Parizekmic temperament?

On the basis of this:

http://launch.dir.groups.yahoo.com/group/tuning/message/77917

I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing. Jacques also is involved, as he concocted a rational version of Parizekmic[5].

If anyone wants to play:

! parizekmic5.scl
!
Parizekmic[5], (676/675 tempering), POTE tuning
5
!
248.88923
453.33232
702.22155
951.11077
2/1

! parizekmic9.scl
!
Parizekmic[9] (676/675 tempering), POTE tuning
9
!
204.44309
248.88923
453.33232
497.77845
702.22155
746.66768
951.11077
995.55691
2/1

! parizekmic14.scl
!
Parizekmic[14] (676/675 tempering), POTE tuning
14
!
44.44614
204.44309
248.88923
293.33536
453.33232
497.77845
657.77541
702.22155
746.66768
906.66464
951.11077
995.55691
1155.55386
2/1

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> On the basis of this:
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/77917
>
> I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing.

A related 2.3.5.13 temperament tempers out 676/675 and 3159/3125; it slices the generator in fourth to an approximate 26/25. Adding 126/125 and 196/195 to 676/675 gives a 2.3.5.7.13 version of this temperament, and adding 144/143 or 176/175 gives your choice of two full 13-limit versions. Here's a 39-note MOS with generator 7/135 octaves; you can interpret it as a no-11s scale if you like, or take it up to the 17-limit. It has, among many other things, 27 1-13/10-3/2 triads.

! quadraparizekmic39.scl
!
Quadraparizekmic[39]; 7/135 octave generator
39
!
17.77778
62.22222
80.00000
124.44444
142.22222
186.66667
204.44444
248.88889
266.66667
311.11111
328.88889
373.33333
391.11111
435.55556
453.33333
497.77778
515.55556
560.00000
577.77778
622.22222
640.00000
684.44444
702.22222
746.66667
764.44444
808.88889
826.66667
871.11111
888.88889
933.33333
951.11111
995.55556
1013.33333
1057.77778
1075.55556
1120.00000
1137.77778
1182.22222
2/1

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> On the basis of this:
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/77917
>
> I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing. Jacques also is involved, as he concocted a rational version of Parizekmic[5].

That's cool, how did you know ?
(but which one do you mean ?)
I like POTE tunings !
(in french "un POTE" is a slang for "a fellow friend")

>
> If anyone wants to play:
>
> ! parizekmic5.scl
> !
> Parizekmic[5], (676/675 tempering), POTE tuning
> 5
> !
> 248.88923
> 453.33232
> 702.22155
> 951.11077
> 2/1
>

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

> > I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing. Jacques also is involved, as he concocted a rational version of Parizekmic[5].
>
> That's cool, how did you know ?
> (but which one do you mean ?)

Due to the wonders of Scala, I was able to find that the following scale is very close to Parizekmic[5]:

! neutr_pent2.scl
!
Quasi-Neutral Pentatonic 2, 15/13 x 52/45 in each trichord, after Dudon
5
!
15/13
4/3
3/2
45/26
2/1

> I like POTE tunings !
> (in french "un POTE" is a slang for "a fellow friend")

It's a pretty friendly sort of pure-octaves tuning in my opinion. Well behaved in various ways.

I see, no idea from where Manuel got this prehistoric thing
that probably inspired this zig-zaguee 5D extension :

! bala_semifo.scl
!
Burkinabe typical semifourth pentatonic bala feast scale
12
!
135/104
26/23
104/69
207/160
13/10
45/26
104/69
2/1
69/40
26/15
52/23
2/1
! harmonic temperament out of primes 3, 5, 13, 23, Dudon 1987

Now concerning Parizeksmic, I have better than 5 notes, with the Mima 7th degree algorithm, that verifies :
x^8 = x + 2
(x = 1.1544230572469 or 248.6064241458 cents)

Starting from the previous pentatonic it goes
[117 135 156 180 208 240 277 320 369 426 492 568 656 757 874 1009 1164 1344 1552 1792
2069 2388 2757 3182 3672 4240 4896 5653 6526 ...
which should go a long way without problem (a 29 notes MOS like in here is a nice choice).
Quite interesting are the just ratios that spring out naturally from the sequences such as
x^2 ~4/3 of course (156/117)
x^3 ~ 20/13 (320/208)
x^4 ~16/9 (208/117)
x^12 ~28/5 (1792/320)
and also x^7 ~41/15 (656/240)
- - - - - - -
Jacques

Gene wrote :

> Due to the wonders of Scala, I was able to find that the following > scale is very close to Parizekmic[5]:
>
> ! neutr_pent2.scl
> !
> Quasi-Neutral Pentatonic 2, 15/13 x 52/45 in each trichord, after > Dudon
> 5
> !
> 15/13
> 4/3
> 3/2
> 45/26
> 2/1

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

> Now concerning Parizeksmic, I have better than 5 notes, with the Mima
> 7th degree algorithm, that verifies :
> x^8 = x + 2
> (x = 1.1544230572469 or 248.6064241458 cents)
>
> Starting from the previous pentatonic it goes
> [117 135 156 180 208 240 277 320 369 426 492 568 656 757 874 1009
> 1164 1344 1552 1792
> 2069 2388 2757 3182 3672 4240 4896 5653 6526 ...
> which should go a long way without problem (a 29 notes MOS like in
> here is a nice choice).

Your generator is actually the largest root in absolute value, so it shouldn't give rise to problems if you start it off right. That's a heck of a recurrence for musical purposes, it looks to me.

On Fri, Jan 7, 2011 at 2:14 PM, genewardsmith
<genewardsmith@...> wrote:
>
> On the basis of this:
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/77917
>
> I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing. Jacques also is involved, as he concocted a rational version of Parizekmic[5].

I guess it's up to Petr, ultimately, but I wish we could give this one
some kind of programmatic name. The 9-note MOS sounds like you're on
some kind of tropical island off the coast of Barbados playing
jubilant sun-music with the natives or something. More specifically, I
feel like it takes the "island" feel of 5-equal and expands it out
into diatonic and chromatic versions. I like island temperament
myself, but it's up to Petr since he seems to have been the first to
come up with it.

Furthermore, 24-equal does this one really well - so this could be a
great start for the average guy out there to delve into regular
mapping and use 24-tet to boot.

This really does sound like some kind of 5-equal meets meantone scale,
there are lots of parallels:

> Parizekmic[5], (676/675 tempering), POTE tuning
> 5
> !
> 248.88923
> 453.33232
> 702.22155
> 951.11077
> 2/1

This parallels meantone[5], and has a very 5-equal-ish feel to it.
Probably because the 13/10, which bisects the fourth, also has a
fourth under it, so you hear a kaleidoscopic mix of 4/3's and 3/2's
everywhere you go.

> ! parizekmic9.scl
> !
> Parizekmic[9] (676/675 tempering), POTE tuning
> 9
> !
> 204.44309
> 248.88923
> 453.33232
> 497.77845
> 702.22155
> 746.66768
> 951.11077
> 995.55691
> 2/1

^This is the "diatonic" version, and also has kind of a "tonal" feel
to it. An "island" tonal feel. It's kind of hard to find instruments
whereby this works - marimbas and such seem to work really well.
There's a pythagorean pentatonic scale is contained in here as well.

