Temperaments where two consecutive superparticular ratios become equivalent

This is an interesting pattern that I've noticed recently.

A lot of the most common temperaments take two consecutive
superparticular ratios and make them equivalent. That is, they take
three consecutive harmonics and alter the middle one so that the
interval between it and the ends is the same. This would make for a
particularly useful tuning for a number of (I hope obvious) conceptual
and even psychoacoustic reasons.

The obvious canonical example of this is meantone, where 9/8 and 10/9
become the same. This means that the 8:9:10 intervallic structure
becomes 8:sqrt(10/8):10.

This pattern isn't limited to only meantone though, but it looks like
a lot of the most "successful" temperaments have used it. Doing the
same thing with 7:8:9 tempers out 64/63, which although it doesn't
look like it has a name of its own as a rank-3 temperament, is used in
pajara and dominant and so on.

Probably the most interesting thing I've noticed is that whereas
"meantone" was the holy grail of music a few hundred years ago, and it
involves merging two sizes of whole tone together, an arguable 21st
century "holy grail" is miracle temperament, which involves merging
two sizes of half step together (14:15:16). Although by itself merging
14:15:16 isn't actually miracle temperament, but rather marvel
temperament, getting rid of 225/224 is admittedly awfully common in a
LOT of rank-2 temperaments.

So a natural question in my mind is, is there some temperament where
if you merge two sizes of "quarter-tone" together, you get another
"holy grail" temperament?

Has anyone systematically explored all of these from 1:2:3 on up? Is
there some kind of pattern as to which ones produce the lowest error?
Or in general, might this approach lead to the discovery of new
temperaments that haven't really been delved into yet?

-Mike

Another idea I was just throwing around is to run this process on
20:21:22, which has only factors of 3, 5, 7, and 11 in it. This would
temper out 441/440, which I see is also tempered out in miracle.
Running this interval into Graham's temperament finder leads to a lot
of unfamiliar looking temperaments beyond that and I see it doesn't
have a name.

The other question is, what happens if the pattern is extended?

24:25:26 has factors of 2, 3, 5, and 13 in it, and tempers out 625/624.
25:26:27 also has factors of 2, 3, 5, and 13 in it, and tempers out 676/675.
26:27:28 has factors of 2, 3, 7, and 13 in it, and tempers out 729/728.

(note the pattern: superparticular triad x:y:z tempers out y^2/(y^2-1)
if x:y is equated to y:z)

Tempering out all of those yields a rank-3 "Catakleismic++"
temperament, which encapsulates 34, 53, and 72-tet and has an error of
0.253 cents. See here:
http://x31eq.com/cgi-bin/rt.cgi?ets=34_72_53&error=0.770&limit=13&invariant=0_0_1_0_6_5_22_0_14_1_0_1_-3_0_0

If you treated it as a 2.3.5.7.13 subgroup temperament, it would be rank-2.

Anyway I'm not good enough with the math to know if this is really
worth anything but it seems to be an interesting approach.
Conceptually, it's certainly a useful one. It might also be useful to
break the harmonic series up into different "blocks" and apply this
algorithm separately to each one. Or perhaps try for
non-superparticular triads, but intervals that are 2 or 3 harmonics
equidistant, etc.

I'm not sure how this applies to other extremely accurate
temperaments, like schismatic, which don't seem to follow this pattern
at all.

