The blackjackisma

As has been noted, you can get a comma pump along a line of 15/14 root movements using the 2/(15/14)^10 comma. However, long before I noticed this I had fun with a similar pump in miracle temperament. If for example you start with [0 0 0] and proceed by [0 0 i] until [0 0 15], then in miracle you get a comma pump, returning to [0 0 0] as the equivalent chord [1 0 15]. The set of all the notes of such chords, tempered in miracle, spans the miracle gamut from -7 to 13 secors, which gives 21 successive miracle generators, in other words Blackjack (Miracle[21].) This means the corresponding comma, (15/4)^7/(8/5) = |-10 7 8 -7>, is a comma of miracle which I think should be called the blackjackisma. Adding 2401/2400 to the blackjackisma leads to 7-limit miracle, and if for some reason that isn't good enough for you, no appreciable tuning accuracy is lost by tossing in 4375/4374 at any rate.

Gene wrote:

> This means the corresponding comma, (15/4)^7/(8/5) = |-10 7 8 -7>, is a > comma of miracle which I think
> should be called the blackjackisma.

With all due respect, I would dare to call this the "blackjackisma #1".
The other very important factor, which certainly would also work as a "blackjackisma", is |10,5,8,-13> which is the same as (12/7)^13 * (5/12)^8. If you treat it exactly this way, you get a 21-tone "equal" scale of ~116.60811-cent steps in non-monotonic order.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> With all due respect, I would dare to call this the "blackjackisma #1".
> The other very important factor, which certainly would also work as a
> "blackjackisma", is |10,5,8,-13> which is the same as (12/7)^13 * (5/12)^8.
> If you treat it exactly this way, you get a 21-tone "equal" scale of
> ~116.60811-cent steps in non-monotonic order.

I don't see why you think it is very important. The corresponding comma pump is pretty elaborate. If b is my blackjackisma, then your version number 2 is b^3/(225/224)^8, which shows that if you take both together you get torsion, by the way.

I wrote:

> > The other very important factor, which certainly would also work as a
> > "blackjackisma", is |10,5,8,-13> which is the same as (12/7)^13 * > > (5/12)^8.
> > If you treat it exactly this way, you get a 21-tone "equal" scale of
> > ~116.60811-cent steps in non-monotonic order.
>
> I don't see why you think it is very important. The corresponding comma > pump is pretty elaborate.

If I assign "a=12/7, b=5/12", divide both by the 21st root of the comma, and then multiply something like "abaababaababaabaababa", for example, I get 21 pitches in non-monotonic order which is in fact a 21-tone chain of "secors". The starting approximated triad in question is actually 5:7:12, which means that something like 3/2 doesn't come out as one of the approximants but rather as a product of some powers of "a" and "b".

BTW: It's interesting that it's "a^13 * b^8" here and the amity comma has in my example was also "a^13 * b^8", namely "a=5/24, b=64/5". This means I could actually do the same pump in this version of miracle as I did in amity, in terms of generator patterns. I think this is what you've done by retuning some stuff a few years ago, if I'm not mistaken

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> BTW: It's interesting that it's "a^13 * b^8" here and the amity comma has in
> my example was also "a^13 * b^8", namely "a=5/24, b=64/5". This means I
> could actually do the same pump in this version of miracle as I did in
> amity, in terms of generator patterns. I think this is what you've done by
> retuning some stuff a few years ago, if I'm not mistaken

Aha! If I put the matter in terms of cubic lattice coordinates, then an otonal tetrad with root 81/80 is represented by [-1 4 3] and with root 1029/1024 by [3 4 1], so that one can be transformed to the other. In the same way, amity is represented by [5 -13 -8] and your comma by [-5 -8 13], so they are indeed transformable. Both amity and meantone comma pumps may be converted into miracle comma pumps. Unfortunately, if we put amity and meantone together we get the 7edo patent val, not too interesting as a way of transforming back out of miracle.

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

> Aha! If I put the matter in terms of cubic lattice coordinates, then an otonal tetrad with root 81/80 is represented by [-1 4 3] and with root 1029/1024 by [3 4 1], so that one can be transformed to the other. In the same way, amity is represented by [5 -13 -8] and your comma by [-5 -8 13], so they are indeed transformable. Both amity and meantone comma pumps may be converted into miracle comma pumps. Unfortunately, if we put amity and meantone together we get the 7edo patent val, not too interesting as a way of transforming back out of miracle.
>

Here are some more examples:

81/80 ~ 135/128 ~ 1029/1024
525/512 ~ 175/162 ~ 2401/2400
250/243 ~ 3125/3087 ~ 3136/3125
27/25 ~ 64/63 ~ 126/125
1029/1000 ~ 875/864 ~ 1728/1715
16/15 ~ 25/24 ~ 36/35 ~ 49/48

I've made use of 81/80~135/128~1029/1024 and 64/63~126/125. 250/243~3125/3087~3136/3125 fairly cries aloud to get used, but I don't know that it ever has been.

Gene wrote:

> I've made use of 81/80~135/128~1029/1024 and 64/63~126/125.

Have you? Is it possible to hear it somewhere?

> 250/243~3125/3087~3136/3125 fairly cries aloud to get used, but I don't > know
> that it ever has been.

Okay, I'll try to make a short chord progression in porcupine first and then I'll see if I manage to retune it appropriately. The idea definitely sounds interesting.

BTW: about 5 years ago, I used a slightly different approach -- not sure if you heard the examples. I took the 12-tone "Euler's monochord" scale, which is essentially 5-limit JI, and replaced the 3/2 with one interval and the 5/4 with another. In one of them, I left the 3/2 alone and swapped 5/4 and 6/5. In others, I even replaced the 2/1 with something else -- when I wanted to get things like 3:5:7:9, for example.
Maybe this 3D way of retuning is even more flexible than the 2D one, at least that's what common sense tells me that should be the case.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > I've made use of 81/80~135/128~1029/1024 and 64/63~126/125.
>
> Have you? Is it possible to hear it somewhere?

Here's 64/63~126/125:

http://www.archive.org/details/MusicForYourEars

Here's 81/80~126/125:

http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3
http://xenharmonic.wikispaces.com/36edo#Music

I mentioned Mysterious Mush recently, which was 81/80~135/128, but it seems to be lost.