonic purity) in 28-
equal. see if you can run through the above analyses with the negri
system, which the table associates with a period and generator of
[1200, 126.238272] and, in terms of which, the primes 2, 3, and 5 are
associated with [1,0], [2,-4], and [2,3] respectively. trace out 9-
and 10-note scales on the 28-equal bingo card, and convince yourself
that all the chords wouldn't be simultaneously possible if not for
the 28-equal bingo card repeating itself at the vector shown
as "(negri)" on the second graph back at

http://sonic-arts.org/dict/eqtemp.htm --

a vector which corresponds to the nameless 16875:16384 interval in
just intonation.

all these analyses could be carried out in the 7-limit as well, but
all the charts and diagrams would need an additional dimension, and
it's hard to show complicated 3-dimensional structures in 2
dimensions. it's been done on a case-by-case basis for some simple
cases, though.

hopefully this makes sense, you can have fun looking at other equal
temperaments and systems, and hopefully make some neat music from it
all!

questions?

Sweet post Paul, worth quoting in full.

-Carl

>> I'd also like to discuss some musical and structural
>> possibilities inherent in these temperaments!!!
>
>well, i guess you've come to the right place! here's a
>"bingo card" for 28-equal, showing the 5-limit consonances that can
>be formed with each pitch as its immediate neighbors:
>
>/tuning/files/perlich/28.gif
>
>28-equal is one i've discussed early and often -- it's got nice
>octatonic (diminished) scales, and you can modulate them around a
>circle of 7-equal fifths if you want. 28-equal is not only a member
>of the "diminished" family, it's also a member of the "negri" family
>(which involves 9- and 10-note scales, all of whose steps except one
>are equal), as you can see on the first graph here:
>
>http://sonic-arts.org/dict/eqtemp.htm
>
>the data for this graph is here:
>
>/tuning/database?
>method=reportRows&tbl=10&sortBy=4
>
>the ninth row of this table, which begins "major diesis", corresponds
>to the diminished, 'octatonic' system of constructing scales and
>chords. as you can see, the optimal period and generator for this
>scale are [300, 94.1] -- meaning the scale repeats itself every 300
>cents, 4 times per octave, and that the generator for new pitches is
>about 94.1 cents. in 28-equal this would be approximated by 85.7
>cents, while in 12-equal it's 100 cents.
>
>as you can also see in that row, the the first three primes 2, 3, and
>5 are approximated in this system as [4,0] [6,1] and [9,1] in terms
>of the generator. this means that the prime 2 is approximated by
>
>4*300 + 0*94.1 cents = 1200 cents, or in 28-equal,
>4*300 + 0*85.7 cents = 1200 cents;
>
>the prime 3 is approximated by
>
>6*300 + 1*94.1 cents = 1894.1 cents, or in 28-equal,
>6*300 + 1*85.7 cents = 1885.7 cents;
>
>the prime 5 is approximated by
>
>9*300 + 1*94.1 cents = 2794.1 cents, or in 28-equal,
>9*300 + 1*85.7 cents = 2785.7 cents.
>
>the approximations other consonant intervals like 6:5 can be found by
>vector addition: the represenation of 6:5 is the representation of 2
>plus the represenation of 3 minus the representation of 5, since 6/5
>= 2*3/5. the represenation of 6:5 is therefore
>
>[4,0] + [6,1] - [9,1] = [1,0], so the approximation to 6:5 is always
>300 cents in the diminished, 'octatonic' system.
>
>the 8-tone-per octave scale created by applying the period an
>infinite number of times and the generator once, is called the
>octatonic or diminished scale and is probably familiar to you from
>its 12-equal incarnation. trace it out on the 28-equal bingo card to
>see that it contains 4 major triads and 4 minor triads per octave.
>notice that these chords wouldn't all be possible if the bingo card
>didn't repeat itself at the vector shown as "major diesis" on the
>second graph back at
>
>http://sonic-arts.org/dict/eqtemp.htm --
>
>this is the reason the relevant row in the table is designated "major
>diesis" (which is the name of the interval formed by the vector in
>question on the just intonation bingo card, ratio 648:625).
>
>the thirteenth row of the table corresponds to the "negri" system.
>negri actually used the system within a 19-equal tuning, but it's
>perfectly applicable (though with not as much harmonic purity) in 28-
>equal. see if you can run through the above analyses with the negri
>system, which the table associates with a period and generator of
>[1200, 126.238272] and, in terms of which, the primes 2, 3, and 5 are
>associated with [1,0], [2,-4], and [2,3] respectively. trace out 9-
>and 10-note scales on the 28-equal bingo card, and convince yourself
>that all the chords wouldn't be simultaneously possible if not for
>the 28-equal bingo card repeating itself at the vector shown
>as "(negri)" on the second graph back at
>
>http://sonic-arts.org/dict/eqtemp.htm --
>
>a vector which corresponds to the nameless 16875:16384 interval in
>just intonation.
>
>all these analyses could be carried out in the 7-limit as well, but
>all the charts and diagrams would need an additional dimension, and
>it's hard to show complicated 3-dimensional structures in 2
>dimensions. it's been done on a case-by-case basis for some simple
>cases, though.
>
>hopefully this makes sense, you can have fun looking at other equal
>temperaments and systems, and hopefully make some neat music from it
>all!
>
>questions?