-Mike

On Sun, May 23, 2010 at 3:02 AM, Mike Battaglia <battaglia01@...> wrote:
> This is an interesting pattern that I've noticed recently.
>
> A lot of the most common temperaments take two consecutive
> superparticular ratios and make them equivalent. That is, they take
> three consecutive harmonics and alter the middle one so that the
> interval between it and the ends is the same. This would make for a
> particularly useful tuning for a number of (I hope obvious) conceptual
> and even psychoacoustic reasons.
>
> The obvious canonical example of this is meantone, where 9/8 and 10/9
> become the same. This means that the 8:9:10 intervallic structure
> becomes 8:sqrt(10/8):10.
>
> This pattern isn't limited to only meantone though, but it looks like
> a lot of the most "successful" temperaments have used it. Doing the
> same thing with 7:8:9 tempers out 64/63, which although it doesn't
> look like it has a name of its own as a rank-3 temperament, is used in
> pajara and dominant and so on.
>
> Probably the most interesting thing I've noticed is that whereas
> "meantone" was the holy grail of music a few hundred years ago, and it
> involves merging two sizes of whole tone together, an arguable 21st
> century "holy grail" is miracle temperament, which involves merging
> two sizes of half step together (14:15:16). Although by itself merging
> 14:15:16 isn't actually miracle temperament,  but rather marvel
> temperament, getting rid of 225/224 is admittedly awfully common in a
> LOT of rank-2 temperaments.
>
> So a natural question in my mind is, is there some temperament where
> if you merge two sizes of "quarter-tone" together, you get another
> "holy grail" temperament?
>
> Has anyone systematically explored all of these from 1:2:3 on up? Is
> there some kind of pattern as to which ones produce the lowest error?
> Or in general, might this approach lead to the discovery of new
> temperaments that haven't really been delved into yet?
>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This is an interesting pattern that I've noticed recently.
>
> A lot of the most common temperaments take two consecutive
> superparticular ratios and make them equivalent. That is, they take
> three consecutive harmonics and alter the middle one so that the
> interval between it and the ends is the same.

If you look at mediants between 1/1 and n/(n-1), you get
(n+1)/n, (n+2)/(n+1) ... with ratios between them of
P^2/(P^2-1) for the various integers P. You mentioned the
squares produced by P = 8, 9, and 15, which are 5 or 7-limit. There's also P = 49, which gives 2401/2400, where the numerator is a 4th power, not just a square. Going up to the 11th limit gives us more:
100/99, 121/120, 441/440, 3025/3024, 9801/9800. There are more such patterns involving figurate numbers, starting from where the numerator is a triangular rather than a square number: 55/54, 66/65, 78/77, 91/90, 105/104, 325/324, 351/350, 2080/2079 for the 13-limit. By a theorem of Stormer, for any p-limit there are only a finite number of superparticular ratios of any kind, and so looking at all of them really makes the most sense. However, it doesn't hurt to notice that ratios between successive low-limit superparticulars, such as 81/80, 225/224, 2401/2400 or 9801/9800 seem to be specially magical.

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

> However, it doesn't hurt to notice that ratios between successive low-limit superparticulars, such as 81/80, 225/224, 2401/2400 or 9801/9800 seem to be specially magical.
>

I didn't say what I intended, which is ratios between a superparticular of the form n^/(n^2-1) and an adjacent low-limit superparticular seem to have that extra mojo. (9/8)/(10/9) = 81/80,
(15/14)/(16/15) = 225/224 where a square numerator is adjacent to a triangular numerator, and (49/48)/(50/49) = 2401/2400, (99/98)/(100/99) = 9801/9800 where you'd have to drag in other figurate numbers if you want to keep worrying about that. In any case, lot's more of these out there but you rapidly get into the area of microtempering.

On Sun, May 23, 2010 at 4:12 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > However, it doesn't hurt to notice that ratios between successive low-limit superparticulars, such as 81/80, 225/224, 2401/2400 or 9801/9800 seem to be specially magical.
> >
>
> I didn't say what I intended, which is ratios between a superparticular of the form n^/(n^2-1) and an adjacent low-limit superparticular seem to have that extra mojo. (9/8)/(10/9) = 81/80,
> (15/14)/(16/15) = 225/224 where a square numerator is adjacent to a triangular numerator, and (49/48)/(50/49) = 2401/2400, (99/98)/(100/99) = 9801/9800 where you'd have to drag in other figurate numbers if you want to keep worrying about that. In any case, lot's more of these out there but you rapidly get into the area of microtempering.

The triangular numbers show up as the difference between two
superparticulars that are one step away from being adjacent? so take
8:9:10:11 - the difference between 8:9 and 10:11 is going to be 45/44,
the numerator of which is 9*10/2 or T(9).

