Rank 3 temperaments?

Hi there,

Perhaps someone here could suggest a webpage discussing rank 3 temperaments?
I've been trying to find something like that for days. I've made some scales
recently and I'd like to know if I could call them rank 3 temperaments or
not. Thanks a lot in advance.

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Hi there,
>
> Perhaps someone here could suggest a webpage discussing rank 3
> temperaments?
> I've been trying to find something like that for days. I've made
> some scales recently and I'd like to know if I could call them
> rank 3 temperaments or not. Thanks a lot in advance.
>
> Petr

If you can connect all the pitches in the tuning with r different
intervals, you have a rank-r temparement.

I'm not aware of much work on rank 3. On a few occasions there's
been talk of "hyper MOS" and stuff like that, and it was conjectured
such concepts may be linked to rank > 2 temperaments (just as MOS
is related to rank 2 temperaments).

Some of Gene's writing on temperaments is generalized wrt rank.
See for example:
http://lumma.org/tuning/gws/regular.html

-Carl

Petr Pa��zek wrote:
> Hi there,
> > Perhaps someone here could suggest a webpage discussing rank 3 temperaments?
> I've been trying to find something like that for days. I've made some scales
> recently and I'd like to know if I could call them rank 3 temperaments or
> not. Thanks a lot in advance.
> > Petr

Any temperament that tempers out a 7-limit comma (like 126/125 or 4375/4374), if none of the exponents are zero, could be considered a rank 3 temperament. You could also use two 11-limit commas, three 13-limit commas, etc. The resulting temperament would have three generators -- typically one would be an octave and another can be set to a different consonant interval like 3/2 or 4/3. My original version of starling temperament had the octave, major and minor thirds as generators.

Two rank 3 temperaments that have had some discussion are marvel and starling. The easiest way to describe a tuning for temperaments like these is to specify a tuning for each of the prime intervals (2/1 3/1 5/1 7/1 11/1). Starling tempers out 126/125, and has a TOP tuning for the prime intervals as follows:

<1199.010636, 1900.386896, 2788.610946, 3366.048410]

Marvel tempers out 225/224 and 385/384, with a TOP tuning of the primes like this:

<1200.508704, 1901.148724, 2785.132538, 3370.019002, 4149.558115]

Somewhere I have a document with data for other rank 3 temperaments, which have names like Baldur, Odin, and Portent. It's been so long since these came up on tuning-math that I don't even know where to start searching.

Petr Pa��zek wrote:
> Hi there,
> > Perhaps someone here could suggest a webpage discussing rank 3 temperaments?
> I've been trying to find something like that for days. I've made some scales
> recently and I'd like to know if I could call them rank 3 temperaments or
> not. Thanks a lot in advance.

Sorry, there's no single resource. I was planning to write an article this year but the schedule's completely fallen by the wayside. Certainly you can call them rank 3 temperaments and the only contentious term is "temperament". I'd like to talk about "planar temperaments" but that term's in the dog house along with "linear temperaments".

I did discuss them with Gene on tuning-math. You can search the archives. The theory has advanced since then so it should be possible to search for them much faster. But I haven't even implemented those ideas let alone written them up fully. The basic algorithms are in

http://x31eq.com/primerr.pdf

The rank 3 searches I have implemented should still be good enough to find the interesting temperament classes. You'll have to work out the Python code though

http://x31eq.com/temper/regular.zip

The main reason there hasn't been much progress is that the world in general hasn't shown much interest. Keyboards are naturally 2-dimensional beasts. You may use a rank 3 system for notation or as a harmonic lattice but there's no particular need to compare it to other rank 3 systems.

Still, some systems that have come up (and Gene probably gave names):

- Tempering out 225:224 from 7-limit JI. This is the union of a lot of common temperament classes. It could be applied using a kcyboard or system of notation designed for 5-limit JI. It's essentially what my tripod notation (also not written up) is based around.

- Tempering out 2401:2400 from 7-limit JI. Gets you very close to JI. The 7-limit neutral thirds lattices I use for miracle temperament generalize to this rank 3 class. The searches Gene and I have done confirm that it's an optimal rank3 temperament FWIW. You can also extended into the 11-limit by associating 11:9 with the neutral thirds. It's far less accurate but still a stand-out system.

- Using miracle temperament with an additional accidental to show the difference between 21:16 and 13:10, for example. This came up as an optimal 13-limit system on some search I did.

- There are other interesting notations you can get by starting with schismatic (which is a simple chain of fifths) and adding an accidental for the 7- and 11-limits.

Graham