Paul,

I've created a web page for your post. It's available
here, with some others...

http://lumma.org/tuning/erlich/

...check it out!

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Paul,
>
> I've created a web page for your post. It's available
> here, with some others...
>
> http://lumma.org/tuning/erlich/
>
> ...check it out!
>
> -Carl

thanks, carl! most of the links on john's page aren't valid at this
point, but it's nice that yours is up! note that

http://lumma.org/tuning/erlich/1999.10.04.DecatonicScalesAsPBs.txt

is already included in http://www.sonic-arts.org/dict/pblock.htm

but that _the forms of tonality_ does a better job giving ji versions
of the symmetrical decatonics (they have a larger number of consonant
chords there), and note that the pentachordal decatonic scale is a
periodicity block, it just isn't of the fokker (parallelepiped) type.

>but that _the forms of tonality_ does a better job giving ji versions
>of the symmetrical decatonics (they have a larger number of consonant
>chords there), and note that the pentachordal decatonic scale is a
>periodicity block, it just isn't of the fokker (parallelepiped) type.

Oh, is that the one where you say you don't think the pentachordal
decs are pbs, but you could be wrong?

Good thing you were wrong. :)

We need to get tFoT up on tha web. Write me off-list.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Paul,
>
> I've created a web page for your post. It's available
> here, with some others...
>
> http://lumma.org/tuning/erlich/
>
> ...check it out!

Why don't you make this stuff more accessible by having a link from
your home page?

>Why don't you make this stuff more accessible by having a link
>from your home page?

The short answer: One day, I will.

The medium answer: This page has been up for over a year.
If people would link to it, it'd be findable with Google.

The long answer: Some use links to navigate the web, I use
urls. A site with a proper path scheme is often easier to
navigate this way than by following links. It is, in fact,
one of the things Yahoo got right.

The final answer: The only way to make it really easy is
to buy a domain, or subnet domain. If you want to pony up
for that I can provide hosting.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Why don't you make this stuff more accessible by having a link
> >from your home page?
>
> The short answer: One day, I will.
>
> The medium answer: This page has been up for over a year.
> If people would link to it, it'd be findable with Google.

I would like to link to your theory page, but you don't have one,
unless your Paul page is it.

> The final answer: The only way to make it really easy is
> to buy a domain, or subnet domain. If you want to pony up
> for that I can provide hosting.

All you need to do is stick in a link, surely? This makes no sense to
me at all.