I wonder if there's any psychoacoustic significance to any of this. I
wonder if the particular way that the harmonic series is warped has
some sort of effect here on the perception of the temperament. I
suppose that on an intuitive level, it clearly does (flat fifths have
a certain "sound" to them, wide fifths have a different one, etc).

-Mike

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

> I didn't say what I intended, which is ratios between a
> superparticular of the form n^/(n^2-1) and an adjacent low-limit

Can you check that formula?

Is this the jumping jacks formula?

-Carl

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

> The triangular numbers show up as the difference between two
> superparticulars that are one step away from being adjacent? so take
> 8:9:10:11 - the difference between 8:9 and 10:11 is going to be 45/44,
> the numerator of which is 9*10/2 or T(9).

You can look at it in terms of mediants: 7/4, 9/5, 11/6 ... with 2/1, leading to 36/35, 55/54... from below, and 9/4, 11/5, 13/6 ... with 2/1, leading to 45/44, 66/65, ... from above. Then there's 7/5, 10/7, 13/9 with 3/2, blah blah.

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

> I wonder if there's any psychoacoustic significance to any of this.
> I wonder if the particular way that the harmonic series is warped
> has some sort of effect here on the perception of the temperament.
> I suppose that on an intuitive level, it clearly does (flat fifths
> have a certain "sound" to them, wide fifths have a different one,
> etc).

The significance is indirect. Temperaments are good if they
provide many consonances with few notes, without introducing a
lot of tuning error. Tempering out a comma makes it a "unison
vector", which cuts the infinite prime-limit lattice into
periodic sections. The simpler a comma's ratio, the closer it
lies to the lattice's origin and the narrower the periodic
slices it will define. So "few notes" is satisfied by simple
commas because they provide access to the entire lattice via
periodic slices or tiles containing fewer lattice points.
Low tuning error, on the other hand, is satisfied by commas
that are small in magnitude, since it is their magnitude that
will vanish in the temperament. It's straightforward that
superparticular ratios tend to have the smallest magnitude
for a given complexity. Probably you knew all this and are
wondering why cutting ratios of the form (n+2)/n seems so
fruitful. A few years ago, I had an inkling why this might be
so, but it escapes me at the moment.

-Carl

>"The significance is indirect. Temperaments are good if they provide many consonances with few notes, without introducing a lot of tuning error."
What do you mean by "tuning error"? Does that describe how much comma-tic "build up" there is?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > I didn't say what I intended, which is ratios between a
> > superparticular of the form n^2/(n^2-1) and an adjacent low-limit

> Is this the jumping jacks formula?

The infamous jumping jacks. Yes, 81/80, 225/224 and 2401/2400 are jumping jacks, "jumping" between two jacks.

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

> The infamous jumping jacks. Yes, 81/80, 225/224 and 2401/2400 are jumping jacks, "jumping" between two jacks.
>

For those playing along at home, suppose n>p+1 is a p-limit integer, and n-1 and n+1 are also, so we have three successive p-limit integers: n-1, n, n+1. Then n^2/(n^2-1) = (n/(n-1))/(n+1)/n is a "jumping jack". The restriction n>p+1 is just to weed out the easy, small stuff, and is kind of arbitrary.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> So a natural question in my mind is, is there some temperament where
> if you merge two sizes of "quarter-tone" together, you get another
> "holy grail" temperament?

Ennealimmal does this with 48:49:50 I believe, and 49/48
is about 36 cents...

By shrinking this from whole to half to quarter, you're just
asking for more accurate temperaments. And you could certainly
argue that that's the future of music. But others would argue
that perhaps more variety is the future. Or more dramatic puns
(which means less accurate temperaments). Anyway, there's a
limit to the error we can perceive, so it runs out pretty quick.
Ennealimmal is already basically there.

