Threeless temperaments

Hi there tuners.

A while ago, I was thinking of various prime restrictions in temperaments and I found something interesting.

There's the Bohlen-Pierce temperament whose approximations lack the number 2 and some of the "hyper-BP" variants may happily include virtually any other prime while still excluding 2.

Then, there are temperaments of Margo Schulter or Hudson Lacerda (or whoever else was involved at that time) whose approximations lack the number 5 but happily include virtually any prime above or below that.

Now my question is: Has it been really just me so far who has tried to make music in temperaments excluding the number 3 from their approximations?

Thanks.

Petr

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

> Now my question is: Has it been really just me so far who has tried to make
> music in temperaments excluding the number 3 from their approximations?

LOL. Hi, Petr. Welcome to my life! "No 3's" is my middle name. Okay, granted--some of my favorite temperaments aren't "non-3" but just "really BAD 3", but yes, I am very much interested in these approaches. I'd love to hear some of your favorites, as I'm compiling a list of the best subgroup temperaments that I can find based on the following triads (not all of which are non-3):

5:6:7
6:7:9
7:8:9
4:5:7
4:6:7
9:11:15
8:10:11
7:9:11
6:9:11
18:22:27
9:11:13
10:13:15
8:10:13

I think all of these triads could, to a greater or lesser extent, by used as analogs to the familiar 4:5:6 as bases for tonal music. Right now I'm looking for all the lowest-complexity lowest-error temperaments for each of these and indexing EDOs which do a good job with them. I'll post my own results when I've concluded the survey, but in the meantime I'd welcome suggestions.

-Igs

Petr,

You might find that 11-equal is to 4:7:9:11 what 12-equal is to 4:5:6.
I've been playing with that a lot lately.

-Mike

2011/2/26 Petr Pařízek <petrparizek2000@...>
>
> Hi there tuners.
>
> A while ago, I was thinking of various prime restrictions in temperaments
> and I found something interesting.
>
> There's the Bohlen-Pierce temperament whose approximations lack the number 2
> and some of the "hyper-BP" variants may happily include virtually any other
> prime while still excluding 2.
>
> Then, there are temperaments of Margo Schulter or Hudson Lacerda (or whoever
> else was involved at that time) whose approximations lack the number 5 but
> happily include virtually any prime above or below that.
>
> Now my question is: Has it been really just me so far who has tried to make
> music in temperaments excluding the number 3 from their approximations?
>
> Thanks.
>
> Petr

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

> Now my question is: Has it been really just me so far who has tried to make
> music in temperaments excluding the number 3 from their approximations?

I doubt it. But when you speak of such things, note that there is a difference between 9s but no 3s, and no 3s at all. The former has 7-9-11 and 9-11-13 to play with. With the latter, I wonder if you could tell us what your favorite chords are?

Igs wrote:

> Now my question is: Has it been really just me so far who has tried to
> make
> music in temperaments excluding the number 3 from their approximations?

> LOL. Hi, Petr. Welcome to my life! "No 3's" is my middle name. Okay,
> granted--some of my favorite
> temperaments aren't "non-3" but just "really BAD 3", but yes, I am very
> much interested in these approaches.

:-)

> I'd love to hear some of your favorites, as I'm compiling a list of the
> best subgroup temperaments that I can find
> based on the following triads (not all of which are non-3):

So far I've made three recordings. I never thought I would one day link to
all of them in a single message; but Nevertheless, here they are:
http://dl.dropbox.com/u/8497979/13LimitSpectrum.mp3
http://dl.dropbox.com/u/8497979/13limitImpro.mp3
http://dl.dropbox.com/u/8497979/2200_Over_2197_Temperament.mp3

The first temperament used is a 2D system with a generator of (32/5)^(1/9). This means there's an "unintended" fifth of ~714 cents but I've used it anyway (mainly in the introduction). My primary target approximation was 4:5:7:13.

The second one is also a 2D tuning with a generator equal to 130^(1/13) where the primary approximation was 8:10:11:13.

The third one is a 3D tuning whose primary approximation is also 8:10:11:13 and which I'm describing here:
/tuning/topicId_86834.html#86863

Now onto your suggestions.

> 5:6:7

Add 10 to that and you're on a way to myna.

> 6:7:9

If you add 8, this may lead to a "fiveless" pentatonic like 12:14:16:18:21:24. Another possibility is to temper out 1029/1024 or 64/63 or 17496/16807 or 65536/64827 or, in a very good case, 118098/117649.
The untempered non-octave chains inside the plain 6:7:9 are something which I still have to examine more yet.

> 7:8:9

Add 6 and it's the same for this.

> 4:5:7

The most obvious example is to temper out 3136/3125 -- i.e. 2 steps for 5/4, 5 steps for 7/4. This actually makes it possible to approximate 4:5:7:11 using 9 steps to substitute 11/4.

> 4:6:7

This offers the same possibilities as 6:7:9 since 9/6 = 6/4. This is true not only for the temperaments with octaves but for the untempered non-octave chains as well.

> 9:11:15

Not explored so far.

> 8:10:11

Add 16 and you get a possibility of some sort of "fifthless orwell".

> 7:9:11
> 6:9:11

Again, not explored too much so far.

> 18:22:27

The most obvious choice is to temper out 243/242.

> 9:11:13
> 10:13:15

Maybe you will find more of a point in these than I would.

> 8:10:13

Add 16 and you soon get possible 13-limit mappings for the temperaments I used in the recordings.

> I think all of these triads could, to a greater or lesser extent, by used > as
> analogs to the familiar 4:5:6 as bases for tonal music.

To a certain extent, they could.

> Right now I'm
> looking for all the lowest-complexity lowest-error temperaments for each > of
> these and indexing EDOs which do a good job with them. I'll post my own
> results when I've concluded the survey, but in the meantime I'd welcome
> suggestions.

I'll be happy to know about your conclusions then.

Petr