Scales without small intervals?

Hi tuners,

I was just thinking what scales I could find which would approximate 5-limit triads fairly well (something like 12 cents of mistuning, at most) and at the same time would not contain smaller intervals than, say, 100 cents or so. To begin with, I could divide 2/1 into 7 equal steps or 3/1 into 11 equal steps. But this tempers out 25/24, which means that both 5/4 and 6/5 are very much away from their approximated intervals here. Having tried some of the 2D temperaments, so far I've been successful with meantone and porcupine (you may argue I could use mavila as well but its fifths are so far from 3/2 that I finally excluded this choice). I was even thinking if I could possibly make something like a 3D tuning out of the 5-limit system, whether tempered or untempered, and find such a kind of scale in it somehow. -- Right now, I still have some other things to do, but maybe I could tell you why about my further aims with this on wednesday.

Thanks in advance, any comments are appreciated.

Petr

I wrote:

> so far I've been successful with meantone and porcupine

Hahah, my list is growing ... Diaschismatic. :-)

Petr

Petr Pařízek wrote:
> Hi tuners,
> > I was just thinking what scales I could find which would approximate 5-limit > triads fairly well (something like 12 cents of mistuning, at most) and at > the same time would not contain smaller intervals than, say, 100 cents or > so. To begin with, I could divide 2/1 into 7 equal steps or 3/1 into 11 > equal steps. But this tempers out 25/24, which means that both 5/4 and 6/5 > are very much away from their approximated intervals here. Having tried some > of the 2D temperaments, so far I've been successful with meantone and > porcupine (you may argue I could use mavila as well but its fifths are so > far from 3/2 that I finally excluded this choice). I was even thinking if I > could possibly make something like a 3D tuning out of the 5-limit system, > whether tempered or untempered, and find such a kind of scale in it > somehow. -- Right now, I still have some other things to do, but maybe I > could tell you why about my further aims with this on wednesday.
> > Thanks in advance, any comments are appreciated.
> > Petr

A quick glance at the horograms in Paul Erlich's "Middle Path" paper will show you how many notes of a temperament you can get without including steps smaller than 100 cents. Amity for instance gives you a 7-note scale -- any more notes and you get a 23-cent step. But a look at the mapping shows that you need at least 14 notes for one 5-limit triad, so that rules out amity as a possibility. (You might find a subset of amity that works by skipping half of the notes or something, but it's probably not worth the effort at this point.)

Right off the bat, you can eliminate possibilities like catler, injera, and ennealimmal, since their generators are less than 100 cents. You can also look at the error chart for 5/4 and 3/2 and eliminate temperaments like august or dominant that have more than 12 cents mistuning for these intervals. That leaves:

size required 5/4 err. 3/2 err. 2/1 err.
meantone 7 5 0.5 -4.4 1.7
negripent 9 8 -7.9 -4.7 1.8
porcupine 7 6 -1.0 8.0 -3.1
ripple 11 9 1.1 -8.6 3.3
srutal 10 8 3.8 2.3 -0.9

> size required 5/4 err. 3/2 err. 2/1 err.
> meantone 7 5 0.5 -4.4 1.7
> negripent 9 8 -7.9 -4.7 1.8
> porcupine 7 6 -1.0 8.0 -3.1
> ripple 11 9 1.1 -8.6 3.3
> srutal 10 8 3.8 2.3 -0.9

Srutal (a.k.a. diaschismic) looks like the non-meantone
winner here.

-Carl

Herman wrote:

> ripple 11 9 1.1 -8.6 3.3

Wow, thanks, haven’t looked at this one. Is that the 6561/6250 temperament?

Petr

Scales with allsortsa intervals. (Large, small, Vast and iotic, (or
should that be idiotic);-)

I have been busy updating and improving my scales directory for those
interested in scales which will work on any meantone + lotsa other
systems.

http://www.lucytune.com/scales/

I managed to get conventional notation and triads for most of them now
(2300-odd) and have midifiles which I am in the slow process of
converting into audios with triads etc.

If you can run FileMaker, you can sort, search, and organise them in
many revealing ways

Also to a lesser extent, using .xls files.

On 7 Jul 2009, at 09:51, Petr Pařízek wrote:

>
>
> 
>
> Herman wrote:
>
> > ripple 11 9 1.1 -8.6 3.3
>
> Wow, thanks, haven’t looked at this one. Is that the 6561/6250
> temperament?
>
> Petr
>
>
>
>
>
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Petr Pařízek wrote:
> Herman wrote:
> >> ripple 11 9 1.1 -8.6 3.3
> > Wow, thanks, haven’t looked at this one. Is that the 6561/6250 temperament?
> > Petr

That's the one.

if meantone works then meta-meantone should also
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Kraig wrote:

> if meantone works then meta-meantone should also

I thought metameantone was used as a more general term -- do you mean something in particular?

