Yahoo Tuning Groups Ultimate Backup tuning The 2200/2197 temperament (AKA 8:10:11:13:16 in 3D)

Petr wrote :

> Hi Jacques.
> Very glad you like the tuning -- I think I had just discovered it
> last year. :-D
> Very few people actually know how "lucky" that small interval is
> (in terms of tempering possibilities). You can think of the
> temperament as two chains of ~455.073 cent intervals (as many tones
> as you like) where one chain is away from the other by ~840.957
> cents -- i.e. a 3D temperament where the period is 2/1, one> generator (let's call it A) is ~455.073 cents and is used as many
> times as desired, and the other one (let's call it B) is ~840.957
> cents and is used only once. Since 2200/2197 is really small, you
> can see how closely this temperament approximates the target ratios:
> A > 13/10
> B > 13/8
> B-A > 5/4
> 2A+B > 11/4
> 3A > 11/5
>
> Petr

Hi Petr,
This is a real treasure you found here ! Yes, these vectors are very
close to JI ratios - there is no 3 or 7, that's what makes it
unusual, but the harmonies it presents between 5, 11 and 13 are
unique - I can't wait to explore it !
One question, why has the B interval to be used only once ? Could it
not be used up to three times, to reach the 275/32 - or 2197/512 -
coïncidence ? Or have you some musical or others reasons for a
choice of only two chains ?
I found that an other coïncidence occurs between 5A+3B (=4798.236 c.)
and 4 octaves (4096000/4084223) - have you noticed that ?
Could it be eventually tempered as well ? (it would just need to
slightly raise the "13/8" ratio, of course it would make sense only
if you extend to at least four A-chains).
Also, have you special reasons to choose 455.073 and 840.957 cents ?
Although it would be a subtle difference to hear, I found that
455.0422 and 840.0102 c. would realize the differential coherence of
all 11/10 and 13/8 intervals, in addition of equal-beating of 11/10 and 13/10 intervals.
On the other hand, a JI version would probably not be bad at all !
Because what is remarquable in this temperament it seems to me, is
that it reveals, in its own way and without much deformation, verynatural harmonies.
Thanks for sharing !
- - - - - - -
Jacques

🔗Jacques Dudon <fotosonix@...>

Jacques wrote :

> Why has the B interval to be used only once ? Could it
> not be used up to three times, to reach the 275/32 - or 2197/512 -
> coïncidence ? Or have you some musical or others reasons for a
> choice of only two chains ?

...I had a better look and found, of course, that the 3rd A-chain
would repeat the first one at a "65/32" higher (2B - A)...
Might be one part of the answer - on the other hand you find also the
same comma between two consecutive A-chains, only after a longer path
(-B -6A)

> I found that an other coïncidence occurs between 5A+3B (=4798.236 c.)
> and 4 octaves (4096000/4084223) - have you noticed that ?
> Could it be eventually tempered as well ? (it would just need to
> slightly raise the "13/8" ratio, of course it would make sense only
> if you extend to at least four A-chains).

... then the 4th A-chain, octave reduced, would repeat the 1st one
and this would close the spiral to a 87-equal-or unequal-divisions of
the octave.
Might be of a certain pedagogic and practical interest - knowing that
already in 87-edo :
A = 33 steps = 455.1724 c.
B = 61 steps = 841.37931 c.
"5/4" = 28 steps = 386.2069 c.
"11/8" = 40 steps = 551.72414 c.
"11/10" = 12 steps = 165.51724 c.
Also a nice "meantone-like" semi-third "143/128" = "160/143" = 14
steps = 193.10345 c.
Plenty of good material in here !
- - - - - - -
Jacques

🔗Petr Parízek <p.parizek@...>

Jacques wrote:

> One question, why has the B interval to be used only once ? Could it not be used up to three times,
> to reach the 275/32 - or 2197/512 - coïncidence ?
> Or have you some musical or others reasons for a choice of only two chains ?

