Hi - tonality diamond

im an aspiring musician interested in microtonality and have read some
literature and find the tonality diamond a v intuitive concept that
sounds great

what are the best ways to make a tonality diamond, with super
particular intervals? are there are other important factors? is
there a method to find how many tonality diamonds there are for n
different intervals with z distinct intervals and p distinct primes?
superparticular? and what arrangement makes this diamond?

what is the general consensus of the usage of the diamond, and for
what reasons would one use a lattice, rectangle, or triangle, other,
instead? as u can probably tell i dotn have a good understanding of
these concepts but i would really like to hear ur response because i
am sick of fucking around with scala and want to make some music!

thankyou!! !

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>
> im an aspiring musician interested in microtonality and have read some
> literature and find the tonality diamond a v intuitive concept that
> sounds great
>
> what are the best ways to make a tonality diamond, with super
> particular intervals? are there are other important factors? is
> there a method to find how many tonality diamonds there are for n
> different intervals with z distinct intervals and p distinct primes?
> superparticular? and what arrangement makes this diamond?
>
> what is the general consensus of the usage of the diamond, and for
> what reasons would one use a lattice, rectangle, or triangle, other,
> instead? as u can probably tell i dotn have a good understanding of
> these concepts but i would really like to hear ur response because i
> am sick of fucking around with scala and want to make some music!
>
> thankyou!! !
>

sorry to double post but i have to correct myself: n intervals, z
distinct with p distinct primes which makes a diamond

eg. the primes 2, 3, 5 make the complete 9-tone diamond with 3
distinct superparticular intervals with the arrangement:

16/15
9/8
25/24
16/15
9/8
16/15
25/24
9/8
16/15

any general information u could provide about tonality diamond, just
intonation, superparticular intervals, or anything thats gonna help be
understand and build more interesting but harmonically consistant
scales would be v much appreciated thanku!

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:

> 16/15
> 9/8
> 25/24
> 16/15
> 9/8
> 16/15
> 25/24
> 9/8
> 16/15

This gives the scale

[16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8, 2]

which isn't the tonality diamond, but contains it. Are you asking
about construction of scales containing tonality diamonds? What other
constraints--for instance, for a p-limit diamond, do you want to
construct it with pi(p) superparicular ratios, any sort of p-limit
superparticular ratios, or don't you care about superparticular ratios
that much really?

What, in other words, is your goal?

> any general information u could provide about tonality diamond, just
> intonation, superparticular intervals, or anything thats gonna help be
> understand and build more interesting but harmonically consistant
> scales would be v much appreciated thanku!

That's a huge question. Do you know about Fokker blocks, for isntance?

i see what you mean, but that 9 tone one is 'better', right, cause it
has only 3 unique superparticular intervals and generates 9 tones
whereas the 5limit diamond has 7 tones and needs 4 unique
superparticular intervals to generate....?

i know this sounds stupid and it probably is but i want to know the
best way to construct a scale using some different primes.

like, what is the best way to make a scale with the primes, 2, 3, 7
or 2,5,11

the way which will give the least different intervals to the number of
tones ratio and will let every harmonic series for each prime factor
be complete for the given primes

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:

> i know this sounds stupid and it probably is but i want to know the
> best way to construct a scale using some different primes.

One approach which seems to produce good results is what I call dwarf
scales: http://66.98.148.43/~xenharmo/dwarf.htm

Your scale is an example--it's the 9-et, 5-limit dwarf. It is also a
Fokker block for 128/125 and 135/128, and an Euler genus. Minimal
scales containing diamonds and satisfying an additional constraint,
such as being epimorphic, have certainly been considered here. I think
I did minimal dwarves containing diamonds at one point.