Petr

>
> > Herman Miller wrote:
> >
> >> size required 5/4 err. 3/2 err. 2/1 err
> >> ripple 11 9 1.1 -8.6 3.3
> >
> >
> > Wow, thanks, haven’t looked at this one. Is that the 6561/6250
> temperament?
> >
> > Petr
>
> That's the one.

Thanks Herman,
Just for my understanding, what means here "size required 11 9 " ?
the number of tones required ?

Funny temperament indeed, where 6521/6250 (73,48 cents) is choosen to
disappear, but not 81/80.
I am curious, will it not make it a bit special, if Petr wants to
avoid small intervals ??

(sorry guys, I am quite ignorant in these temperaments !) - but I
have been wandering one thing :
did anyone already worked on temperaments vanishing some of those
(some of my favorite commas) :

96/95
361/360
513/512
1216/1215

??
- - -
Jak

Jacques wrote:

> Just for my understanding, what means here "size required 11 9 " ?
> the number of tones required ?

Because my original aim was to find scales containing only intervals larger than 100 cents, the answer is that an 11-tone ripple scale meets this requirement and that you need at least an 9-tone scale to approximate one entire 5-limit triad. The thing here is that ripple approximates 4/3 with 5 generators and 8/5 with 8 generators, similarly as meantone approximates 3/2 with 1 generator and 5/1 with 4 generators.

> Funny temperament indeed, where 6521/6250 (73,48 cents) is choosen
> to disappear, but not 81/80.
> I am curious, will it not make it a bit special,
> if Petr wants to avoid small intervals ??

It’s 6561, not 6521. 6521 wouldn’t be very meaningful here, because that’s a prime number. OTOH, 6561/6250 is the distance of 8 perfect fourths minus 5 pure minor sixths, which is tempered out in ripple. The reason why I finally excluded this temperament from my choices was that it reminded me too much of 12-equal.

> 96/95

This factor contains the primes 2, 3, 5, and 19. Combining that with another factor containing the same primes, you could get a particular 2D temperament and find its generator. Since this factor contains 4 different primes, you could either make a 3D temperament out of it (for example, use a period of 2/1 and temper the 3/1 and 5/1 in such a way that one fifth minus one major third makes 19/16) or you could make a 2D temperament ommitting two approximations (for example, by using a period of 2/1 and a generator of somewhere between 6/5 and 19/16, which means that 3/1 and 5/1 are not approximated here).

> 361/360

This is similar. If you want to get a 3D temperament, the best way is to use a period of 2/1, then one generator can be 1/4 of the comma narrower than 19/16, and another generator can be 1/4 of the comma narrower than 4/3 (so that 2 first generators plus 2 second generators make a pure 5/2).

> 513/512

This can be tempered out with a period of 2/1 and a generator slightly wider than 4/3 so that stacking three of these fourths is close or equal to 19/8.

> 1216/1215

Again, this factor contains 4 primes and therefore what comes first to my mind is to make a 3D temperament where the period is 2/1, one generator is 1/6 of the comma wider than 3/2, and the other generator can be either 1/6 of the comma wider than 5/4 (then 5 first + 1 second generator makes a pure 19/2) or it can be 1/6 of the comma narrower than 19/16 (then 5 first minus 1 second generator makes a pure 32/5).

Petr

I wrote:

> one generator can be 1/4 of the comma narrower than 19/16, and another generator
> can be 1/4 of the comma narrower than 4/3 (so that 2 first generators
> plus 2 second generators make a pure 5/2).

Well, for the 361/360 temperament, it would be actually more convenient to have one generator 1/2 of the comma narrower than 19/12 (which equals „sqrt(5/2)“) and the other generator 1/4 of the comma narrower than 19/16. If we call them A and B, then the tempered fourth will be A-B and the tempered twelfth will be 2A+B.

Petr

I wrote:

> the tempered fourth will be A-B
> and the tempered twelfth will be 2A+B.

The fourth will, but the twelfth won't, of course. 2A+B maps to 95/32, not 3/1.

You see, it's 12:25 AM here, what do you want of e? :-D

Petr

I wrote:

> This factor contains the primes 2, 3, 5, and 19. Combining that with > another factor containing the same primes,
> you could get a particular 2D temperament and find its generator.