The reason is rather a question of practical performance. If I want to temper out 2200/2197, I can break it into three powers of some other factors (instead of four powers of prime numbers), which will offer me a possible solution. So if I call the three factors "x, y, z" and I assign, for example, "x = 2/13, y = 5, z = 11", then 2200/2197 equals to "x^3 * y^2 * z^1" or "x*y*x*y*x*z". These multiplications exploit some sort of periodicity and therefore one possible generator of 10/13 (i.e. x*y) can be chained essentially "ad infinitum" while 2/13 (i.e. the remaining x) is the other generator which complements the temperament and allows you to get only "slightly higher" than 1/11. If our aim is to approximate 8:10:11:13 rather than 1/(8:10:11:13), we can invert it all and replace some octave equivalents, so we get generators of 13/10 (a) and 13/8 (b), which makes a*a*b slightly lower than 11/4. Of course, you may decide to approximate other chords as well, but if you wish to have as many locations of the primary target approximants as possible, then using one generator once and another generator more times is the best way of "harnessing" a 3D temperament which I can think of.

> I found that an other coïncidence occurs between 5A+3B (=4798.236 c.)
> and 4 octaves (4096000/4084223) - have you noticed that ?

Actually, I haven't. ;-)

> Could it be eventually tempered as well ?

It could but you would get a pretty complex temperament. And if I try to make it less complex, then one generator has to be something pretty weird -- for example, if "a" is 320/169 and "b" is 13/8, then a*b*b makes 5/1 and a*a*a*b is slightly higher than 11/1.

> Also, have you special reasons to choose 455.073 and 840.957 cents ?

Yes, both 11/4 and 11/5 have the least possible mistuning, which is better than having one interval mistuned twice as much and the other pure.

> Although it would be a subtle difference to hear, I found that 455.0422 and 840.0102 c.
> would realize the differential coherence of all 11/10 and 13/8 intervals,
> in addition of equal-beating of 11/10 and 13/10 intervals.

Not sure what you mean by differential coherence.

BTW: Your mention of 37-equal reminds me of a 2D temperament which I found back in 2006 but haven't been able to realize so far (see message #75311) and another from March 2008 whose period is 2/1 and whose generator is the 13th root of 130.

Petr

🔗Jacques Dudon <fotosonix@...>

Petr wrote :

(Jacques) :
>> Although it would be a subtle difference to hear, I found that
455.0422 and 840.0102 c.
>> would realize the differential coherence of all 11/10 and 13/8
intervals,
>> in addition of equal-beating of 11/10 and 13/10 intervals.

> Not sure what you mean by differential coherence.

The first-order difference tones of all "11/10" and "13/8" intervals
of your temperament,
11 - 10 = 1
13 - 8 = 5
are in tuned with the scale (in the case of octaves = 2/1).
It will become audible with proper sounds of course, like in my
photosonic disks but also with many acoustic or electric instruments.
I use it myself in extended JI mainly, but it does not requires the
tuning to be in any JI.

> BTW: Your mention of 37-equal reminds me of a 2D temperament which
> I found back in 2006 but haven't been able to realize so far (see
> message #75311) and another from March 2008 whose period is 2/1 and
> whose generator is the 13th root of 130.

Of 87-edo you mean ? It's funny that you talk of 37-edo, because 37-
edo would be working quite well also in your 2200/2197 temperament :
A would be 14 steps, and B 26 steps. Quite interesting !
I notice that your 13th root of 130 is very well approximated in both
too : 20 of 37, or 47 of 87.
And the 47 steps can be found very easily in your temperament, by
doing (-B -2A).
There must be more than a simple coïncidence here !

About limiting to two A-chains, I must admit your explanation goes
over my competences. But I arrive to the same conclusion anyway. With
a third comma that can be tempered but without more interest, it's
20480/20449 (equality of the 160/143 and 143/128 meantones).
One last thing I found with this temperament, may be you verified
too, is that it has two interesting DE (even distributions), in 13
and 16 notes/octave. Steps in 87 divisions of the last one are :
0 7 12 16 21 28 33 40 45 49 54 61 66 73 78 82 (87)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

🔗Petr Parízek <p.parizek@...>

Jacques wrote:

> The first-order difference tones of all "11/10" and "13/8" intervals of your temperament,
> 11 - 10 = 1
> 13 - 8 = 5
> are in tuned with the scale (in the case of octaves = 2/1).

Interesting. How would you apply this concept to, let's say, meantone temperament if the primary approximated chord is 3:4:5? And would the result be in any way different if the target chord was 4:5:6?

