Finding 2D temperaments by turning chords into scales

Hi again.

I think this may be of great interest to those who are interested in 2D temperaments in general.

If we take a 5-limit major triad like 3:4:5:6 or 4:5:6:8, we can easily replace the large intervals by smaller ones just by subtracting the cent sizes or dividing the factors (like turning a 4/3 into 6/5 followed by 10/9 or some such) and we can repeat this process as many times as we wish. This eventually gives us a 3D system and we can get a 2D "imitation" if we temper out the smallest interval.

For example, suppose we have "a=4/3, b=5/4, c=6/5" and therefore a*b*c gives us an octave. Since "a" is larger than the other two, we then do either "d=a/c" or "d=a/b"; okay, let's say we go for the former. So then we might have something like "e=b/d", then "f=c/e", then "g=e/d", and all of a sudden, the octave originally consisting of "a*b*c" is now made of "d^5*f^2*g^3" (i.e. 5 large single-step sizes, 2 small single-step sizes, 3 commas).

We may notice that our "g" is so small that we can easily temper it out in such a way that two target intervals and their combinations are perfectly in tune. The easiest way of doing this is to multiply both "d" and "f" by "G^(3/7)" as there are 7 first-class steps and 3 commas in our scale. This results in 2/7-comma meantone and both 2/1 and 25/24 will be in tune. If, OTOH, we wish to have, let's say, 5/2 and 12/5 in tune (which in turn makes 6/1 in tune as well), then we first find the exponents of "d, f, g"; these are "7 2 4" for 5/2 and "6 3 4" for 12/5. Using the method of Gaussian elimination (at least that's what I think it is), we then get "9 0 4" and "0 9 4", which says that our "d" and "f" should both be multiplied by "g^(4/9)". This gives us meantone whose octaves are 1/9-comma sharp and whose major and minor thirds are 1/9-comma flat.

The most interesting fact about this procedure is that it allows you to very easily discover other 2D temperaments within the originally untempered system of 5-limit JI. For example, if, instead of doing "e=b/d", we did "e=b/c", then possible further steps may lead us to completely different results like, for instance, "f=c/d, g=d/f, h=f/e, i=e/h" -- so that "i" is 15625/15552 and if we temper that out, we get hanson with a totally different pattern of "large" and "small" steps than in meantone.

Surprisingly enough, when I first used this method in early 2006, my aim at that time was to find temperaments outside the standard 1:2:3:5 framework. So in the beginning, I just checked how the procedure was applicable to meantone and schismatic, and once I knew that, I then started looking for various non-octave or partial-limit temperaments. That's exactly how I discovered the 2D version of BP which tempers out 245/243 or how I discovered the "triharmonic temperament" where the period is 4/1 and the generator is ~357 cents. Only then did I start looking into the realm of other 5-liit 2D temperaments and only then did I start to understand the actual "incompatibility" of each other and why they weren't musically interchangeable.

I think this method must have been known for ages in the context of scales, if not in the context of temperaments. Nevertheless, I haven't found anything which would support my assumption, nor have I found much discussion about the topic in the "tuning-related" conversations I've run across. Does this mean that it hasn't been given a great deal of attention? And if so, doesn't it deserve more? There now seem to be at least 4 different methods available for finding 2D temperaments, each of which has its own pros and cons. That doesn't necessarily mean that some are more effective and others should be forgotten. I myself was able to find some temperaments by using one method and other temperaments by using a different one.

Petr

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

> For example, suppose we have "a=4/3, b=5/4, c=6/5" and therefore a*b*c gives us an octave. Since "a" is larger than the other two, we then do either "d=a/c" or "d=a/b"; okay, let's say we go for the former. So then we might have something like "e=b/d", then "f=c/e", then "g=e/d", and all of a sudden, the octave originally consisting of "a*b*c" is now made of "d^5*f^2*g^3" (i.e. 5 large single-step sizes, 2 small single-step sizes, 3 commas).

You should take a look at Brun's algorithm.

Gene wrote:

> You should take a look at Brun's algorithm.

Thanks for the suggestion.

I was trying to find something out there on the web and currently I'm not sure what the difference is between Brun's algorithm and the "Crunch algorithm" discussed on Tuning Math some time ago.
Anyway, it looked to me like always dividing the largest factor by the second largest one. Honestly, I think there could be some other criteria for deciding which of the other factors we should divide by.

A great example is the chord of 3:5:7:9 where 5/3 is the largest factor and when you divide this by 9/7, the "similarity" of the new result to 9/7 becomes so obvious that you almost instantly start thinking about finding the comma, which you can happily do -- i.e. "a=5/3, b=7/5, c=9/7, d=a/c, e=b/d, f=d/c". This gives you the BP pentatonic which you can easily extend to the 9-tone diatonic (c/e), or to the 13-tone chromatic (g/e), or even to the 17-tone enharmonic (where the 15625/15309 can serve as a scale step as well).
I usually view it like this:
a|b|c
dc|b|c
dc|ed|c
cfc|ecf|c
gefeg|egef|ge
ehefeeh|eehef|ehe
eieefeeie|eeieef|eiee

Petr

I wrote:

> I usually view it like this:
> a|b|c
> dc|b|c
> dc|ed|c
> cfc|ecf|c
> gefeg|egef|ge
> ehefeeh|eehef|ehe
> eieefeeie|eeieef|eiee

Interestingly enough, when doing it the "usual" way of dividing by the second largest factor, we arrive at a slightly different 13-tone scale which suggests tempering out either 3125/3087 or 16875/16807:

a|b|c
bd|b|c
ecd|ce|c
edfd|fde|df
egefeg|fege|gef

Petr

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > You should take a look at Brun's algorithm.
>
> Thanks for the suggestion.
>
> I was trying to find something out there on the web and currently I'm not sure what the difference is between Brun's algorithm and the "Crunch algorithm" discussed on Tuning Math some time ago.

Me either, as I can't read Norwegian, but I'm told it's pretty much the same thing.

> Anyway, it looked to me like always dividing the largest factor by the second largest one.

That was my "crunch" algorithm, yes.