24-equal does this within 4 cents of error for pretty much everything,
but something interesting to consider is to deliberately detune
everything such that the 50-cent leading tone is made sharper, maybe
closer to 70 cents where it's probably more effective. Alternately,
maybe we need to find a new "optimal" leading tone - maybe the 70 cent
interval in 19-equal works so well because of the intervals it's
approximating, and now that we're using 10:13:15 for our base triad
instead of 4:5:6, there will be a new ideal leading tone.

> ! parizekmic14.scl
> !
> Parizekmic[14] (676/675 tempering), POTE tuning
> 14
> !
> 44.44614
> 204.44309
> 248.88923
> 293.33536
> 453.33232
> 497.77845
> 657.77541
> 702.22155
> 746.66768
> 906.66464
> 951.11077
> 995.55691
> 1155.55386
> 2/1

And this is the chromatic version, which I haven't messed around with much yet.

-Mike

On Mon, Jan 17, 2011 at 2:52 PM, Mike Battaglia <battaglia01@...> wrote:
>
> 24-equal does this within 4 cents of error for pretty much everything,
> but something interesting to consider is to deliberately detune
> everything such that the 50-cent leading tone is made sharper, maybe
> closer to 70 cents where it's probably more effective. Alternately,
> maybe we need to find a new "optimal" leading tone - maybe the 70 cent
> interval in 19-equal works so well because of the intervals it's
> approximating, and now that we're using 10:13:15 for our base triad
> instead of 4:5:6, there will be a new ideal leading tone.

Also, to add to this, it might be good to temper out 91/90 here as
well, and extend this to 2.3.7.13/10 as well; if you could work it out
such that the subminor triads are decent matches for 6:7:9, and the
supermajor triads are still decent approximations to 10:13:15, the
whole thing might end up being even more colorful sounding. As for
naming, maybe we could give all of these individual sub-temperaments
names, and the whole thing could fall under the Parizekmic clan or
Parizekmic family or something. Since the basic generator subdivides
4/3, I think that there's a lot of fertile music territory here, and
I'm pretty sure some kind of ideal related subgroup temperament that
fits this "island" theme I keep hearing.

A systematic subgroup-based exploration of generators that subdivide
the fourth and fifth might be a good idea as well.

-Mike

I always thought this was Bug temperament, or maybe Beep? At any rate, it seems like they all arrive at very similar scales. I've used the 14-EDO, 19-EDO, and 24-EDO versions and they don't sound very much different from one another. You can translate music from one to the next without a lot of change.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Jan 7, 2011 at 2:14 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > On the basis of this:
> >
> > http://launch.dir.groups.yahoo.com/group/tuning/message/77917
> >
> > I'm tentatively calling the linear 2.3.13/5 temperament tempering out 676/675 "parizekmic", though that apparently wasn't what Petr was doing. Jacques also is involved, as he concocted a rational version of Parizekmic[5].
>
> I guess it's up to Petr, ultimately, but I wish we could give this one
> some kind of programmatic name. The 9-note MOS sounds like you're on
> some kind of tropical island off the coast of Barbados playing
> jubilant sun-music with the natives or something. More specifically, I
> feel like it takes the "island" feel of 5-equal and expands it out
> into diatonic and chromatic versions. I like island temperament
> myself, but it's up to Petr since he seems to have been the first to
> come up with it.
>
> Furthermore, 24-equal does this one really well - so this could be a
> great start for the average guy out there to delve into regular
> mapping and use 24-tet to boot.
>
> This really does sound like some kind of 5-equal meets meantone scale,
> there are lots of parallels:
>
> > Parizekmic[5], (676/675 tempering), POTE tuning
> > 5
> > !
> > 248.88923
> > 453.33232
> > 702.22155
> > 951.11077
> > 2/1
>
> This parallels meantone[5], and has a very 5-equal-ish feel to it.
> Probably because the 13/10, which bisects the fourth, also has a
> fourth under it, so you hear a kaleidoscopic mix of 4/3's and 3/2's
> everywhere you go.
>
> > ! parizekmic9.scl
> > !
> > Parizekmic[9] (676/675 tempering), POTE tuning
> > 9
> > !
> > 204.44309
> > 248.88923
> > 453.33232
> > 497.77845
> > 702.22155
> > 746.66768
> > 951.11077
> > 995.55691
> > 2/1
>
> ^This is the "diatonic" version, and also has kind of a "tonal" feel
> to it. An "island" tonal feel. It's kind of hard to find instruments
> whereby this works - marimbas and such seem to work really well.
> There's a pythagorean pentatonic scale is contained in here as well.
>
> 24-equal does this within 4 cents of error for pretty much everything,
> but something interesting to consider is to deliberately detune
> everything such that the 50-cent leading tone is made sharper, maybe
> closer to 70 cents where it's probably more effective. Alternately,
> maybe we need to find a new "optimal" leading tone - maybe the 70 cent
> interval in 19-equal works so well because of the intervals it's
> approximating, and now that we're using 10:13:15 for our base triad
> instead of 4:5:6, there will be a new ideal leading tone.
>
> > ! parizekmic14.scl
> > !
> > Parizekmic[14] (676/675 tempering), POTE tuning
> > 14
> > !
> > 44.44614
> > 204.44309
> > 248.88923
> > 293.33536
> > 453.33232
> > 497.77845
> > 657.77541
> > 702.22155
> > 746.66768
> > 906.66464
> > 951.11077
> > 995.55691
> > 1155.55386
> > 2/1
>
> And this is the chromatic version, which I haven't messed around with much yet.
>
> -Mike
>

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

> I guess it's up to Petr, ultimately, but I wish we could give this one
> some kind of programmatic name.

Petr didn't come up with it, and didn't seem to have an opinion on the name, so it's really still open. I don't object to "island"--anyone else have a comment?

Although my contributions will probably be much rarer these days, I'll try to get in contact at least for some time again.

Gene wrote:

> Petr didn't come up with it, and didn't seem to have an opinion on the > name, so it's really still open. I don't
> object to "island"--anyone else have a comment?

I do have an oppinion on the name but I don't have any objections to the actual temperament.
I've never properly understood what has motivated you to use my name for this temperament. If it's only because of the 676/675 interval, then I'm doubtful about how much the two temperaments really do have to do with each other (I mean, your and mine).
As I've said earlier, if you strip out the possibility to map 5, you take away one of the primary targets I was aiming for when I made that tuning.

BTW: This is what I was refferring to in the old message you were quoting:
http://dl.dropbox.com/u/8497979/ta13b.mp3

Petr

On Mon, Jan 17, 2011 at 5:23 PM, cityoftheasleep
<igliashon@...> wrote:
>
> I always thought this was Bug temperament, or maybe Beep? At any rate, it seems like they all arrive at very similar scales. I've used the 14-EDO, 19-EDO, and 24-EDO versions and they don't sound very much different from one another. You can translate music from one to the next without a lot of change.
>
> -Igs

It isn't bug/beep unless you eliminate 27/25, meaning that either the
syntonic comma or the diatonic half step is going to have to be
reversed. In this case, we aren't trying to represent 5 at all, except
indirectly as part of 13/10. The 24-EDO one is really far from beep,
in particular.

I think the 24-EDO version sounds a lot better; 29-EDO is good too.
19-EDO's is passable. I was pretty excited with 14-EDO, but I didn't
like the ultramajor triads as much as with 24-EDO. 19-EDO was better,
but I think that 24-EDO's are so much better that the 5 extra notes
don't make much of a difference.