To be precise, if you temper out both 96/95 and 361/360, the result is indistinguishable from 12-equal because the period is 400 cents, the generator is ~100.3 cents, 3/1 is approximated with 5 (periods) minus 1 (generator), 5/1 is approximated with 7 periods, and 19/1 is approximated with 13 (periods) minus 1 (generator). So the mapping would be "[(3, 0), (5, -1), (7, 0), (13, -1)]".

Similarly, if you temper out both 361/360 and 1216/1215, you get the octave as the period, a tempered fourth as a generator, and a mapping of "[(1, 0), (2, -1), (-1, 8), (3, 3)]".

Interestingly enough, if you temper out 96/95 together with 1216/1215, you get a 19-limit version of the diaschismatic temperament, where the period is half an octave, the generator is ~99 cents, and the mapping is "[(1, 0), (3, 1), (5, -2), (8, 3)]". It's debatable whether this can be accepted as a variant of diaschismatic, since the optimum diaschismatic generator for good 5-limit approximations should be ~106 cents.

Petr

Petr Parizek wrote:
>
> > This factor contains the primes 2, 3, 5, and 19. Combining that with
> > another factor containing the same primes,
> > you could get a particular 2D temperament and find its generator.
>
> To be precise, if you temper out both 96/95 and 361/360, the result is
> indistinguishable from 12-equal because the period is 400 cents, the
> generator is ~100.3 cents, 3/1 is approximated with 5 (periods) > minus 1
> (generator), 5/1 is approximated with 7 periods, and 19/1 is > approximated
> with 13 (periods) minus 1 (generator). So the mapping would be > "[(3, 0),
> (5, -1), (7, 0), (13, -1)]".
>
> Similarly, if you temper out both 361/360 and 1216/1215, you get > the octave
> as the period, a tempered fourth as a generator, and a mapping of > "[(1, 0),
> (2, -1), (-1, 8), (3, 3)]".
>
> Interestingly enough, if you temper out 96/95 together with > 1216/1215, you
> get a 19-limit version of the diaschismatic temperament, where the > period is
> half an octave, the generator is ~99 cents, and the mapping is > "[(1, 0), (3,
> 1), (5, -2), (8, 3)]". It's debatable whether this can be accepted > as a
> variant of diaschismatic, since the optimum diaschismatic generator > for good
> 5-limit approximations should be ~106 cents.
>
> Petr

Hi Petr,

Thanks for these very precise and instructive answers to my question about tempering
96/95, 361/360, 513/512, and 1216/1215.
It will take me some time to fully understand everything, as I am not familiar with these processes.
I knew that these 19-limit commas would allow many clever chromatic temperaments, and now you made it clear.

Here are a few complements :
Your solution for 513/512 is the simplest thing, one I did thought of, to keep pure 2/1 and 19/16. As you said it can be done by using a fourth slighty wider than 4/3 as generator (or reversely a slightly narrow fifth of 1.499024707 = 700.829 c.).
Assuming the major third as 24/19 (here 403.316 c.), it dissolves also 96/95 and leads to a quasi-pythagorean tuning but based on pure 19/16s.
As a fractal variation, the "Melkis" recurrent sequence, already explained in this list, does the same but with a slight modification of 19/16 recovers the perfect differential coherence of "19-16 = 3" (lost by tempering the fifth to 1.4990247 in the precedent tempering)
Melkis fifth = 1.499073445 = 700.8852814 c. and its factor 19 is lowered only by 0.17 c. (to 18.9981469).
(reversely the Melkis fourth = 1.33415744713 can be used as a more convenient generator)
So Melkis can be seen as a temperament dissolving 513/512 and 96/95, but is mainly a meta-tuning, that generalizes an infinity of harmonic tunings.
If I express the Melkis fourth by v, it verifies v^4 = 2v + 1/2
for example in this series : 152 203 270,75 361 482 643..., 482 = v^4 = 203 + 203 + 76
Both Melkis and its attractor friend, the cubic root of 19/8, lead to simple but very nice sounding diatonic minor scales ; a TOP-optimized temperament based on 513/512, using same mecanisms, would certainly be possible, but would loose some of their specific acoustic qualities.