>> The medium answer: This page has been up for over a year.
>> If people would link to it, it'd be findable with Google.
>
>I would like to link to your theory page, but you don't have one,
>unless your Paul page is it.

Yes, please link to that. There's also tctmo at...

http://lumma.org/tuning/tctmo/

>> The final answer: The only way to make it really easy is
>> to buy a domain, or subnet domain. If you want to pony up
>> for that I can provide hosting.
>
>All you need to do is stick in a link, surely? This makes no sense
>to me at all.

"You are entering a world of pain" ... my unfinished website.

:)

-Carl

Carl Lumma wrote:

>>Why don't you make this stuff more accessible by having a link
>> >>
>>from your home page?
>
>The short answer: One day, I will.
>
>The medium answer: This page has been up for over a year.
>If people would link to it, it'd be findable with Google.
>

It's too much trouble to submit it yourself?

http://www.google.com/addurl.html

--
* David Beardsley
* microtonal guitar
* http://biink.com/db

>>The medium answer: This page has been up for over a year.
>>If people would link to it, it'd be findable with Google.
>>
>
>It's too much trouble to submit it yourself?
>
>http://www.google.com/addurl.html

Say, I didn't know they did that.

But that still won't help the PageRank in the same way as
other web sites linking to it.

-Carl

this is just a test ...
i've been having some email problems for
a few days.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_46408.html#46554

>> >
> >the 8-tone-per octave scale created by applying the period an
> >infinite number of times and the generator once, is called the
> >octatonic or diminished scale and is probably familiar to you from
> >its 12-equal incarnation. trace it out on the 28-equal bingo card
to
> >see that it contains 4 major triads and 4 minor triads per octave.

***Hi Paul,

I'm not "getting" this... What exactly is the "period" and how is it
applied an "infinite number of times...?" I thought maybe it was
where a scale could start over again...

Help pls!

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> /tuning/topicId_46408.html#46554
>
> >> >
> > >the 8-tone-per octave scale created by applying the period an
> > >infinite number of times and the generator once, is called the
> > >octatonic or diminished scale and is probably familiar to you
from
> > >its 12-equal incarnation. trace it out on the 28-equal bingo
card
> to
> > >see that it contains 4 major triads and 4 minor triads per
octave.
>
> ***Hi Paul,
>
> I'm not "getting" this... What exactly is the "period" and how is
it
> applied an "infinite number of times...?"

well, not quite an infinite number of times -- in practice, only from
the lowest registers of human hearing to the highest.

> I thought maybe it was
> where a scale could start over again...

yes, exactly. for example, the diatonic scale has a period of 1
octave, and this octave repetition is applied as many times as
audible. the diminished scale has a period of 300 cents or 1/4
octave, also repeated as many times as audible.

> Help pls!

i'm glad you're actually trying to read and understand that post! let
me know if you need any more help. perhaps the "infinity" part was
the only thing that confused you (i hope so), perhaps i should have
just kept everything within the frame of 1 octave to keep it
simpler . . .

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> yes, exactly. for example, the diatonic scale has a period of 1
> octave, and this octave repetition is applied as many times as
> audible. the diminished scale has a period of 300 cents or 1/4
> octave, also repeated as many times as audible.

And in any event, never more than the Plack frequency. :)

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

/tuning/topicId_46408.html#46616
>
> > I thought maybe it was
> > where a scale could start over again...
>
> yes, exactly. for example, the diatonic scale has a period of 1
> octave, and this octave repetition is applied as many times as
> audible. the diminished scale has a period of 300 cents or 1/4
> octave, also repeated as many times as audible.
>

***Ok, that would make sense...

> > Help pls!
>
> i'm glad you're actually trying to read and understand that post!
let
> me know if you need any more help.

***Yes

perhaps the "infinity" part was
> the only thing that confused you (i hope so)

***no

, perhaps i should have
> just kept everything within the frame of 1 octave to keep it
> simpler . . .