There are lots of 5-limit commas that could vanish here. 81/80 seems
to be a decent one; Gene had some other ideas as well. This makes the
fifths flat, and 19-equal and 24-equal are good choices for this.
14-tet is economical, but I don't get the same bright, sunny feel from
the 9-note MOS that I do with 24-tet. Trying to work in the 7-limit
and crossbreed this with 91/90 accomplishes the same thing.

-Mike

On 18 January 2011 07:11, Mike Battaglia <battaglia01@...> wrote:

> It isn't bug/beep unless you eliminate 27/25, meaning that either the
> syntonic comma or the diatonic half step is going to have to be
> reversed. In this case, we aren't trying to represent 5 at all, except
> indirectly as part of 13/10. The 24-EDO one is really far from beep,
> in particular.

There is, at least, a theme that Bug, Beep, and Bogey all start with a
B. So Barbados belongs better than Island.

Graham

I was going to reply to this thread last week, but I was so happy with what I found that it took some of my time to play with it.
To resume, I did several experiences around these ideas, that arrived to very different things :

I used for that the recurrent sequence x^8 = x + 2
(x = 1.1544230572469 or 248.6064241458 cents) that I renamed, I think now definitively, "Bala", because the first five notes of the seed itself contains the most evident model I know of the African pentatonic balafon tuning :
[117 135 156 180 208 240 277 320 369 426 492 568 656 757 874 1009 1164 1344 1552 1792]
... continuing in a 29 notes MOS with 2069 2388 2757 3182 3672 4240 4896 5653 6526]
(any group of 5 consecutive terms in any Bala sequence will work for a superb pentatonic Balafon tuning)
This precise sequence is quite interesting as it perfectly integrates the 7/5, 14/13, 28/15 and 21/20 ratios, in addition to the more local 15/13, 13/10, 4/3 and 16/9 ratios (example 1792/320 here = 28/5).
But at no point this sequence passes near by a power of 2, which would have defined a mapping for a 3. 5. 7. 13 temperament. If I want to take an image, it's something like a house with a bunch of orphan kids playing together (5, 7, 13, 15, 39, 41, 45, etc. and others), while nobody knows what happened with the parents (1, 3, 9...).
After the 6/29 notes MOS of the sequence a relatively small interval increases the precedents every 29 generations, and the first 2^n approximation is found only 83 generators after the "7" point, himself 12 generators after the "5" point. And the next more satisfying MOS after 29 if my calculations are correct, is found with 251 udo (in which the generator is 52 steps).
So instead, the first direction I tested was to combine two Bala sequences, one of which would bring back the "missing fundamentals" of the initial sequence.
It is very easy to generate such a sequence from the first one, as many -c properties would offer it, such as :
15 - 13 = 2
7 - 5 = 2
13 - 10 = 3, etc.
but the most precise technique I found was to use the Fibonacci numbers triad 13 - 8 = 5 to define a minor sixth above a term (n), by the difference between 4 times the term (n - 3) generations and the term (n ),
ex. in 208 240 277 320,
(4*208) - 320 = 512 = (8/5)*320.
Applying this simple -c algorithm the associated ribbon sequence goes :
216 250 288 332 384 443 512 ... etc., which basically transposes the first scale by 12/13, along with extraordinary precise 16/13, 16/15, 8/5, 6/5 etc. intervals and complements.

This second sequence divides each step of the 29 notes cycle of the first sequence in approximatively 2/3 + 1/3 step, this means the whole thing could be roughly approximated by a 87-edo.
This is a good example of what I call a "ribbon temperament" which could be translated - I think - as a rank 3 temperament using the compilation of a linear temperament, and its single transposition by a complementary interval.
The whole thing is incredible, as it combines African pentatonics, Eolian modes, Ethiopian and Japanese scales, Pelogs, Mohajiras, etc., etc., in superb kaleidoscopic harmonies with many [9:6:4] chords.
Even with only 12 notes such a scale I experimented with delight is :

! bala_ribbon.scl
!
Parizekmic scale based on a double Bala sequence
12
!
25/24
9/8
6/5
13/10
4/3
83/60
3/2
8/5
26/15
9/5
39/20
2/1
! Interleaved Bala -c recurrent sequences, x^8 - 2 = x
; = 1.1544230572469 or 248.6064241458 cents (5-29-251 notes MOS)
! first sequence [117 135 156 180 208 240 (277) 320 369 ...
! second sequence transposed by 4 - x^3 (~32/13) on black keys :
! [108 125 144 166 192... (125 = C#)
! Dudon 2011

(note that 277/240, a 15/13 equivalent instead of 9/8 brings a possible variation, as shown in the next tunings ; I will also certainly propose a tuning with more notes next)

Now when I found this idea of temperament I had not noticed, because I didn't went carefully through Petr's original message, that it arrives - I suppose - more or less to Petr's 3D system. This is a truly fantastic temperament !

Then I went another direction to experiment the single semifourth chain such as suggested by Gene, that I also extended to reach the zone where 7 appears. In my sense the african grounds then meets with Turkish thirds there (plenty of 56/45 intervals) and Hijaz modes along with 15/14, also a few pure 7/5 here and there, and both systems would have superb microtonal Jazz harmonies and chords (like [28:21:15]...) etc., but with only discrete septimal color.
Here are two of these experiments (note that they use another sequence in theory, without discernible difference)

! balasept-under.scl
!
5.7.13.15.21 tuning based on a single Balasept sequence
12
!
21/20
277/240
97/80
13/10
4/3
7/5
3/2
97/60
26/15
28/15
39/20
2/1
! Balasept -c recurrent sequence, x^15 = 4x + 4
; x = 1,15440763765962 or 248.58329996 cents (5 then 29 notes MOS)
! first segment [117 (135) 156 180 208 240 277 320 ...
! second and higher segment on black keys :
! [252 291 336 388 448 ... (252 = C#)

! balasept-above.scl
!
5.7.13.15 tuning based on a single Balasept sequence
12
!
517/480
277/240
597/480
13/10
4/3
689/480
3/2
53/32
26/15
28/15
39/20
2/1
! Balasept -c recurrent sequence, x^15 = 4x + 4
; x = 1,15440763765962 or 248.58329996 cents (5 then 29 notes MOS)
! first segment [117 (135) 156 180 208 240 277 320 ...
! second and higher segment on black keys :
! [448 517 597 689 795 ... (517 = C#)

So back to the names discussion I can only agree that what Gene developped, and Petr's original idea, from the glimpse I had of it, are two completly different temperaments (that's also what Gene precised from the beginning).
It belongs to Petr to say if he agrees with that, but I would advocate for calling "Parizekmic" Petr's original system only.
(... except that he has invented, it seems, many other "Parizekmic temperaments" of the same quality !)
Concerning a name for these last Gene's POTE 676/675 linear temperaments or other versions of similar semifourth generators, and perhaps more true to its roots, "Bala" (current african name for the pentatonic balafon), or other references to this pentatonic tuning would be my suggestion.
- - - - - - -
Jacques

Petr wrote :

> Although my contributions will probably be much rarer these days, > I'll try
> to get in contact at least for some time again.
>
> Gene wrote:
>
> > Petr didn't come up with it, and didn't seem to have an opinion > on the
> > name, so it's really still open. I don't
> > object to "island"--anyone else have a comment?
>
> I do have an oppinion on the name but I don't have any objections > to the
> actual temperament.
> I've never properly understood what has motivated you to use my > name for
> this temperament. If it's only because of the 676/675 interval, > then I'm
> doubtful about how much the two temperaments really do have to do > with each
> other (I mean, your and mine).
> As I've said earlier, if you strip out the possibility to map 5, > you take
> away one of the primary targets I was aiming for when I made that > tuning.
>
> BTW: This is what I was refferring to in the old message you were > quoting:
> http://dl.dropbox.com/u/8497979/ta13b.mp3
>
> Petr

> (Petr wrote) :
>
> BTW: This is what I was refferring to in the old message you were > quoting:
> http://dl.dropbox.com/u/8497979/ta13b.mp3
>
> Petr

...Kaleidoscopic !
( I confirm what I said )
I am very fond of these pluri-cultural bridges.
Thanks for sharing !
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> It isn't bug/beep unless you eliminate 27/25, meaning that either the
> syntonic comma or the diatonic half step is going to have to be
> reversed. In this case, we aren't trying to represent 5 at all, except
> indirectly as part of 13/10. The 24-EDO one is really far from beep,
> in particular.

