Octagar (4000/3969) planar temperament

If you examine the 7-limit ranks 3 planar temperaments, the ones which can be obtained by tempering out a single 7-limit comma, a sort of bias in favor of flat over sharp tending systems can be discerned. Unfortunately, the opposite bias seems to be in evidence with respect to people's stated preferences, at least on these lists, with more people expressing a fondness for slightly sharp fifths over slightly flat ones. One reason I used hemifamity (5120/5103 planar) temperament was just its slightly sharp tendency, with the 3 and 7 tending sharp.

Another planar temperament with 3 and 7 tending sharp is octagar (4000/3969) temperament. If you orthogonally project in Euclidean interval/tuning space perpendicular to both 4000/3969 and 2, you project the 7-limit lattice onto a two-dimensional one. The two closest lattice points to the origin are 21/20 and either 63/50 or 63/40. Choosing the former, which is nearly perpendicular (97.7 degrees) to 21/20, we obtain a lattice basis. Using the facts that (21/20)(63/50)(4000/3969) = 4/3, (63/50)^2 * (4000/3969) = 8/5 and (21/20)^2/(63/50) = 8/7 we can locate the 7-limit pitch classes on the lattice, mapping them to lattice points via the mapping [<0 -1 0 2|, <0 -1 -2 -1|].

If you sharpen both 3 and 7 by (4000/3969)^(1/6) and leave 2 and 5 alone, you obtain a fifth in the 704-705 cent range promoted by Margo, and you also obtain the 7-limit minimax tuning. 162, 189 or 230 equal also give tunings in that range. If you want something with the fifth a little less sharp, the 9-limit minimax tuning has a 3 sharp by only (4000/3969)^(1/8) and penalizes 7 by having it sharp by twice that amount.

Another feature of octagar temperament is that the 5/4-sqrt(8/5)-sqrt(8/5) version of the augmented triad is all over the place, being so low in complexity. Has this chord received any attention?

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

I intended to post this to tuning-math, but Fate decreed otherwise. It's perfectly appropriate here also, I think.

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

> Another feature of octagar temperament is that the 5/4-sqrt(8/5)->sqrt(8/5) version of the augmented triad is all over the place, being >so low in complexity. Has this chord received any attention?
>

The square root of 8/5 is almost precisely the Pythagorean ditone, so if you do vertical harmony using maquam-like tunings, using interlocking tetrachords, you're bound to get this sonority. It's the chord and implied movements (strongest movements are via Pythagorean relationships, sweetest sonorities via Just) I had in mind when suggesting to Mike that Magen Abot should be tuned 7-limit without tempering out 81/80.

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

> The square root of 8/5 is almost precisely the Pythagorean ditone, so if you do vertical harmony using maquam-like tunings, using interlocking tetrachords, you're bound to get this sonority. It's the chord and implied movements (strongest movements are via Pythagorean relationships, sweetest sonorities via Just) I had in mind when suggesting to Mike that Magen Abot should be tuned 7-limit without tempering out 81/80.

Hmmm...they are separated by sqrt(32805/32768), and if you temper that out along with 4000/3969, you get garibaldi temperament. But then the sharp tendency is quite a bit less for optimal tuning.