Now in these same 513/512 tempering variants, defining the major third rather as 8 fourths - 2 octaves, as would be 8192/6561 in a 3-limit tuning (instead of previously 4 fifths - 2 octaves), would have the effect on these temperaments to dissolve both 361/360 and 1216/1215, instead of 96/95. There are only 4 major thirds of this softer kind in a 12 tones scale, so one could need more notes and therefore would also have to add as many commas.
In Melkis, one single comma will show up to express either 76/75, 96/95, 361/360, 1216/1215 whenever they occur and its value is 2^5 / v^12 = 10.623 c. (about 327/325).
In the (19/8)^(1/3) temperament, the same thing applies and its value would be 9.948 c.(about 175/174).

About the other commas, to dissolve 1216/1215 if I had to, the simplest way I could find was to only augment only 5/4 by 1216/1215... to reach 304/243 = 387.738 c. : then "5" . 3^5 = 19 . 2^6
To dissolve 361/360, the simplest way I would think of would be to augment 5/4 by 361/360, to reach 361/288= 391.116 c. : then "5" . 3^2 . 2^3= 19^2
And to temper 96/95, I was happy with dissolving 513/512 and let the major third be generated by a chain of fifths.
But your solutions seem much more refined.

Another idea I had of a temperament based on 19 would be to temper 19683/19456 (or 3^9 / 19 . 2^10).
Without touching 19 nor 2, this is done by using a fifth of 1.498067943
Again a fractal version (I call "Falafile") does the same thing but recovers the exact differential coherence of "19-16 = 3", with a fifth of 1.498033484 = 699.6838456 c.
We may think that this goes a long way to arrive to something so close to 12ET, but actually 12ET minor thirds are very far from having the acoustic qualities we have here, at least on the level of difference tones.

One last different type of tuning, very chromatic as well 19 I thought of would be to use a period of a third of an octave (again it could be TOP-optimised), with an adapted "19/16" as generator. If we take more precisely 1.186929717 = 296.6814 c., we keep the "19 -16 = 3" coherence between many minor thirds and fifths, whose values here are 1.495437735 = 696.6814 c.
This leads to a strange 12-tone scale of three times three semitones103.3186 c. plus one 90.0442 c. in chain and eventually later up to a 93 tones tuning.

I see that you mention also a period of 400 cents for tempering both 96/95 and 361/360, but I did not grasp how you arrive at this period because of these commas.

- - - - - -
Jacques

Jacques wrote:

> Assuming the major third as 24/19 (here 403.316 c.), it dissolves also > 96/95
> and leads to a quasi-pythagorean tuning but based on pure 19/16s.

Provided that the "model" major thirds are 5/4 and not 24/19, which I find highly debatable for such a kind of temperament (and I wouldn't even think about that in this case). Many people would certainly agree with you about this possibility (especially the well-temperament promoters) but to me it sounds like comparing uncomparable amounts of mistuning. To be honest with you, I was not sure what you meant until I took my calculator and discovered that multiplying the two commas gave me the ordinary syntonic comma.

>Now in these same 513/512 tempering variants, defining the major third
> rather as 8 fourths - 2 octaves, as would be 8192/6561 in a 3-limit tuning
> (instead of previously 4 fifths - 2 octaves), would have the effect
> on these temperaments to dissolve both 361/360 and 1216/1215, instead of > 96/95.

This is the second of the three temperaments I talked about in my previous post. The original 5-limit version of this, the schismatic/Helmholtz temperament, tempers out the schisma of 32805/32768. For this tuning, the most favored generator was the 8th root of 10 (or 1/8 of the schisma wider than a pure fourth). This also raises the question if adding the 19th harmonic is worth the mistuning since the maximum amount of mistuning in the 5-limit schismatic temperament is just a fraction of a cent.

> About the other commas, to dissolve 1216/1215 if I had to, the simplest > way
> I could find was to only augment only 5/4 by 1216/1215... to reach 304/243
> = 387.738 c. : then "5" . 3^5 = 19 . 2^6

I would be very very doubtful whether I should say this "tempers out" the comma. You haven't tempered it out. :-) It's like if you said "To temper out 81/80, I would just widen the 5/4 by 81/80 and then 5/1 will be approximated by (3/2)^4".

> To dissolve 361/360, the simplest way I would think of would be to augment > 5/4
> by 361/360, to reach 361/288= 391.116 c. : then "5" . 3^2 . 2^3= 19^2

Ahum ... I'm very curious how you would map 19/1 then.

> And to temper 96/95, I was happy with dissolving 513/512
> and let the major third be generated by a chain of fifths.