Ah, right--27/25 is the difference between two 6/5's and a 4/3, and in both 19 and 24 we have better 6/5's elsewhere than the ~250-cent intervals, so those tunings are not consistent with Bug. In 14-EDO we don't, and because 7/6, 6/5, and 15/13 are all approximated by the same ~257-cent interval, we can say that all three of 27/25, 49/48, and 676/675 are tempered out. What's the temperament where 49/48 is tempered out? Is that not Beep?

> I think the 24-EDO version sounds a lot better; 29-EDO is good too.
> 19-EDO's is passable. I was pretty excited with 14-EDO, but I didn't
> like the ultramajor triads as much as with 24-EDO. 19-EDO was better,
> but I think that 24-EDO's are so much better that the 5 extra notes
> don't make much of a difference.

Yeah, 24-EDO gives basically-Just 10:13:15 triads. If that's the target, then 24's a definite keeper. Seems to be one of the few somewhat-consonant chords that 24 is really good for.

> There are lots of 5-limit commas that could vanish here. 81/80 seems
> to be a decent one; Gene had some other ideas as well. This makes the
> fifths flat, and 19-equal and 24-equal are good choices for this.
> 14-tet is economical, but I don't get the same bright, sunny feel from
> the 9-note MOS that I do with 24-tet. Trying to work in the 7-limit
> and crossbreed this with 91/90 accomplishes the same thing.

14 is better for tempering 49/48, to get 6:7:9 triads. Not nearly as good as tempering 64/63, but I kind of like that 9-note MOS better than the superpyth diatonic scale.

-Igs

P.S. I like the name Barbados. I also agree with the "island" feel.

On Tue, Jan 18, 2011 at 3:19 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > It isn't bug/beep unless you eliminate 27/25, meaning that either the
> > syntonic comma or the diatonic half step is going to have to be
> > reversed. In this case, we aren't trying to represent 5 at all, except
> > indirectly as part of 13/10. The 24-EDO one is really far from beep,
> > in particular.
>
> Ah, right--27/25 is the difference between two 6/5's and a 4/3, and in both 19 and 24 we have better 6/5's elsewhere than the ~250-cent intervals, so those tunings are not consistent with Bug. In 14-EDO we don't, and because 7/6, 6/5, and 15/13 are all approximated by the same ~257-cent interval, we can say that all three of 27/25, 49/48, and 676/675 are tempered out. What's the temperament where 49/48 is tempered out? Is that not Beep?

I didn't think that beep was a 7-limit temperament, but maybe. The
temperament I described didn't involve the 7-limit - it just mapped 2,
3, and 13/10. However, as I suggested before, if you also eliminate
91/90, you equate 9/7 and 13/10, so then you'd get to the temperament
where two 7/6's map to a 4/3. I thought this would be cool, but I
don't like it as much, since it tends to make the fifths really flat,
and I like better fifths for this tuning. For a good 91/90 tuning,
check out the "driftwood" temperament I posted in the other thread,
which is basically blackwood with supermajor triads. The supermajor
triads work out to 10:13:15 and/or 14:18:21, and the subminor triads
work out to 6:7:9 and/or whatever the opposite of 10:13:15 is.

You also get pretty good 4:7:9' and 4:7:8:9' tetrads over 5 of the
roots, where 9' (read "nine prime") refers to the alternate mapping
for 9.

27-edo is also pretty ideal for a tuning that eliminates 91/90.

> > I think the 24-EDO version sounds a lot better; 29-EDO is good too.
> > 19-EDO's is passable. I was pretty excited with 14-EDO, but I didn't
> > like the ultramajor triads as much as with 24-EDO. 19-EDO was better,
> > but I think that 24-EDO's are so much better that the 5 extra notes
> > don't make much of a difference.
>
> Yeah, 24-EDO gives basically-Just 10:13:15 triads. If that's the target, then 24's a definite keeper. Seems to be one of the few somewhat-consonant chords that 24 is really good for.

And 10:12:13:15 chords too, and 10:11:12:13:14:15 chords... :) Check
out the 16-note super-diminished MOS I posted in the other thread.

> 14 is better for tempering 49/48, to get 6:7:9 triads. Not nearly as good as tempering 64/63, but I kind of like that 9-note MOS better than the superpyth diatonic scale.
>
> -Igs
>
> P.S. I like the name Barbados. I also agree with the "island" feel.

So let's call 676/675 the "island" clan or family or whatever, and
then the temperament eliminating 676/675 and 91/90 (which also
eliminates 49/48) "barbados" temperament then?

-Mike

Jacques wrote:

> It belongs to Petr to say if he agrees with that, but I would advocate
> for calling "Parizekmic" Petr's original system only.

This seems like a wise idea. Although both temperaments turn the 676/675 into unison, one of the primary tasks of my 3D tuning was to include 5/4 or 5/3 in the mapping. Gene's 2D version seems to map 20/13 with 3 generators without offering 13/8 or 13/12, which makes it difficult to find audibly "approximately periodic" triads. OTOH, the periodicity of triads like 8:10:13 or even 6:10:13 is clearly audible and approximating it can bring a much greater variety of interval characteristics to the temperament.

Petr

On Tue, Jan 18, 2011 at 5:26 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> Jacques wrote:
>
> > It belongs to Petr to say if he agrees with that, but I would advocate
> > for calling "Parizekmic" Petr's original system only.
>
> This seems like a wise idea. Although both temperaments turn the 676/675
> into unison, one of the primary tasks of my 3D tuning was to include 5/4 or
> 5/3 in the mapping. Gene's 2D version seems to map 20/13 with 3 generators
> without offering 13/8 or 13/12, which makes it difficult to find audibly
> "approximately periodic" triads. OTOH, the periodicity of triads like
> 8:10:13 or even 6:10:13 is clearly audible and approximating it can bring a
> much greater variety of interval characteristics to the temperament.

We were trying to approximate 10:13:15 actually. Or rather, my
original subgroup temperament was, and Gene expanded it out into some
other really useful non-subgroup temperaments. Here's the original
thread that sparked my interest:

/tuning/topicId_95430.html#95432

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
What's the temperament where 49/48 is tempered out? Is that not Beep?

Nope! Want to name it?

> I didn't think that beep was a 7-limit temperament, but maybe.

It is, but not that one. Tempers out 21/20 and 27/25.

> 27-edo is also pretty ideal for a tuning that eliminates 91/90.

What's 46 then? You can put 46 together with 102, and get an echidna-like temperament tempering out {91/90, 169/168, 385/384, 441/440} which is completely out of 27's league in terms of accuracy. Sharp 8/7 generator, 1/2 octave period, MOS of size 10, 18 or 28. You can tune the generator as, for instance, 67\342.