In the interest of making more music using these temperaments, it would be good to have the generator/period etc. information. Rank 3 temperaments could use some examples too, as I, and I would guess most of us, don't know how you're using the generators. I'd like to see an "octagar" example scale for instance.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> If you examine the 7-limit ranks 3 planar temperaments, the ones which can be obtained by tempering out a single 7-limit comma, a sort of bias in favor of flat over sharp tending systems can be discerned. Unfortunately, the opposite bias seems to be in evidence with respect to people's stated preferences, at least on these lists, with more people expressing a fondness for slightly sharp fifths over slightly flat ones. One reason I used hemifamity (5120/5103 planar) temperament was just its slightly sharp tendency, with the 3 and 7 tending sharp.
>
> Another planar temperament with 3 and 7 tending sharp is octagar (4000/3969) temperament. If you orthogonally project in Euclidean interval/tuning space perpendicular to both 4000/3969 and 2, you project the 7-limit lattice onto a two-dimensional one. The two closest lattice points to the origin are 21/20 and either 63/50 or 63/40. Choosing the former, which is nearly perpendicular (97.7 degrees) to 21/20, we obtain a lattice basis. Using the facts that (21/20)(63/50)(4000/3969) = 4/3, (63/50)^2 * (4000/3969) = 8/5 and (21/20)^2/(63/50) = 8/7 we can locate the 7-limit pitch classes on the lattice, mapping them to lattice points via the mapping [<0 -1 0 2|, <0 -1 -2 -1|].
>
> If you sharpen both 3 and 7 by (4000/3969)^(1/6) and leave 2 and 5 alone, you obtain a fifth in the 704-705 cent range promoted by Margo, and you also obtain the 7-limit minimax tuning. 162, 189 or 230 equal also give tunings in that range. If you want something with the fifth a little less sharp, the 9-limit minimax tuning has a 3 sharp by only (4000/3969)^(1/8) and penalizes 7 by having it sharp by twice that amount.
>
> Another feature of octagar temperament is that the 5/4-sqrt(8/5)-sqrt(8/5) version of the augmented triad is all over the place, being so low in complexity. Has this chord received any attention?
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> In the interest of making more music using these temperaments, it would be good to have the generator/period etc. information.

It's rank 3 so it doesn't exactly have a period. Generators can be taken to be tempered versions of 2, 63/50 and 21/20, which for 7-limit minimax tuning would be

2: 1200 cents
63/50: 406.843 cents
21/20: 88.957 cents

Mapping: [<1 2 3 3|, <0 -1 -2 -1|, <0 -1 0 2|]

Here's a scale using it:

! octasquare25.scl
5x5 generator square octagar tempered scale
25
!
68.4275
88.9570
157.3845
177.9139
208.3998
228.9292
297.3567
317.8862
386.3137
406.8431
475.2707
495.8001
564.2277
584.7571
615.2429
704.1999
793.1569
882.1138
971.0708
1001.5566
1022.0861
1090.5136
1111.0430
1179.4706
1200.0000

That is pretty trippy. The approximations to Just are obviously all extremely close but the layout is bizarre. Which might not be a bad thing at all, and I assume that you can make a large number of different layouts for different purposes, by planning ahead.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > In the interest of making more music using these temperaments, it would be good to have the generator/period etc. information.
>
> It's rank 3 so it doesn't exactly have a period. Generators can be taken to be tempered versions of 2, 63/50 and 21/20, which for 7-limit minimax tuning would be
>
> 2: 1200 cents
> 63/50: 406.843 cents
> 21/20: 88.957 cents
>
> Mapping: [<1 2 3 3|, <0 -1 -2 -1|, <0 -1 0 2|]
>
> Here's a scale using it:
>
> ! octasquare25.scl
> 5x5 generator square octagar tempered scale
> 25
> !
> 68.4275
> 88.9570
> 157.3845
> 177.9139
> 208.3998
> 228.9292
> 297.3567
> 317.8862
> 386.3137
> 406.8431
> 475.2707
> 495.8001
> 564.2277
> 584.7571
> 615.2429
> 704.1999
> 793.1569
> 882.1138
> 971.0708
> 1001.5566
> 1022.0861
> 1090.5136
> 1111.0430
> 1179.4706
> 1200.0000
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> That is pretty trippy. The approximations to Just are obviously all extremely close but the layout is bizarre. Which might not be a bad thing at all, and I assume that you can make a large number of different layouts for different purposes, by planning ahead.

The layout can be whatever you want it to be, but the point is to use something which makes use of 4000/3969 tempering. If you grab something 5-limit at random to temper, you will likely find 225/224, 1029/1024, 126/125 or 2401/2400 offering themselves as prospects.