tonality diamonds and hopes for a more general understaind of JI

hi again, gene you mentioned in a post that it was not a good idea to
try building scales from a selection of primes rather than all primes
< p, why is that?

you also mentioned that it needs to be considered whether having
superparticular intervals is important. well in the scales i have
been using i notice the superparticular intervals are generally of the
form

x^2/((x+1)(x-1)), where x is a factor of the denominator of the
previous interval so in the scenario of for instance the 5lim 9 tone
dwarf this gives the result of each superparticular interval being a
power of a prime divided by the product of the other prime factors.

so this is one harmonic simplicity in the scale, however it lacks
others, like 'trivalence'.

so in order for me to answer that question you asked myself, i need to
know, what are all the general harmonic principals that should be
taken into consideration for a just intonation scale, what are your
thoughts on a general udnerstanding of JI scales

like being epimorphic, trivalent, being a tonal diamond, and how do
these properties relate to each other - eg. does having one mean it is
impossible to have another, etc?

like i mentioned before i am interested in finding a selection of
playable JI scales.

i have searched the web for information and have come across the
specific information for specific techniques but nothing from a more
general approach

the only one that i have been able to use so far is the 5lim 9tone
dwarf. i would like to find something with other primes like 7,
17..etc, but having a scale of more than 15 tones is not attractive
for me and i find it confusing if for instance there is 5 and 7 in a
scale because the intervals get very small.

> x^2/((x+1)(x-1)), where x is a factor of the denominator of the
> previous interval

i meant to say previous note

there is some stuff on diamonds () here
http://anaphoria.com/lamb.PDF
and
http://anaphoria.com/dia.PDF
more important than epimorphic intervals is to fill in the gaps close to the 1/1 with intervals that makes the appearance of an interval be subtended by the same number of steps , otherwise known as a constant structure.
Not only did Partch do this by also Novaro did the same thing

tfllt wrote:
>
> hi again, gene you mentioned in a post that it was not a good idea to
> try building scales from a selection of primes rather than all primes
> < p, why is that?
>
> you also mentioned that it needs to be considered whether having
> superparticular intervals is important. well in the scales i have
> been using i notice the superparticular intervals are generally of the
> form
>
> x^2/((x+1)(x-1)), where x is a factor of the denominator of the
> previous interval so in the scenario of for instance the 5lim 9 tone
> dwarf this gives the result of each superparticular interval being a
> power of a prime divided by the product of the other prime factors.
>
> so this is one harmonic simplicity in the scale, however it lacks
> others, like 'trivalence'.
>
> so in order for me to answer that question you asked myself, i need to
> know, what are all the general harmonic principals that should be
> taken into consideration for a just intonation scale, what are your
> thoughts on a general udnerstanding of JI scales
>
> like being epimorphic, trivalent, being a tonal diamond, and how do
> these properties relate to each other - eg. does having one mean it is
> impossible to have another, etc?
>
> like i mentioned before i am interested in finding a selection of
> playable JI scales.
>
> i have searched the web for information and have come across the
> specific information for specific techniques but nothing from a more
> general approach
>
> the only one that i have been able to use so far is the 5lim 9tone
> dwarf. i would like to find something with other primes like 7,
> 17..etc, but having a scale of more than 15 tones is not attractive
> for me and i find it confusing if for instance there is 5 and 7 in a
> scale because the intervals get very small.
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>
> hi again, gene you mentioned in a post that it was not a good idea to
> try building scales from a selection of primes rather than all primes
> < p, why is that?

It's a perfectly fine idea, but not if you want superparticular ratio
steps, where you will run into problems.

> well in the scales i have
> been using i notice the superparticular intervals are generally of the
> form
>
> x^2/((x+1)(x-1)), where x is a factor of the denominator of the
> previous interval...

That is the square numerator form, the most common type of
superparticular comma, arising from (n/(n-1))/(n/(n+1)). The second
most common kind is the triangular numberator kind, arising from
((n+1)/(n-1))/((n+2)/n).

> so in order for me to answer that question you asked myself, i need to
> know, what are all the general harmonic principals that should be
> taken into consideration for a just intonation scale, what are your
> thoughts on a general udnerstanding of JI scales

There are lots of interesting properties, and it depends on what you
are looking for. Do you want to keep the ratio of the largest over the
smallest step small? Do you want to give the scale a coherent
structure, so that a certain number of scale steps starting from one
place is in some sense like the same number of steps starting from
another place? Do you want to maximize the quantity of consonant
intervals or chords on some list of such intervals or chords? Do you
care if the scale is theoretically just intonation, or is it enough
that it sounds like just intonation? About how many steps should it have?

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:

> more important than epimorphic intervals is to fill in the gaps
close to
> the 1/1 with intervals that makes the appearance of an interval be
> subtended by the same number of steps , otherwise known as a constant
> structure.

Epimorphic implies constant structure. Epimorphic is a good condition
on JI scales, and requires a more coherent scale structure than simply
constant structure, but probably the most interesting condition along
these lines is Rothenberg propriety.

Sometimes scales skate very close to the line:

! octone.scl
octone around 60/49-7/4 interval
8
!
15/14
60/49
5/4
10/7
3/2
12/7
7/4
2

This has all superparticular ratios, and is epimorphic, constant
structure, and strictly proper--but just barely. Does that matter? It
is a JI scale, but it can't be fully understood that way because it
has a near-just 10/7 from 60/49 to 7/4, not to mention the neutral thirds.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>

> It's a perfectly fine idea, but not if you want superparticular ratio
> steps, where you will run into problems.
>

i have found quite a lot of superparticular scales being selective
with the primes, just not many of them make complete diamonds.

> There are lots of interesting properties, and it depends on what you
> are looking for. Do you want to keep the ratio of the largest over the
> smallest step small? Do you want to give the scale a coherent
> structure, so that a certain number of scale steps starting from one
> place is in some sense like the same number of steps starting from
> another place? Do you want to maximize the quantity of consonant
> intervals or chords on some list of such intervals or chords? Do you
> care if the scale is theoretically just intonation, or is it enough
> that it sounds like just intonation? About how many steps should it
have?
>

okay. well generally speaking i guess on the one hand i want
consistency, symmetry, simplicity, and completeness, an internal and
perceived coherency and on the other i wanted a minimum number of
distinct tones and intervals. so i guess whatever my take on it will
be it will be some sort of compromise. see i am not a pianist in the
first place and having a scale with a lot of tones makes it just
impossible for me to jam on.

i guess the ultimate solution for me would be to have a usb-tonality
pressure pad diamond midi controller but somehow i think im not gonna
hold my breath for that. another workaround would be to have a vst
compliant midi application so that the keyboard can be in one line of
the diamond at a time and can be moved up and down it with the mod
wheel or something. that would be seriously cool.

if there was even a way to get scala to be able to record to midi that
would be useful.

back to the scales. i sort of care if it is really just intonation or
not, for some reason i really dont like the whole concept of
temperaments and edo stuff but i guess it is an important reminder
that all this shit only matters so far as we can make a distinction.

for the purpose of my own music i am sort of torn at the moment. i
think what i will probably do is try and have a complete tonal diamond
with a 7lim plus include some squares and keep it to a minimum number
of superparticular intervals, then save each line of the diamond as an
individual scale and automate the loading of the different files in my
vst instruments.

thanks for sharing all this info has been very helpful!