On Tue, Jan 18, 2011 at 6:50 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> What's the temperament where 49/48 is tempered out? Is that not Beep?
>
> Nope! Want to name it?
>
> > I didn't think that beep was a 7-limit temperament, but maybe.
>
> It is, but not that one. Tempers out 21/20 and 27/25.

And 49/48 is included in that, right? Graham's temperament spits out
"beep" as one of the options if 49/48 is tempered out.

To be honest, I'm really not the biggest fan of tempering out 49/48 -
7/6 and 8/7 still sound too different to me to temper together. It's
the same with tempering out 25/24. The first triangular number I'm
down to eliminate is 64/63 - before that I can't handle it. I also
don't mind eliminating 36/35, I guess, although I think of that as
81/80 and 64/63 both disappearing. Probably it has to do with my
tendency to group 7/6 and 8/7 in different perceptual categories from
a lifetime of diatonic listening and 6/5 and 7/6 together.

Since Igs seems to be a fan of this maybe he can name it.

> > 27-edo is also pretty ideal for a tuning that eliminates 91/90.
>
> What's 46 then? You can put 46 together with 102, and get an echidna-like temperament tempering out {91/90, 169/168, 385/384, 441/440} which is completely out of 27's league in terms of accuracy.

Good call! 46 does it really well. I guess 27-edo is just ideal for
the "ultrapyth" 2.3.7.13 tuning that eliminates 91/90 and 64/63.

Echidna is the temperament equating 36/35 and 49/48?

> Sharp 8/7 generator, 1/2 octave period, MOS of size 10, 18 or 28. You can tune the generator as, for instance, 67\342.

I like this, but I guess that 5 doesn't really do too hot in this tuning?

-Mike

On 19 January 2011 11:29, Mike Battaglia <battaglia01@...> wrote:

>> > I didn't think that beep was a 7-limit temperament, but maybe.
>>
>> It is, but not that one. Tempers out 21/20 and 27/25.
>
> And 49/48 is included in that, right? Graham's temperament spits out
> "beep" as one of the options if 49/48 is tempered out.

Right, Beep is the one that tempers out 49:48, because 21:20 and 27:25
imply 49:48. It works out as:

[0, 3, -2, 0> + [-4, -1, 0, 2> = [-4, 2, -2, 2> = 2[-2,1,-1,1>

Which is the same as 27/25 * 49/48 = (21/20)^2

> To be honest, I'm really not the biggest fan of tempering out 49/48 -
> 7/6 and 8/7 still sound too different to me to temper together. It's
> the same with tempering out 25/24. The first triangular number I'm
> down to eliminate is 64/63 - before that I can't handle it. I also
> don't mind eliminating 36/35, I guess, although I think of that as
> 81/80 and 64/63 both disappearing. Probably it has to do with my
> tendency to group 7/6 and 8/7 in different perceptual categories from
> a lifetime of diatonic listening and 6/5 and 7/6 together.

Ah, 7:6 and 8:7 become the same. That's what I called Bogey by
analogy with Bug. I didn't realize Bug had a different 7-limit
extension. It doesn't come up in the unison vector searches because
they don't cover sub-prime limits.

http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=2.3.7

> Since Igs seems to be a fan of this maybe he can name it.

What's the new thing to name?

Graham

On Wed, Jan 19, 2011 at 2:55 AM, Graham Breed <gbreed@...> wrote:
>
> Ah, 7:6 and 8:7 become the same. That's what I called Bogey by
> analogy with Bug. I didn't realize Bug had a different 7-limit
> extension. It doesn't come up in the unison vector searches because
> they don't cover sub-prime limits.
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=2.3.7

I am still completely clueless as to how to determine the 7-limit
comma that's being tempered out here. I've read every page on your
site, I've read Carl's stuff, I've read Gene's stuff on the
xenharmonic wiki, and I've read through random stuff on tuning and
tuning-math plenty of times, and I am still completely useless when it
comes to this simple facet of tuning theory. Can someone help me out
here? It has something to do with the reduced mapping?

> > Since Igs seems to be a fan of this maybe he can name it.
>
> What's the new thing to name?

Just 49/48 temperaments in general. Ones in which the fourth is
bisected equally and 7/6 and 8/7 are equated. I for some reason really
hate this tuning, so someone else can name it, because I'm going to
call it "this temperament sucks" temperament otherwise.

-Mike

On 19 January 2011 12:01, Mike Battaglia <battaglia01@...> wrote:
> On Wed, Jan 19, 2011 at 2:55 AM, Graham Breed <gbreed@...> wrote:
>>
>> Ah, 7:6 and 8:7 become the same. That's what I called Bogey by
>> analogy with Bug. I didn't realize Bug had a different 7-limit
>> extension. It doesn't come up in the unison vector searches because
>> they don't cover sub-prime limits.
>>
>> http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=2.3.7
>
> I am still completely clueless as to how to determine the 7-limit
> comma that's being tempered out here. I've read every page on your
> site, I've read Carl's stuff, I've read Gene's stuff on the
> xenharmonic wiki, and I've read through random stuff on tuning and
> tuning-math plenty of times, and I am still completely useless when it
> comes to this simple facet of tuning theory. Can someone help me out
> here? It has something to do with the reduced mapping?

49:48 is [-4, -1, 0, 2>. With a 2.3.7 basis, that becomes [-4, -1,
2>. The reduced mapping of Bogey is

[< 1 2 3 ]
< 0 -2 -1 ]>

<1, 2, 3 | -4, -1, 2> = -4 - 2 + 6 = 0
<0, -2, -1 | -4, -1, 2> = 0 +2 - 2 = 0

>> > Since Igs seems to be a fan of this maybe he can name it.
>>
>> What's the new thing to name?
>
> Just 49/48 temperaments in general. Ones in which the fourth is
> bisected equally and 7/6 and 8/7 are equated. I for some reason really
> hate this tuning, so someone else can name it, because I'm going to
> call it "this temperament sucks" temperament otherwise.

Bogey characterizes the family, but it's a name I made up and isn't
generally accepted. With a particular 5-limit approximation, it
becomes Beep.

It looks like Bug is the 5-limit subset of Beep. So Beep, Bug, and
Bogey will all sound the same, but Bogey has a reasonable level of
accuracy.

Semaphore is another member of the family:

http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=7

It's a shame all these names were made up without reference to the
2.3.7 subgroup. I was using "Wonneg" privately, as a combination of
Wonder and Negri, prior to MMM Day.

Graham

Mike wrote:

> > http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=2.3.7
>
> I am still completely clueless as to how to determine the 7-limit
> comma that's being tempered out here. I've read every page on your
> site, I've read Carl's stuff, I've read Gene's stuff on the
> xenharmonic wiki, and I've read through random stuff on tuning and
> tuning-math plenty of times, and I am still completely useless
> when it comes to this simple facet of tuning theory. Can someone
> help me out here? It has something to do with the reduced mapping?

I can only do it for codimension 1 cases, which fortunately
this is. It should be evident from the mapping that the period
is an octave and the generator is an approx 8/7. From this we
can write

2^2 * (8/7)^-2 = 3

4 * 49/64 = 3

4/3 = 64/49

so 49/48 vanishes.

To do more than one comma you need something like LLL
reduction...

> > What's the new thing to name?
>
> Just 49/48 temperaments in general.