***Please!

***Seriously, it seems you are trying to compare 28-equal with a
scale that is *not* an ET. (Yes, no....buzz...)

I think I could use a bit of review as to how this all relates to
the "honeycombs" that you've presented...

I'd like to say, Paul, that your posts, even the complex ones, always
seem like they're something that one *should* know, and you always
write them as clearly as possible... which makes a person feel it is
well worth the time to study and learn from them!

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> /tuning/topicId_46408.html#46616
> >
> > > I thought maybe it was
> > > where a scale could start over again...
> >
> > yes, exactly. for example, the diatonic scale has a period of 1
> > octave, and this octave repetition is applied as many times as
> > audible. the diminished scale has a period of 300 cents or 1/4
> > octave, also repeated as many times as audible.
> >
>
> ***Ok, that would make sense...
>
>
> > > Help pls!
> >
> > i'm glad you're actually trying to read and understand that post!
> let
> > me know if you need any more help.
>
> ***Yes
>
> perhaps the "infinity" part was
> > the only thing that confused you (i hope so)
>
> ***no
>
> , perhaps i should have
> > just kept everything within the frame of 1 octave to keep it
> > simpler . . .
>
> ***Please!
>
> ***Seriously, it seems you are trying to compare 28-equal with a
> scale that is *not* an ET. (Yes, no....buzz...)

umm . . . the 8-note scale i was referring to, an "octatonic"
or "diminished" scale, is in this context a subset of 28-equal. the
period of the diminished system in 28-equal is (as always) 300 cents
(7 steps of 28-equal), while the period is 85.7 cents (2 steps of 28-
equal). so applying the period yields the pattern of steps 7 7 7 7
within the octave, so the pattern of pitches is 0 7 14 21 (28), and
then applying the generator once yields the pattern 0 0+2 7 7+2 14
14+2 21 21+2 (28), or 0 2 7 9 14 16 21 23 (28).

> I think I could use a bit of review as to how this all relates to
> the "honeycombs" that you've presented...

hopefully, you can now map the scale on the "honeycomb" lattice, go
back to the context of my original post, and put the pieces together.
let me know if you need more hints!

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

/tuning/topicId_46408.html#46633

>
> umm . . . the 8-note scale i was referring to, an "octatonic"
> or "diminished" scale, is in this context a subset of 28-equal. the
> period of the diminished system in 28-equal is (as always) 300
cents
> (7 steps of 28-equal), while the period is 85.7 cents (2 steps of
28-
> equal). so applying the period yields the pattern of steps 7 7 7 7
> within the octave, so the pattern of pitches is 0 7 14 21 (28), and
> then applying the generator once yields the pattern 0 0+2 7 7+2 14
> 14+2 21 21+2 (28), or 0 2 7 9 14 16 21 23 (28).
>

***I think you mean here, Paul, that the *generator* is 85.7 cents,
yes? If so, I'm understanding this... It's a very clear explanation.
Thanks!

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> /tuning/topicId_46408.html#46633
>
> >
> > umm . . . the 8-note scale i was referring to, an "octatonic"
> > or "diminished" scale, is in this context a subset of 28-equal.
the
> > period of the diminished system in 28-equal is (as always) 300
> cents
> > (7 steps of 28-equal), while the period is 85.7 cents (2 steps of
> 28-
> > equal). so applying the period yields the pattern of steps 7 7 7
7
> > within the octave, so the pattern of pitches is 0 7 14 21 (28),
and
> > then applying the generator once yields the pattern 0 0+2 7 7+2
14
> > 14+2 21 21+2 (28), or 0 2 7 9 14 16 21 23 (28).
> >
>
> ***I think you mean here, Paul, that the *generator* is 85.7 cents,
> yes?

oh yes -- sorry!!

> If so, I'm understanding this... It's a very clear explanation.

whew! now, what happened to "ajmicro", who posted the original
question?

> Thanks!

you're welcome!