I wrote:

> > BTW: Your mention of 37-equal reminds me of a 2D temperament which I found
> > back in 2006 but haven't been able to realize so far (see message #75311) and another
> > from March 2008 whose period is 2/1 and whose generator is the 13th root of 130.
>
> Of 87-edo you mean ?

I'm not sure I understand you. I wasn't talking about any equal tuning, I was just saying that your mention of 37-equal reminded me of these two regular temperaments.

> It's funny that you talk of 37-edo, because 37-edo would be working quite well also
> in your 2200/2197 temperament : A would be 14 steps, and B 26 steps. Quite interesting !

Yes, 37-equal is good at approximating almost any non-3 factor or prime--including 5, 7, 11, or 13.

> I notice that your 13th root of 130 is very well approximated
> in both too : 20 of 37, or 47 of 87.

Carl has suggested the same. I'll have to really explore 87-equal one day.

> And the 47 steps can be found very easily in your temperament, by doing (-B -2A).

Hmmm, I'll probably také all of these strange non-3 commas, sooner or later, and see what happens if I temper out at least two of them at the same time.

> 0 7 12 16 21 28 33 40 45 49 54 61 66 73 78 82 (87)

Not sure about this but don't DE scales have only two one-step interval sizes?

Petr

🔗Jacques Dudon <fotosonix@...>

Petr wrote :

> (Jacques) :
> > The first-order difference tones of all "11/10" and "13/8"
> intervals of your temperament,
> > 11 - 10 = 1
> > 13 - 8 = 5
> > are in tune with the scale (in the case of octaves = 2/1).
>
> Interesting. How would you apply this concept to, let's say,
> meantone temperament if the primary approximated chord is 3:4:5?
> And would the result be in any way different if the target chord
> was 4:5:6?

Apart from the major third's auto-coherence (5 -4 = 1, achieved only
in the quarter-syntonic comma meantone), and knowing that the
coherence of 4/3 or 3/2 does not generate meantone temperaments,
three intervals can be differentially coherent in the three possible
versions of a meantone major triad (3:4:5, 4:5:6, 5:6:8), 5/3,
6/5, 8/5 that have 3 different solutions :
"5 - 3 = 2" is achieved with a Wilson meantone fifth of 1.4945301805
"6 - 5 = 1" is achieved with a Diatess meantone fifth of 1.49335855656
"8 - 5 = 3" is achieved with a Skisni meantone fifth of 1.495953506243

>>> BTW: Your mention of 37-equal reminds me of a 2D temperament
which I found
>>> back in 2006 but haven't been able to realize so far (see
message #75311) and another
>>> from March 2008 whose period is 2/1 and whose generator is the
13th root of 130.

>> Of 87-edo you mean ?

> I'm not sure I understand you. I wasn't talking about any equal
> tuning, I was just saying that your mention of 37-equal reminded me
> of these two regular temperaments.

I asked : "my mention of 87-equal, you mean ?" because you said I
mentionned 37-equal, but I didn't - I only mentionned 87-equal.

>> It's funny that you talk of 37-edo, because 37-edo would beworking quite well also
>> in your 2200/2197 temperament : A would be 14 steps, and B 26
steps. Quite interesting !

> Yes, 37-equal is good at approximating almost any non-3 factor or
> prime--including 5, 7, 11, or 13.

Nice ! - oops, I mean too bad for factor 3 ... ;-(

> I notice that your 13th root of 130 is very well approximated
> in both too : 20 of 37, or 47 of 87.

> Carl has suggested the same. I'll have to really explore 87-equal
> one day.

I did not know Carl did. I found that 87-equal for your 2200/2197
temperamentis better than any other smaller number (I don't know for
130^(1/13)).
37-equal here is correct but does not make the difference between 2A
- B and B - 7A (5 and 4 steps in 87 tones) - or tempers them, as you
like.
With very small numbers 10, 13, 16 also give some honest
approximations - 13 and 16, at least, are logical DE subsets for your
temperament, in the sense they don't break the A-chains.

>> And the 47 steps can be found very easily in your temperament, by
doing (-B -2A).

> Hmmm, I'll probably také all of these strange non-3 commas, sooner
> or later, and see what happens if I temper out at least two of them
> at the same time.