It's a very popular comma but I don't see a family or clan
for it on the wiki. Perhaps that's because the families are
set up to start with a 5-limit comma, so 49/48 is always
showing up as added.

-Carl

On Wed, Jan 19, 2011 at 4:07 AM, Graham Breed <gbreed@...> wrote:
>
>
> 49:48 is [-4, -1, 0, 2>. With a 2.3.7 basis, that becomes [-4, -1,
> 2>. The reduced mapping of Bogey is
>
> [< 1 2 3 ]
> < 0 -2 -1 ]>
>
> <1, 2, 3 | -4, -1, 2> = -4 - 2 + 6 = 0
> <0, -2, -1 | -4, -1, 2> = 0 +2 - 2 = 0

Wait, that's how it works? I just solve the system of equations for

1x+2y+3z = 0
0x-2y-1z = 0

And then for any combination of x, y, and z that work here, the
corresponding commas are tempered out?

Does this work for the non-reduced mapping as well? Is this what the
point of a val is?

> Bogey characterizes the family, but it's a name I made up and isn't
> generally accepted. With a particular 5-limit approximation, it
> becomes Beep.
>
> It looks like Bug is the 5-limit subset of Beep. So Beep, Bug, and
> Bogey will all sound the same, but Bogey has a reasonable level of
> accuracy.
>
> Semaphore is another member of the family:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=7
>
> It's a shame all these names were made up without reference to the
> 2.3.7 subgroup. I was using "Wonneg" privately, as a combination of
> Wonder and Negri, prior to MMM Day.

I don't mind the name Bogey. What was MMM day...?

-Mike

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> 49:48 is [-4, -1, 0, 2>. With a 2.3.7 basis, that becomes
> [-4, -1, 2>. The reduced mapping of Bogey is
>
> [< 1 2 3 ]
> < 0 -2 -1 ]>
>
> <1, 2, 3 | -4, -1, 2> = -4 - 2 + 6 = 0
> <0, -2, -1 | -4, -1, 2> = 0 +2 - 2 = 0

Searching for simple intervals that map to zero is one
method I've used... a bit brutish though. -Carl

On Wed, Jan 19, 2011 at 4:25 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > http://x31eq.com/cgi-bin/rt.cgi?ets=19+24&limit=2.3.7
> >
> > I am still completely clueless as to how to determine the 7-limit
> > comma that's being tempered out here. I've read every page on your
> > site, I've read Carl's stuff, I've read Gene's stuff on the
> > xenharmonic wiki, and I've read through random stuff on tuning and
> > tuning-math plenty of times, and I am still completely useless
> > when it comes to this simple facet of tuning theory. Can someone
> > help me out here? It has something to do with the reduced mapping?
>
> I can only do it for codimension 1 cases, which fortunately
> this is. It should be evident from the mapping that the period
> is an octave and the generator is an approx 8/7.

How are you getting this from the mapping? You mean under the
"Generator Tunings" section? If I didn't know 49/48 was vanishing, I
wouldn't automatically assume 250 cents was a sharp 8/7.

From this we can write

> 2^2 * (8/7)^-2 = 3
>
> 4 * 49/64 = 3
>
> 4/3 = 64/49
>
> so 49/48 vanishes.

How did you get this first line - just by playing around with the
scale and noticing that moving downward by 2 of the sharp 8/7's makes
for a pretty obvious 3/2 mapping, and choosing that?

-Mike

Mike wrote:
> How are you getting this from the mapping? You mean under the
> "Generator Tunings" section? If I didn't know 49/48 was vanishing,
> I wouldn't automatically assume 250 cents was a sharp 8/7.

The tuning is irrelevant. The octave is mapped with one
period and no generators, so the period is an octave. 7 is
mapped with -1 generators and some 3 octaves, etc. -Carl

On 19 January 2011 13:29, Mike Battaglia <battaglia01@...> wrote:

> Wait, that's how it works? I just solve the system of equations for
>
> 1x+2y+3z = 0
> 0x-2y-1z = 0
>
> And then for any combination of x, y, and z that work here, the
> corresponding commas are tempered out?

That's it! Provided you solve for integers, of course. It's a kernel
or null space, and looking up those terms in a linear algebra book or
Wikipedia or whatever might help.

> Does this work for the non-reduced mapping as well? Is this what the
> point of a val is?

Yes, but there's more to vals.

> I don't mind the name Bogey. What was MMM day...?

I'm thinking of changing it to Semaphore. The Semaphore pentatonic is
essentially what I called Bogey so I don't think there's a need for
another name for a subset. Or we could call it "Stroke" because it's
already Godzilla/Hemifourths/Mothra/Semaphore/Semifourths. There's a
new temperament called Casablanca that isn't related.

MMM Day was when we made a piece of music on one day. This is what I
came up with:

http://x31eq.com/music/MMMDay.mp3

It's a mixture of Semaphore/Bogey and Slendric/Wonder.

Graham

On Wed, Jan 19, 2011 at 4:41 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
> > How are you getting this from the mapping? You mean under the
> > "Generator Tunings" section? If I didn't know 49/48 was vanishing,
> > I wouldn't automatically assume 250 cents was a sharp 8/7.
>
> The tuning is irrelevant. The octave is mapped with one
> period and no generators, so the period is an octave. 7 is
> mapped with -1 generators and some 3 octaves, etc. -Carl

So you're talking about this then:

Reduced Mapping
2 3 7
[< 1 2 3 ]
< 0 -2 -1 ]>

And I assume the top val is the period, and the bottom one is the
generator. So what does this have to do with the non-reduced mapping?

Equal Temperament Mappings
2 3 7
[< 19 30 53 ]
< 24 38 67 ]>

The octave is 19 periods and 24 generators? I assume that, in this
case, we interpret the bival differently, so as to say that 2 can be
either 19 "periods" or 24 "periods," and hence this is the 19 and 24
temperament?

-Mike

On Wed, Jan 19, 2011 at 4:46 AM, Graham Breed <gbreed@...> wrote:
>
> That's it! Provided you solve for integers, of course. It's a kernel
> or null space, and looking up those terms in a linear algebra book or
> Wikipedia or whatever might help.

OK, thanks.

> > Does this work for the non-reduced mapping as well? Is this what the
> > point of a val is?
>
> Yes, but there's more to vals.

Oh. Damn.

> > I don't mind the name Bogey. What was MMM day...?
>
> I'm thinking of changing it to Semaphore. The Semaphore pentatonic is
> essentially what I called Bogey so I don't think there's a need for
> another name for a subset. Or we could call it "Stroke" because it's
> already Godzilla/Hemifourths/Mothra/Semaphore/Semifourths. There's a
> new temperament called Casablanca that isn't related.

LOL, we have both a "hemifourths" and a "semifourths?"

> MMM Day was when we made a piece of music on one day. This is what I
> came up with:
>
> http://x31eq.com/music/MMMDay.mp3
>
> It's a mixture of Semaphore/Bogey and Slendric/Wonder.

Very nice! Sounds kind of like 5-equal, I assume because of the split fourths.

-Mike

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

> So you're talking about this then:
>
> Reduced Mapping
> 2 3 7
> [< 1 2 3 ]
> < 0 -2 -1 ]>
>
> And I assume the top val is the period, and the bottom one is
> the generator. So what does this have to do with the non-reduced
> mapping?
>
> Equal Temperament Mappings
> 2 3 7
> [< 19 30 53 ]
> < 24 38 67 ]>
>
> The octave is 19 periods and 24 generators? I assume that, in this
> case, we interpret the bival differently, so as to say that 2 can
> be either 19 "periods" or 24 "periods," and hence this is the 19
> and 24 temperament?

yes :)

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

> > > I didn't think that beep was a 7-limit temperament, but maybe.
> >
> > It is, but not that one. Tempers out 21/20 and 27/25.
>
> And 49/48 is included in that, right?