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> > --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> >
> > /tuning/topicId_46408.html#46633
> >
> > >
> > > umm . . . the 8-note scale i was referring to, an "octatonic"
> > > or "diminished" scale, is in this context a subset of 28-equal.
> the
> > > period of the diminished system in 28-equal is (as always) 300
> > cents
> > > (7 steps of 28-equal), while the period is 85.7 cents (2 steps of
> > 28-
> > > equal). so applying the period yields the pattern of steps 7 7 7
> 7
> > > within the octave, so the pattern of pitches is 0 7 14 21 (28),
> and
> > > then applying the generator once yields the pattern 0 0+2 7 7+2
> 14
> > > 14+2 21 21+2 (28), or 0 2 7 9 14 16 21 23 (28).
> > >
> >
> > ***I think you mean here, Paul, that the *generator* is 85.7 cents,
> > yes?
>
> oh yes -- sorry!!
>
> > If so, I'm understanding this... It's a very clear explanation.
>
> whew! now, what happened to "ajmicro", who posted the original
> question?

Here I am...it's getting late on Sept 11th, 2003...I need to go to bed-I'm seeing
cross-eyed, and to digest a new concept like "honeycombs", as fascinating as
they sound, would be unwise!

So I'll look foward to reading up on it tomorrow ....you guys are like mad scientist
gurus!

BTW, can anyone point to any interesting mp3s or audio files on the web in 7-tet?
I have a percussionist I play with, and I've been doing some improvs with hiim in
7-tet...I use very inharmonic FM moog-like sounds which I think work best for such
an inharmonic tuning...it's great!

-Aaron.

>
> BTW, can anyone point to any interesting mp3s or audio files on the
web in 7-tet?
> I have a percussionist I play with, and I've been doing some improvs
with hiim in
> 7-tet...I use very inharmonic FM moog-like sounds which I think work
best for such
> an inharmonic tuning...it's great!
>
> -Aaron.

Hi Aaron, I have one piece on the web in 7-tet,
its at:

http://eceserv0.ece.wisc.edu/~sethares/otherperson/all_mp3s.html

click on "Pagan's Revenge"

I have a couple of others in 7-tet that also use a similar sound set -
in this case sound with the partials of a bar (like a marimba or
glockenspeil) but some with more sustain as well...

--- In tuning@yahoogroups.com, "Bill Sethares" <sethares@e...> wrote:
>
> >
> > BTW, can anyone point to any interesting mp3s or audio files on the
> web in 7-tet?
> > I have a percussionist I play with, and I've been doing some improvs
> with hiim in
> > 7-tet...I use very inharmonic FM moog-like sounds which I think work
> best for such
> > an inharmonic tuning...it's great!
> >
> > -Aaron.
>
> Hi Aaron, I have one piece on the web in 7-tet,
> its at:
>
> http://eceserv0.ece.wisc.edu/~sethares/otherperson/all_mp3s.html
>
> click on "Pagan's Revenge"

Very nice work(s), Bill! I've seen your web page before, and really like all the music
on it quite alot. You certainly present a budding microtonalist with a broad array of
what can be done structuarally and timbrally. I think your instincts about timbre
and which to use with which tunings are right on! Listening to 'Pagan's
Revenge'strengthened my resolve to go in the direction of gamelan-esque type
sounds-your use of these is very effective...Bravo!

> I have a couple of others in 7-tet that also use a similar sound set -
> in this case sound with the partials of a bar (like a marimba or
> glockenspeil) but some with more sustain as well...

I'd love to hear 'em--point the way!