Good idea, it seems they could be musically complementary.

> 0 7 12 16 21 28 33 40 45 49 54 61 66 73 78 82 (87)

> Not sure about this but don't DE scales have only two one-step
> interval sizes ?

MOS scales have been said to have 2 interval sizes. DE to my sense
are not limited to 2, but that's only my personal definition of DE
structures !!
By pure coïncidence, I have been using this same acronym in french
for years ("DE" for "divisions équilibrées"), before I ever heard of
Erv Wilson's MOS concept, and I discovered recently of the use of
"DE" as "distributionally even", in "The Regular Mapping Paradigm"
from Graham's pages :
http://x31eq.com/paradigm.html#cons
It does not says that DE are limited to 2 interval sizes, but
specialists can correct me if I assume something that is wrong here.

If DE were limited to 2 interval sizes, then 3D temperaments scale
structures would need special definitions of their own that would
only add complexity. In the sense I use "DE" (and which is not
limited to scales but covers also waveforms, fractals, etc.), DE
means "divisions that shows semi-periodicities" and 2 interval sizes
is the simplest option, among other ones.

Here your 2200/2197 temperament subsets clearly show semi-
periodicities of the 4 and 9 steps intervals (based on A = 33 steps),
in other words, that are not isolated accidents :

7 5 4 5 7 5 7 5 4 5 7 5 7 5 4 5

7 5 9 7 5 7 5 9 7 5 7 5 9

(33+33+21 = 87)

- - - - - - -
Jacques

🔗Petr Parízek <p.parizek@...>

Jacques wrote:

> "5 - 3 = 2" is achieved with a Wilson meantone fifth of 1.4945301805

> "6 - 5 = 1" is achieved with a Diatess meantone fifth of 1.49335855656

> "8 - 5 = 3" is achieved with a Skisni meantone fifth of 1.495953506243

Around 2002, I was doing an awful lot of experiments with meantone temperament but never ever have I thought of this! Rather than following difference tones, I was mainly focused on comparing beat rates. Therefore, the Wilson meantone was the one where C-E, C-G, and E-G all have the same beat rates. My favorite version, at that time, was the one which is close to 7/25-comma meantone and where C-E beats the opposite of G-E. But when I realized that the fifths can actually have slightly different sizes within the chain, I finally stopped using these irrational intervals and started aiming for absolute beat rates (see the "parizek_qmeb" files in Manuel's archive). The disadvantage then was, of course, that if I originally made the temperament with A4=440Hz, then shifting the entire pitch to A4=442Hz or A4=415Hz made the beat rates more difficult to track down (or vice versa). But even today, it still belongs to my most favorite methods of making meantone temperaments and I used it later to approximate other meantones like quarter-comma (parizek_meanqr) or 2/7-comma (mean2sevr).

> I asked : "my mention of 87-equal, you mean ?" because you said I mentionned 37-equal,
> but I didn't - I only mentionned 87-equal.

Aha, that's strange. Well, maybe I thought 37 because I found that back in 2006.

> Good idea, it seems they could be musically complementary.

Well, I've just tried to temper out both 20480/20449 and 2200/2197 at the same time. And what has come out? The same 2D temperament which I found in March 2008 where I used the 13th root of 130 as the generator. :-D

> It does not says that DE are limited to 2 interval sizes, but specialists can correct me
> if I assume something that is wrong here.

Maybe I was confusing it with "maximally even" scales, I don't know.

Petr

🔗Graham Breed <gbreed@...>

🔗Jacques Dudon <fotosonix@...>

Petr wrote :

(Jacques) :
>> "5 - 3 = 2" is achieved with a Wilson meantone fifth of 1.4945301805
>> "6 - 5 = 1" is achieved with a Diatess meantone fifth of 1.49335855656
>> "8 - 5 = 3" is achieved with a Skisni meantone fifth of 1.495953506243

> Around 2002, I was doing an awful lot of experiments with meantone > temperament but never ever have I thought of this! Rather than > following difference tones, I was mainly focused on comparing beat > rates. Therefore, the Wilson meantone was the one where C-E, C-G, > and E-G all have the same beat rates. My favorite version, at that > time, was the one which is close to 7/25-comma meantone and where C-> E beats the opposite of G-E. But when I realized that the fifths > can actually have slightly different sizes within the chain, I > finally stopped using these irrational intervals and started aiming > for absolute beat rates (see the "parizek_qmeb" files in Manuel's > archive). The disadvantage then was, of course, that if I > originally made the temperament with A4=440Hz, then shifting the > entire pitch to A4=442Hz or A4=415Hz made the beat rates more > difficult to track down (or vice versa). But even today, it still > belongs to my most favorite methods of making meantone temperaments > and I used it later to approximate other meantones like quarter-> comma (parizek_meanqr) or 2/7-comma (mean2sevr).