-Carl

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

> The significance is indirect. Temperaments are good if they
> provide many consonances with few notes, without introducing a
> lot of tuning error. Tempering out a comma makes it a "unison
> vector", which cuts the infinite prime-limit lattice into
> periodic sections. The simpler a comma's ratio, the closer it
> lies to the lattice's origin and the narrower the periodic
> slices it will define. So "few notes" is satisfied by simple
> commas because they provide access to the entire lattice via
> periodic slices or tiles containing fewer lattice points.
> Low tuning error, on the other hand, is satisfied by commas
> that are small in magnitude, since it is their magnitude that
> will vanish in the temperament.

Carl, that right there is the most concise and accessible explanation of temperaments I've EVER read. This belongs in the Encyclopedia, and probably on the Xenharmonic Wiki as well.

-Igs

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

> Carl, that right there is the most concise and accessible
> explanation of temperaments I've EVER read. This belongs in
> the Encyclopedia, and probably on the Xenharmonic Wiki as well.

Feel free to put it up, or let me know if you have an idea as
to to which page it should go on. I registered with the wiki
recently and have been correcting a few typos here and there...

-Carl

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

> Feel free to put it up, or let me know if you have an idea as
> to to which page it should go on. I registered with the wiki
> recently and have been correcting a few typos here and there...

Is there an easy way to back the wiki up? I'm afraid of what might happen if the wiki farm goes bye-bye, though I checked and at least the Wayback Machine seems to be keeping some sort of record.

On Sun, May 23, 2010 at 4:52 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The triangular numbers show up as the difference between two
> > superparticulars that are one step away from being adjacent? so take
> > 8:9:10:11 - the difference between 8:9 and 10:11 is going to be 45/44,
> > the numerator of which is 9*10/2 or T(9).
>
> You can look at it in terms of mediants: 7/4, 9/5, 11/6 ... with 2/1, leading to 36/35, 55/54... from below, and 9/4, 11/5, 13/6 ... with 2/1, leading to 45/44, 66/65, ... from above. Then there's 7/5, 10/7, 13/9 with 3/2, blah blah.

Ah, that makes sense. So getting successive mediants to 2/1 basically
yields mediants that are based off of the triangular numbers, as gene
said. I was basically getting successive mediants to 1/1.

Perhaps another interesting approach to take would be to get the
intervals between successive convergents of a certain interval, say
7/4. That might be even more psychoacoustically useful.

-Mike

On Thu, May 27, 2010 at 10:54 AM, Mike Battaglia <battaglia01@...> wrote:
> On Sun, May 23, 2010 at 4:52 AM, genewardsmith
> <genewardsmith@...> wrote:
>> You can look at it in terms of mediants: 7/4, 9/5, 11/6 ... with 2/1, leading to 36/35, 55/54... from below, and 9/4, 11/5, 13/6 ... with 2/1, leading to 45/44, 66/65, ... from above. Then there's 7/5, 10/7, 13/9 with 3/2, blah blah.
>
> Ah, that makes sense. So getting successive mediants to 2/1 basically
> yields mediants that are based off of the triangular numbers, as gene
> said.

Er, you're Gene. I thought for a second I was replying to Carl.

-Mike

On Sun, May 23, 2010 at 5:47 AM, Carl Lumma <carl@...> wrote:
>
> Ennealimmal does this with 48:49:50 I believe, and 49/48
> is about 36 cents...
>
> By shrinking this from whole to half to quarter, you're just
> asking for more accurate temperaments. And you could certainly
> argue that that's the future of music. But others would argue
> that perhaps more variety is the future. Or more dramatic puns
> (which means less accurate temperaments). Anyway, there's a
> limit to the error we can perceive, so it runs out pretty quick.
> Ennealimmal is already basically there.
>
> -Carl

More variety in the sense that no one temperament predominates?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Ennealimmal does this with 48:49:50 I believe, and 49/48
> > is about 36 cents...
> > By shrinking this from whole to half to quarter, you're just
> > asking for more accurate temperaments. And you could certainly
> > argue that that's the future of music. But others would argue
> > that perhaps more variety is the future. Or more dramatic puns
> > (which means less accurate temperaments). Anyway, there's a
> > limit to the error we can perceive, so it runs out pretty quick.
> > Ennealimmal is already basically there.
> > -Carl
>
> More variety in the sense that no one temperament predominates?

Yes. -C.