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> Here I am...it's getting late on Sept 11th, 2003...I need to go to
bed-I'm seeing
> cross-eyed, and to digest a new concept like "honeycombs", as
fascinating as
> they sound, would be unwise!

sorry, i was too abstract as usual. the "honeycombs" are really no
different from the "lattices" you're used to seeing . . . or maybe
not . . .

basically, each cell is labeled with a pitch class from the equal
temperament in question, ranging from 0 to n-1 for n-equal. each cell
is adjacent to all the cells that form a (approximate) 5-limit
consonance with it. if you move one cell to the east, you're going up
a "perfect fifth" or down a "perfect fourth". if you move one cell to
the northeast, you're going up a "major third" or down a "minor
sixth". if you move one cell to the southeast, you're going up
a "major sixth" or down a "minor third". each consonant triad is
represented by a triangle of cells, pointing upward in the case of
the major triad and pointing downward in the case of the minor triad.
so these "honeycombs" make it easy to see what pitches will result if
you play music based on progressions of triads connected by common
tones (or just having notes in common).

here's the "honeycomb" for 12-equal, which should be easy to
understand:

/tuning-math/files/Paul/12p.gif

(the bigger hexagons enclose 12 smaller honeycombs each)

and for 19-equal, since you've been working with that:

/tuning-math/files/Paul/19p.gif

> So I'll look foward to reading up on it tomorrow ....you guys are
like mad scientist
> gurus!

just trying to help (maybe give you new ideas for your music) . . .
let me know if i'm succeeding.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
>
> > Here I am...it's getting late on Sept 11th, 2003...I need to go to
> bed-I'm seeing
> > cross-eyed, and to digest a new concept like "honeycombs", as
> fascinating as
> > they sound, would be unwise!
>
> sorry, i was too abstract as usual. the "honeycombs" are really no
> different from the "lattices" you're used to seeing . . . or maybe
> not . . .
>
> basically, each cell is labeled with a pitch class from the equal
> temperament in question, ranging from 0 to n-1 for n-equal. each cell
> is adjacent to all the cells that form a (approximate) 5-limit
> consonance with it. if you move one cell to the east, you're going up
> a "perfect fifth" or down a "perfect fourth". if you move one cell to
> the northeast, you're going up a "major third" or down a "minor
> sixth". if you move one cell to the southeast, you're going up
> a "major sixth" or down a "minor third". each consonant triad is
> represented by a triangle of cells, pointing upward in the case of
> the major triad and pointing downward in the case of the minor triad.
> so these "honeycombs" make it easy to see what pitches will result if
> you play music based on progressions of triads connected by common
> tones (or just having notes in common).
>
> here's the "honeycomb" for 12-equal, which should be easy to
> understand:
>
> /tuning-math/files/Paul/12p.gif
>
> (the bigger hexagons enclose 12 smaller honeycombs each)
>
> and for 19-equal, since you've been working with that:
>
> /tuning-math/files/Paul/19p.gif
>
> > So I'll look foward to reading up on it tomorrow ....you guys are
> like mad scientist
> > gurus!
>
> just trying to help (maybe give you new ideas for your music) . . .
> let me know if i'm succeeding.

Very cool...yes, I've seen JI lattices. The honeycombs are better, because you
can keep major and minor chord on the same tonic, and the shape is just a
mirror--very cool!

Of course, adding the 7 limit makes it 3D and very complex......

Actually, I think I might try to print out a honeycomb the next time I write. It will
help me save time a little bit, I think. I did write out a pitch circle for 'Juggler', so
that when I went beyond an octave, I could do clock arithmetic with my finger
instead of the math in my head......

Thanks, Paul!

Best
Aaron.

Aaron - great piece (the juggler). Very clever use of the 19-tet semitones, the way the
melody repeats figures offset by the small intervals... striking ending as well. Also,
thanks for the kind words. There is really a bit more theory than intuition, though,
behind the choice of timbres... but surely, gamelonesque timbres are a natural for 7-
tet.

> >
> > I have a couple of others in 7-tet that also use a similar sound set -
> > in this case sound with the partials of a bar (like a marimba or
> > glockenspeil) but some with more sustain as well...
>
> I'd love to hear 'em--point the way!