I found that differential coherence always have consequent equal-beating /or synchronous-beating properties, while the reverse is not true.
Your observation on the Wilson meantone is right, and as far as I know, not mentionned by Erv Wilson. But I found the same, and I found a few other meantones presenting such "triple-equal beating" properties, possibly already known, I don't know. I was going to post that on the Tuning List last year, but at that time I had only five, and I had the feeling that one was missing... Recently I have found the sixth guy, and perhaps an ancient one could be a seventh, anyway they are a good number and I will take the time to present them some day : soon in your TL mails...
There is no contradiction between having different sizes of fifths and following equal (or synchronous)-beating algorithms. On the contrary, once you know the right recurrent algorithms (whose solutions are irrational), you have total freedom to use all the rational frequencies you want, as long as they follow these algorithms, and each has an infinity of versions. Of course you don't need to follow any sequence at all, and I do it also.
I will have to take a look at your tunings, because I don't understand why a pitch transposition would affect the beat rates proportions, unless you want to keep the same beat rates, or I don't catch the problem.

> I've just tried to temper out both 20480/20449 and 2200/2197 at the > same time. And what has come out? The same 2D temperament which I > found in March 2008 where I used the 13th root of 130 as the > generator. :-D

I'm glad my intuition was right, then : they're connected ! - funny how we re-discover the same things from different paths sometimes, I do that all the time.
Now you have the choice between 2D, 3D and why not 4D then ? ;-)

>> It does not says that DE are limited to 2 interval sizes, but specialists can correct me
>> if I assume something that is wrong here.

> Maybe I was confusing it with "maximally even" scales, I don't know.

Well, no one replied but it does not tell if "DE" scales are defined as limited to 2 interval sizes or not, I just shared my experience that the idea can be extended to more than 2 sizes.
- - - - - - -
Jacques

🔗Jacques Dudon <fotosonix@...>

Graham wrote :

>> (Jacques) : It does not says that DE are limited to 2 interval
sizes, but specialists can correct me
>> if I assume something that is wrong here.

> (Petr) : Maybe I was confusing it with „maximally even“ scales, I
don’t know.

> They're both limited to two interval sizes. "Distributionally even"
> is "maximally even" generalized to non equal temperaments.

Thanks for the precision. If this is the common use of these terms
then some more general terms have to be found to refer for example to
the octave divisions I gave for Petr's 2200/2197 temperament, andother 3D temperaments.
I suggest "3 sizes-DE" ... then it would keep the idea of even
distribution, while it precises the number of interval sizes it
refers to.

An example of "3 sizes-DE" easy to understand in 37 steps (and a
potential 13-limit meta-temperament) :

(octave = 37, generators = 9 and 16)

16 12 9
9 7 12 9
9 7 5 7 9
4 5 7 5 7 4 5
4 5 3 4 5 3 4 4 5

- - - - - - -
Jacques

The 2200/2197 temperament (AKA 8:10:11:13:16 in 3D)

🔗Petr Pařízek <p.parizek@...>

🔗Carl Lumma <carl@...>

🔗Jacques Dudon <fotosonix@...>

🔗Petr Parízek <p.parizek@...>

🔗Jacques Dudon <fotosonix@...>

🔗Jacques Dudon <fotosonix@...>

🔗Petr Parízek <p.parizek@...>

🔗Jacques Dudon <fotosonix@...>

🔗Petr Parízek <p.parizek@...>

🔗Jacques Dudon <fotosonix@...>

🔗Petr Parízek <p.parizek@...>

🔗Graham Breed <gbreed@...>

🔗Jacques Dudon <fotosonix@...>

🔗Jacques Dudon <fotosonix@...>