Right. Tempers out 36/35, 49/48 and, in case it matters, 1728/1715.

Graham's temperament spits out
> "beep" as one of the options if 49/48 is tempered out.
>
> To be honest, I'm really not the biggest fan of tempering out 49/48

It's not that great, but pajara tempers out 50/49 and we mostly seem willing to live with that. 49/48 gets tempered out by 19et and some less accurate temperaments--keemun, negri, blacksmith, semaphore etc.

> 7/6 and 8/7 still sound too different to me to temper together. It's
> the same with tempering out 25/24. The first triangular number I'm
> down to eliminate is 64/63 - before that I can't handle it.

64 is more in the square department, like 49. 36 is both square and triangular. How does 45/44 grab you?

> Since Igs seems to be a fan of this maybe he can name it.

I forgot what "this" is.

> > > 27-edo is also pretty ideal for a tuning that eliminates 91/90.
> >
> > What's 46 then? You can put 46 together with 102, and get an echidna-like temperament tempering out {91/90, 169/168, 385/384, 441/440} which is completely out of 27's league in terms of accuracy.
>
> Good call! 46 does it really well. I guess 27-edo is just ideal for
> the "ultrapyth" 2.3.7.13 tuning that eliminates 91/90 and 64/63.
>
> Echidna is the temperament equating 36/35 and 49/48?

Echidna is rank two, tempering out 2048/2025 as well as 1728/1715. If you just temper out the latter, it's orwellismic. I just finished a piece in an 11-limit extension of that, orwellic, which tempers out that and 176/175.

> > Sharp 8/7 generator, 1/2 octave period, MOS of size 10, 18 or 28. You can tune the generator as, for instance, 67\342.
>
> I like this, but I guess that 5 doesn't really do too hot in this tuning?

It's more complex than 3, 7, 13 or 17 but less so than 11, so I'm not sure why you've singled it out.

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

> I am still completely clueless as to how to determine the 7-limit
> comma that's being tempered out here.

My Maple stuff spits it out when I feed it [<19 30 0 53|, <24 38 0 67|], but whether this approach can be adapted to what you are using/doing I don't know.

> > What's the new thing to name?
>
> Just 49/48 temperaments in general.

To start with, the planar temperament and the no-fives temperment, I presume.

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

> > > I don't mind the name Bogey. What was MMM day...?
> >
> > I'm thinking of changing it to Semaphore. The Semaphore pentatonic is
> > essentially what I called Bogey so I don't think there's a need for
> > another name for a subset. Or we could call it "Stroke" because it's
> > already Godzilla/Hemifourths/Mothra/Semaphore/Semifourths. There's a
> > new temperament called Casablanca that isn't related.
>
> LOL, we have both a "hemifourths" and a "semifourths?"

Not really. We have one temperament which is called either semaphore or godzilla. We've also got mothra/cynder, which doesn't seem to have anything to do with this stuff, dividing the meantone fifth into thirds to get an 8/7. If we want to free up semaphore for other purposes, we could try to bury it as a name for a linear temperament.

In the two following "Parizekmic" scales, I focused on a Mohajira subset of the same ribbon system I exposed.
By adding two specific notes to a Mohajira heptaphone you complete it with a pentatonic Bala (semifourths) scale, and this was the combination I wanted to test.

The Mohajira subset (2^n reduced) here is :
[13 117 1 9 39 351 3]

and by adding 45 and 135 you get the main Bala starting sequence :
[117 135 39 45 13]...

This ensemble was completed here with 3 more notes that differ in the two tunings.
"Mohaj-Bala_443" has some of the characteristics of the Bala_ribbon.scl of my last mail, such as an additional Bala pentatonic mapped on the black keys. It has the same jazzy feeling, with even more jazz chords and in a more oriental fashion.
"Mohaj-Bala_213" is more dissymetric in term of quartertones, resulting in a more important 5-limit area that features acceptable forms of many indian ragas such as Bhupali, Malkauns, Yaman, Todi, Jog, Bhairavi and much more ; it has less Bala transpositions but seven [9:6:4] (or [9:8:6]) chords. Using the 4 quartertones it has a Bayati type of scale, a Burmese other and several variants of Kora and Valiha scales. It has also, in its longer main chain of semifourths, two pentatonic Balinese scales (which I suspect to be related to the "Island feeling" observed by Mike and Igs !).
On the aspect of its diversity, this second scale looks great ; it shows also the limits of the system, as one syntonic comma will appear next between the two chains after 8 successive semifourth transpositions of a major third.
The Mohajira scale is very well integrated in both, and the nice thing with this system is that all these scales make always very musical transitions.

! mohaj-bala_443.scl
!
Parizekmic Mohajira+Bala scale, based on a double Bala sequence
12
!
16/15
52/45
443/360
13/10
4/3
64/45
3/2
8/5
26/15
83/45
39/20
2/1
! Interleaved Bala -c recurrent sequences, x^8 - 2 = x
; = 1.1544230572469 or 248.6064241458 cents (5-29-251 notes MOS)
! first sequence [351 (203) 117 135 156 180 208 240 (277) 320...
! second sequence transposed by 4 - x^3 (~32/13) :
! [(216) (250) 288 332 384 443 512 ... (512 = F#)

! mohaj-bala_213.scl
!
Parizekmic Mohajira+Bala scale, based on a double Bala sequence
12
!
16/15
52/45
71/60
13/10
4/3
64/45
3/2
8/5
26/15
16/9
39/20
2/1
! Interleaved Bala -c recurrent sequences, x^8 - 2 = x
; = 1.1544230572469 or 248.6064241458 cents (5-29-251 notes MOS)
! first sequence [351 (203) 117 135 156 180 208 240 (277) 320 (369) 426...
! second sequence transposed by 4 - x^3 (~32/13) :
! [(216) (250) 288 (332) 384 (443) 512 ... (512 = F#)

- - - - - - -
Jacques

Here is the final touch after my first Parizekmic experiments, exposed in my last two posts.
As I said, one syntonic comma (81/80 or even bigger with my -c temperament, because of the slightly extended fifths) appears between the two chains, not surprisingly after 8 successive semifourth transpositions of the major thirds.
Another comma appears in the other diagonal of the two chains, this time after 15 successive semifourth transpositions of the major thirds :
in pure JI this comma would be 13^2 * 3^5 / 5 * 2^13, or 41607/40960 ~4.5166 cents.
Avoiding these two commas, plus "maximally even" considerations result in the structures of the following 19 and 24 notes tunings.
The main Bala sequence I used is :
[609 702 812 936 1080 1248 1440 1664 1920 2216 2560]
Its ribbon-sequence (y = 1 - (x^3)/4) is :
[375 432 500 576 664 768 886 1024 1182 1364 1576 1816 2096]
(and stops at 1024 for the 19 notes version)
The full scale in order (2^n reduced) is then :
[3 197 203 13 27 443 227 117 15 125 1 131 135 277 9 591 609 39 5 83 341 351 45 375]
(for these two scala files I choose 3 (768) as 1/1, so certain inversions might appear with my previous files, that used 15 or 45).
The 19 notes has three intervals sizes of 3, 5, 7 of roughly 1/87 octave ;
The 24 notes (with the additional 1182 1364 1576 1816 2096) splits the five 7-steps limmas (~135/128) in 3 + 4 steps, then has three intervals sizes of 3, 4, 5 of roughly 1/87 octave.
The 24 notes has among other things 16 Bala pentaphones, 12 Balinese pentaphones, 5 regular Mohajiras heptaphones.
The 19 notes has a litlle less but should be largely enough for a start !