Well, also at

http://eceserv0.ece.wisc.edu/~sethares/otherperson/all_mp3s.html

is the "Incomplete Portrait of Gertrude Stein" (also in 7-tet).

-Bill Sethares

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

/tuning/topicId_46408.html#46910

> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
> >
> > > Here I am...it's getting late on Sept 11th, 2003...I need to go
to
> > bed-I'm seeing
> > > cross-eyed, and to digest a new concept like "honeycombs", as
> > fascinating as
> > > they sound, would be unwise!
> >
> > sorry, i was too abstract as usual. the "honeycombs" are really
no
> > different from the "lattices" you're used to seeing . . . or
maybe
> > not . . .
> >
> > basically, each cell is labeled with a pitch class from the
equal
> > temperament in question, ranging from 0 to n-1 for n-equal. each
cell
> > is adjacent to all the cells that form a (approximate) 5-limit
> > consonance with it. if you move one cell to the east, you're
going up
> > a "perfect fifth" or down a "perfect fourth". if you move one
cell to
> > the northeast, you're going up a "major third" or down a "minor
> > sixth". if you move one cell to the southeast, you're going up
> > a "major sixth" or down a "minor third". each consonant triad is
> > represented by a triangle of cells, pointing upward in the case
of
> > the major triad and pointing downward in the case of the minor
triad.
> > so these "honeycombs" make it easy to see what pitches will
result if
> > you play music based on progressions of triads connected by
common
> > tones (or just having notes in common).
> >
> > here's the "honeycomb" for 12-equal, which should be easy to
> > understand:
> >
> > /tuning-math/files/Paul/12p.gif
> >
> > (the bigger hexagons enclose 12 smaller honeycombs each)
> >
> > and for 19-equal, since you've been working with that:
> >
> > /tuning-math/files/Paul/19p.gif
> >
> > > So I'll look foward to reading up on it tomorrow ....you guys
are
> > like mad scientist
> > > gurus!
> >
> > just trying to help (maybe give you new ideas for your
music) . . .
> > let me know if i'm succeeding.
>
> Very cool...yes, I've seen JI lattices. The honeycombs are better,
because you
> can keep major and minor chord on the same tonic, and the shape is
just a
> mirror--very cool!
>
> Of course, adding the 7 limit makes it 3D and very complex......
>
> Actually, I think I might try to print out a honeycomb the next
time I write. It will
> help me save time a little bit, I think. I did write out a pitch
circle for 'Juggler', so
> that when I went beyond an octave, I could do clock arithmetic with
my finger
> instead of the math in my head......
>
> Thanks, Paul!
>
> Best
> Aaron.

***I use Paul's visual lattice (as found on Dave Keenan's website)
all the time when composing in Blackjack. In fact, I have an
analysis of all the main harmonic moments in the piece (a new
Blackjack woodwind quintet)... some of that I was doing *as* I was
composing, and some afterwards, to make sure I was getting what I
wanted...

J. Pehrson

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

> Very cool...yes, I've seen JI lattices. The honeycombs are better,
because you
> can keep major and minor chord on the same tonic, and the shape is
just a
> mirror--very cool!

well, that's true for lattices too . . .

> Of course, adding the 7 limit makes it 3D and very complex......

have you seen any of my 7-limit lattices? there are some in my
papers, and also in the files folder for this group, also i did lots
of ascii ones in posts here, but to view those posts correctly you
now have to click on "message index" and then "expand messages".

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

/tuning/topicId_46408.html#46940

> --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
>
> > Very cool...yes, I've seen JI lattices. The honeycombs are
better,
> because you
> > can keep major and minor chord on the same tonic, and the shape
is
> just a
> > mirror--very cool!
>
> well, that's true for lattices too . . .
>
> > Of course, adding the 7 limit makes it 3D and very complex......
>
> have you seen any of my 7-limit lattices? there are some in my
> papers, and also in the files folder for this group, also i did
lots
> of ascii ones in posts here, but to view those posts correctly you
> now have to click on "message index" and then "expand messages".