Of course, many people, if they want to temper out the syntonic comma, are just happy about tempering the Pythag. comma and adding the 5/4 as another approximant. But you'll certainly admit that if you only temper out the syntonic comma and nothing else (which makes a 2D temperament instead of a 1D temperament), you'll get much less mistuning with less tones.

> Another idea I had of a temperament based on 19 would be to temper > 19683/19456
> (or 3^9 / 19 . 2^10).
> Without touching 19 nor 2, this is done by using a fifth of 1.498067943

This is the "model" version of the temperament and it makes both 3/1 and 19/3 mistuned by the same amount. If you want to have only one of the three main approximants (meaning 3/1, 19/1, and 19/3) mistuned by the "highest" amount, then at least for me personally, the best way is to use a generator of ~699.592 cents, which makes the fifth mistuned by ~2.36 cents and the 19/1 and 19/3 will both be mistuned by only half that amount. It's similar to the concept of 2/7-comma meantone where the major and minor thirds are both away by half the amount of mistuning in the fifth.

> I see that you mention also a period of 400 cents for tempering both 96/95 > and 361/360,
> but I did not grasp how you arrive at this period because of these commas.

When I tempered out 96/95 together with 361/360, I was also inevitably tempering out 128/125, which is the basis for 3-equal.

Hey, Jacques, I see you're really interested in this 19-limit topic. You know what? I'll take my small QBasic utility which I wrote about half a year ago, I'll reprogram the approximants from my original "2, 3, 5, 7" to "2, 3, 5, 19", and we'll see what it finds, right? FYI: There won't be any 4D factors in this list (like 96/95 or 361/360) so if you want me to try some of these as well, let me know and I'll do something about it. Okay, for now, let's assume we only want to find intervals smaller than the syntonic comma and we'll stop at the 80000th harmonic -- and just see what happens. I'll simply copy and paste the program's output into the message, which means there will also be factors that don't contain the number 19. I'll leave them there. But anyway, at the very beginning of this list, I'll show you the example with the syntonic comma in order you could understand the format in the case of a simple temperament like meantone. -- Okay, let's do that.

==============================

"-4 4 -1 0" = 81/80, SIZE IS 21.50629 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 1, 1), ( 0, 4)
BEST GENERATOR WITH PURE 2/1: 695.8104 CENTS.
HIGHEST ERROR: 6.144654 CENTS.

"-9 3 0 1" = 513/512, SIZE IS 3.378019 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 2, -1), ( 3, 3)
BEST GENERATOR WITH PURE 2/1: 499.0102 CENTS.
HIGHEST ERROR: .9651482 CENTS.

"-1 6 0 -2" = 729/722, SIZE IS 16.70397 CENTS.
PERIOD AND GENERATOR MAPPING:
( 2, 0), ( 3, 1), ( 8, 3)
BEST GENERATOR WITH PURE 2/1: 98.6142 CENTS.
HIGHEST ERROR: 3.340795 CENTS.

" 11 -4 -2 0" = 2048/2025, SIZE IS 19.55257 CENTS.
PERIOD AND GENERATOR MAPPING:
( 2, 0), ( 4, -1), ( 3, 2)
BEST GENERATOR WITH PURE 2/1: 494.1345 CENTS.
HIGHEST ERROR: 3.910514 CENTS.

" 8 3 0 -3" = 6912/6859, SIZE IS 13.32595 CENTS.
PERIOD AND GENERATOR MAPPING:
( 3, 0), ( 4, 1), ( 12, 1)
BEST GENERATOR WITH PURE 2/1: 299.734 CENTS.
HIGHEST ERROR: 4.441985 CENTS.

"-6 -5 6 0" = 15625/15552, SIZE IS 8.107279 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 0, 6), ( 1, 5)
BEST GENERATOR WITH PURE 2/1: 317.1153 CENTS.
HIGHEST ERROR: 1.474051 CENTS.

"-10 9 0 -1" = 19683/19456, SIZE IS 20.08199 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 1, 1), (-1, 9)
BEST GENERATOR WITH PURE 2/1: 699.5924 CENTS.
HIGHEST ERROR: 2.362587 CENTS.

"-15 8 1 0" = 32805/32768, SIZE IS 1.953721 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 2, -1), (-1, 8)
BEST GENERATOR WITH PURE 2/1: 498.2748 CENTS.
HIGHEST ERROR: .2298495 CENTS.

" 0 -10 5 1" = 59375/59049, SIZE IS 9.531577 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 2, -1), ( 0, 5)
BEST GENERATOR WITH PURE 3/1: 1019.329 CENTS.
HIGHEST ERROR: 1.733014 CENTS.