! bala_ribbon19.scl
!
Parizekmic scale based on a double Bala sequence
19
!
203/192
13/12
9/8
443/384
39/32
5/4
125/96
4/3
45/32
277/192
3/2
203/128
13/8
5/3
83/48
117/64
15/8
125/64
2/1
! Interleaved Bala -c recurrent sequences, x^8 - 2 = x
; = 1.1544230572469 or 248.6064241458 cents (5-29-251 notes MOS)
! first sequence [609 702 812 936 1080 1248 1440 1664 1920 2216 2560]
! second sequence transposed by 1 - (x^3)/4 (~32/13) :
! [375 432 500 576 664 768 886 1024]
! Dudon 2011, after Petr Parizek TL 77917 Jul 8, 2008

! bala_ribbon24.scl
!
Parizekmic scale based on a double Bala sequence
24
!
197/192
203/192
13/12
9/8
443/384
227/192
39/32
5/4
125/96
4/3
131/96
45/32
277/192
3/2
197/128
203/128
13/8
5/3
83/48
341/192
117/64
15/8
125/64
2/1
! Interleaved Bala -c recurrent sequences, x^8 - 2 = x
; = 1.1544230572469 or 248.6064241458 cents (5-29-251 notes MOS)
! first sequence [609 702 812 936 1080 1248 1440 1664 1920 2216 2560]
! second sequence transposed by 1 - (x^3)/4 (~32/13) :
! [375 432 500 576 664 768 886 1024 1182 1364 1576 1816 2096]
! Dudon 2011, after Petr Parizek TL 77917 Jul 8, 2008

- - - - - - - -
Jacques

"genewardsmith" <genewardsmith@...> wrote:

> Not really. We have one temperament which is called
> either semaphore or godzilla. We've also got
> mothra/cynder, which doesn't seem to have anything to do
> with this stuff, dividing the meantone fifth into thirds
> to get an 8/7. If we want to free up semaphore for other
> purposes, we could try to bury it as a name for a linear
> temperament.

Mothra was linked to Semaphore on tuning-math on July 5th
2004. http://tinyurl.com/6jn8h52 Yes, there's a reply from
Herman saying it was wrong that I'd missed before.
Semifourths is mentioned on May 8th 2004. So we only have
four names for this temperament class -- is that a record?

I'm calling the 2.3.7 subset Semaphore now. I've also set
the Wikispaces names to take priority. That happens to
mean Mothra disappeared as well, because I only take the
first name in a list.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I'm calling the 2.3.7 subset Semaphore now. I've also set
> the Wikispaces names to take priority. That happens to
> mean Mothra disappeared as well, because I only take the
> first name in a list.

Where are these Wikispaces names, exactly?

On 21 January 2011 21:39, genewardsmith <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> I'm calling the 2.3.7 subset Semaphore now.  I've also set
>> the Wikispaces names to take priority.  That happens to
>> mean Mothra disappeared as well, because I only take the
>> first name in a list.
>
> Where are these Wikispaces names, exactly?

http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments

Graham

Thank you for these tunings Jacques.

Chris

> The 19 notes has a litlle less but should be largely enough for a start !
> ! bala_ribbon19.scl
> !

> Thank you for these tunings Jacques.
>
> Chris
>
> > The 19 notes has a litlle less but should be largely enough for a > start !
> > ! bala_ribbon19.scl
> > !

I am sure you would be able to do beautiful pluri-cultural xenharmonic music with them.
I only tried my 12 tones subsets with Ethno2, which were already appealing even for a non-keyboard player like me.
Not that I need urgently to do take this step, but I would be curious to know if you can implement easily a 19 tones/octave tuning on a hex keyboard like the axis - as I know you have one, you must have thought of this already ?

These Parizekmic experiments would be interesting to be compared to Petr's original temperament idea, which tempers more stuff, while these tunings are more rationalistic/extended JI-oriented and unequal, but optimises -c.
Also an equal-beating of both the Parizekmic and other Bali-Bala semifourths temperaments could be experimented if it interests some of the fans :
it can be done in pure octave period with a semifourth generator of 248.6523168732 cents.

Here is a scale that copies Gene's POTE tuning's 14 tones similar structure :
I call it Bali-Bala, as it contains plenty of both African Bala and Balinese pentatonic scales :

! bali-balaeb_14.scl
!
Bali-Bala[14] (676/675 tempering), equal-beating version
14
!
43.261584366
205.3907325072
248.6523168732
291.9139012392
454.0430493804
497.3046337464
659.4337818876
702.6953662536
745.9569506196
908.0860987608
951.3476831268
994.6092674928
1 156.738415634
2/1
! Eq-Bala triple equal-beating recurrent sequence, 4^3 = x + 5
! triple equal-beating of 15/13, 20/13, and the 4/3 in between
! Dudon 2010

- - - - - - - -
Jacques

I've not tried 19 anything with my axis but had an fantastic amount of fun
with 17.

I'll put it on my list for the near future.

Chris

On Mon, Jan 24, 2011 at 8:12 AM, Jacques Dudon <fotosonix@...> wrote:

>
>
> Thank you for these tunings Jacques.
>
> Chris
>
> > The 19 notes has a litlle less but should be largely enough for a start !
> > ! bala_ribbon19.scl
> > !
>
>
>
> I am sure you would be able to do beautiful pluri-cultural xenharmonic
> music with them.
> I only tried my 12 tones subsets with Ethno2, which were already appealing
> even for a non-keyboard player like me.
> Not that I need urgently to do take this step, but I would be curious to
> know if you can implement easily a 19 tones/octave tuning on a hex keyboard
> like the axis - as I know you have one, you must have thought of this
> already ?
>
> These Parizekmic experiments would be interesting to be compared to Petr's
> original temperament idea, which tempers more stuff, while these tunings are
> more rationalistic/extended JI-oriented and unequal, but optimises -c.
> Also an equal-beating of both the Parizekmic and other Bali-Bala
> semifourths temperaments could be experimented if it interests some of the
> fans :
> it can be done in pure octave period with a semifourth generator
> of 248.6523168732 cents.
>
> Here is a scale that copies Gene's POTE tuning's 14 tones similar structure
> :
> I call it Bali-Bala, as it contains plenty of both African Bala and
> Balinese pentatonic scales :
>
> ! bali-balaeb_14.scl
> !
> Bali-Bala[14] (676/675 tempering), equal-beating version
> 14
> !
> 43.261584366
> 205.3907325072
> 248.6523168732
> 291.9139012392
> 454.0430493804
> 497.3046337464
> 659.4337818876
> 702.6953662536
> 745.9569506196
> 908.0860987608
> 951.3476831268
> 994.6092674928
> 1 156.738415634
> 2/1
> ! Eq-Bala triple equal-beating recurrent sequence, 4^3 = x + 5
> ! triple equal-beating of 15/13, 20/13, and the 4/3 in between
> ! Dudon 2010
>
> - - - - - - - -
> Jacques
>
>
>
>
>
>