***Well, the _Blackjack_ lattices are certainly readable, and they're
7-limit...

JP

hi Aaron,

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
>
> Very cool...yes, I've seen JI lattices. The honeycombs are
> better, because you can keep major and minor chord on the
> same tonic, and the shape is just a mirror--very cool!
>
> Of course, adding the 7 limit makes it 3D and very complex......
>
> Actually, I think I might try to print out a honeycomb the
> next time I write. It will help me save time a little bit,
> I think. I did write out a pitch circle for 'Juggler', so
> that when I went beyond an octave, I could do clock arithmetic
> with my finger instead of the math in my head......

i think you're going to love my software, which is currently
under development. it starts by asking you to define an
Euler Genus in some prime-space, shows you the lattice of that,
then lets you construct a tuning system from it by defining
unison-vectors and other constraints.

... and of course, the whole point is that it's not just for
playing with tunings, but for composing real microtonal music.

the release of version 1.0 is planned for late January 2004.
if you're not around here at that time, contact me.

BTW -- i really like _The Juggler_ a lot too. ... but my
ego suffered a small wound with so many people calling it
"the best 19edo piece ever". here's my entry, which i
correspondingly think is one of my best tunes:

http://sonic-arts.org/monzo/19tet/19samba.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> i think you're going to love my software, which is currently
> under development. it starts by asking you to define an
> Euler Genus in some prime-space, shows you the lattice of that,
> then lets you construct a tuning system from it by defining
> unison-vectors and other constraints.

I'd still like to see some honeycombs of chords added to its
capabilities, and in particular the lattice, not just honeycomb, of
septimal tetrads.

--- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:

Listening to 'Pagan's
> Revenge'strengthened my resolve to go in the direction of gamelan-
esque type
> sounds-your use of these is very effective...Bravo!

I'd like to see Juggler followed up with harmonic partials and some
other tunings--equal temperaments past 19, or linear temperaments.

But that's just me. :)

Joseph Pehrson wrote:

>***Well, the _Blackjack_ lattices are certainly readable, and they're >7-limit...
> >
Only 7-limit? You should ask for a refund! Mine are 11-limit.

http://x31eq.com/lattice.htm#7neutral

http://x31eq.com/decimal_lattice.htm#mos

Graham

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Aaron,
>
>
> --- In tuning@yahoogroups.com, "akjmicro" <akj@r...> wrote:
> >
> > Very cool...yes, I've seen JI lattices. The honeycombs are
> > better, because you can keep major and minor chord on the
> > same tonic, and the shape is just a mirror--very cool!
> >
> > Of course, adding the 7 limit makes it 3D and very complex......
> >
> > Actually, I think I might try to print out a honeycomb the
> > next time I write. It will help me save time a little bit,
> > I think. I did write out a pitch circle for 'Juggler', so
> > that when I went beyond an octave, I could do clock arithmetic
> > with my finger instead of the math in my head......
>
>
>
> i think you're going to love my software, which is currently
> under development. it starts by asking you to define an
> Euler Genus in some prime-space, shows you the lattice of that,
> then lets you construct a tuning system from it by defining
> unison-vectors and other constraints.
>
> ... and of course, the whole point is that it's not just for
> playing with tunings, but for composing real microtonal music.
>
> the release of version 1.0 is planned for late January 2004.
> if you're not around here at that time, contact me.
>
>
>
> BTW -- i really like _The Juggler_ a lot too. ... but my
> ego suffered a small wound with so many people calling it
> "the best 19edo piece ever". here's my entry, which i
> correspondingly think is one of my best tunes:
>
> http://sonic-arts.org/monzo/19tet/19samba.htm

Joe-

Yes, I checked it out....I dig it...very nice lilt and gentleness to it! I agree that it
also has a very subtle 'xenharmonicity' to it! Bravo!

Cheers,
Aaron.