"-12 0 7 -1" = 78125/77824, SIZE IS 6.682981 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), ( 2, 1), ( 2, 7)
BEST GENERATOR WITH PURE 2/1: 385.2856 CENTS.
HIGHEST ERROR: 1.028151 CENTS.

" 2 9 -7 0" = 78732/78125, SIZE IS 13.39901 CENTS.
PERIOD AND GENERATOR MAPPING:
( 1, 0), (-1, 7), (-1, 9)
BEST GENERATOR WITH PURE 2/1: 443.0168 CENTS.
HIGHEST ERROR: 1.674876 CENTS.

another recurrent sequence beside meta-meantone (which is barely different than what you mention being 1.49453018048 leading to a 88 tone MOS)
http://anaphoria.com/meantone-mavila.PDF

is also is an unnamed one on page 14 of http://anaphoria.com/meruthree.pdf
which is close to what you are calling Melkis

--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Petr wrote :

> Hey, Jacques, I see you're really interested in this 19-limit > topic. You
> know what? I'll take my small QBasic utility which I wrote about > half a year
> ago, I'll reprogram the approximants from my original "2, 3, 5, 7" > to "2, 3,
> 5, 19", and we'll see what it finds, right? FYI: There won't be any 4D
> factors in this list (like 96/95 or 361/360) so if you want me to > try some
> of these as well, let me know and I'll do something about it.

Thanks Petr for sharing your lights on this subject ! and to do all this clarifying -

I am only beginning now to understand better these techniques and it will take me some more time to fully understand the potential uses of these expressions of period and generator mappings, that I never used myself, at least that way.
I understand well that some of my "solutions" were at the "Kirnberger I" level of tempering, that is almost none, and that what is acceptable for a schisma like 1216/1215 become less and less acceptable when reaching a comma such as 96/95...
I am glad that I found at least correct 513/512 and 19683/19456 tempering generators all by myself !
What amazes me here is that such different worlds as my own researches on difference tones and fractal series, and these linear temperaments, may be able to meet. Of course they share locally the same mecanisms, but also for more or less mysterious reasons they only meet on some very special musical applications.

Yes I must admit I find the 19th harmonic fascinating, and it has always been one of the basic ingredients of my photosonic disks. It has been allowing me to create many precious IJ models for both Eastern European and Indian music, and that's why 96/95, 361/360, 513/512 and 1216/1215 are basic commas in my practice, where I actually never temper them but play with them a lot, and this leads I believe to the same harmonies you would find in the tunings tempering them. Also I worked on differential-coherent versions of the 22 (or more) indian shrutis, and therefore of many indian ragas, that made good use of the 19 factor.
The "Dudon scales", that compile the divisors of n and d of any ratio, and their octave transpositions (named that way by Gene Ward Smith after a post of mine many years ago on that list), of those four 19-limit commas have proved to have musical applications in photosonic disks :
D(96/95) = 1 19 5 95-3
D(361 /360) = 1 9 19 5 45-361 3 15
D(513/512) = 1-513 9 19 171 3 27 57 (= nice diatonic minor scale from tone 3)
D(1216/1215) = 1 135 9 1215-19 5-81 45 3 405 27 15-243
(and they can be combined and transposed to produce quite many interesting scales)

So, back to temperaments, certainly it would be of some interest to know what temperings 4D commas 96/95, 361/360, 1216/1215 would suggest, with pure 2/1. But I don't want to bother you with this, unless you would find some special inspiration to do it.
Since you did it already and its seems to be like a game for you, what would interest me perhaps even more, is to understand how you manage in general to temper two commas together - you did it for several of those, but it still goes over my head...
What about 96/95 and 81/80, as an example =
"-4 4 -1 0" = 81/80, SIZE IS 21.50629 CENTS.
"5 1 -1 -1" = 96/95, SIZE IS 18.12827 CENTS.
I understand you want to minimize the highest error on the relevant primes.
But next, how do you know what should be their common period and generator ? Do you try every possibility ?

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Jacques

Kraig wrote :

> is also is an unnamed one on page 14 of http://anaphoria.com/> meruthree.pdf
> which is close to what you are calling Melkis
>

I don't know the reason why this link does not works for me, it says "404 - page not found"
May be in the meantime can you say what it is ?
The meantone-mavila.pdf link works.

BTW, I found myself the same ratio Erv calls "Mavila" long ago, and guess what, not only as a 4th degree, but as a 3rd degree recurrent sequence as well !

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Jacques