LIMIT OF JUSTNESS OR RATIONALITY?

DEAR S

IN TONAL SOFT, IT IS MENTIONED THAT JUST SYSTEMS ARE SUBSETS OF
RATIONAL SYSTEMS AND JUST INTERVALS ARE RATIO OF SMALL NUMBERS.

IS THERE ANY BOARDER TO CALL A SYSTEM RATIONAL OR JUST? WHAT IS THE
MAXIMUM LIMIT OF RATIO SMALLNESS TO BE JUST?

IS IT RELATED TO NUMBER OF SOUND HARMONICS ? ARE 27/16 , 70/64 , 81/75
, 240/128 JUST OR RATIONAL?

THANKS

RI / rational intonation

[Joe Monzo]

A term and acronym used to distinguish musical tunings calculated by
ratios of large numbers, from those with small-integer terms and/or an
audible property of containing many very consonant intervals. The latter
is a special case of RI called "just intonation" or JI, whereas RI is
the more general category of tunings which includes all JI tunings.

A notorious example of an RI which is not synonymous with JI is the
tuning of the Hammond organ, which see.

The rational-intonation acronym RI is not to be confused with the
Sanskrit syllable "ri", which is used in Indian music and music-theory.

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

> DEAR S
>
> IN TONAL SOFT, IT IS MENTIONED THAT JUST SYSTEMS ARE SUBSETS OF
> RATIONAL SYSTEMS AND JUST INTERVALS ARE RATIO OF SMALL NUMBERS.
>
>
> IS THERE ANY BOARDER TO CALL A SYSTEM RATIONAL OR JUST? WHAT IS
> THE MAXIMUM LIMIT OF RATIO SMALLNESS TO BE JUST?

It greatly depends on the context -- instrumentation, frequency
range, what musical material preceded and what material follows.

But a rule of thumb around here is n*d < 100 is just, where
n and d are the numerator and denominator of the ratio in
question. This may be a bit low, since 15:8 is (just barely)
tunable by ear.

> IS IT RELATED TO NUMBER OF SOUND HARMONICS ? ARE 27/16 ,
> 70/64 , 81/75, 240/128 JUST OR RATIONAL?

By the above rule of thumb, they are all rational.

-Carl

> IS THERE ANY BOARDER TO CALL A SYSTEM RATIONAL OR JUST? WHAT IS THE
> MAXIMUM LIMIT OF RATIO SMALLNESS TO BE JUST?
>

Rational means you are using two natural numbers (positive integers)
in division. Just means you are using components of a rational set
limited by musical or numerological (harmonic) significance in the
magnitude of thier divisors. There is no max limit for just scales,
so any subset of rational numbers can be called just.
Anything else besides ratios of natural numbers are irrational scales,
for instance scales derived from the square root of two, ratios
involving the square root of three, phi, pi, e, etc.

>
> IS IT RELATED TO NUMBER OF SOUND HARMONICS ? ARE 27/16 , 70/64 , 81/75
> , 240/128 JUST OR RATIONAL?
>
>

Yes and both. Although Just implies you are preferring small whole
number ratios.

Thanks friends.

But Richard , what do you mean by small ? if we choose denominator as
8,10 , 11 , 15 , the ratio is just?

May be it relates to harmonic numbers in two sound of ratio ? may be
relate to beat number ? the ratio like 964:771 is non-just or
pseudo-just or even just (as carl said n*d=743244>100 so not just)? If
beat per second is a tool for identifying ratio as just what is its
limitation around zero even if the numbers are large?

CENT OF JI

N

D

964

771

386.7627436

refrence frequency(f1)

cent

Frequency(f2)

nearest ji

440

386.7627436

550.1426719

5

4

2200.0000000000

2200.57068742

Beat per second

0.570687419

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Richard Eldon Barber
Sent: Friday, September 16, 2005 10:13 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: LIMIT OF JUSTNESS OR RATIONALITY?

> IS THERE ANY BOARDER TO CALL A SYSTEM RATIONAL OR JUST? WHAT IS THE
> MAXIMUM LIMIT OF RATIO SMALLNESS TO BE JUST?
>

Rational means you are using two natural numbers (positive integers)
in division. Just means you are using components of a rational set
limited by musical or numerological (harmonic) significance in the
magnitude of thier divisors. There is no max limit for just scales,
so any subset of rational numbers can be called just.
Anything else besides ratios of natural numbers are irrational scales,
for instance scales derived from the square root of two, ratios
involving the square root of three, phi, pi, e, etc.

>
> IS IT RELATED TO NUMBER OF SOUND HARMONICS ? ARE 27/16 , 70/64 ,
81/75
> , 240/128 JUST OR RATIONAL?
>
>

Yes and both. Although Just implies you are preferring small whole
number ratios.

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________________________________

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:
> Thanks friends.
>
> But Richard , what do you mean by small ? if we choose denominator as
> 8,10 , 11 , 15 , the ratio is just?
>
> May be it relates to harmonic numbers in two sound of ratio ? may be
> relate to beat number ? the ratio like 964:771 is non-just or
> pseudo-just or even just (as carl said n*d=743244>100 so not just)? If
> beat per second is a tool for identifying ratio as just what is its
> limitation around zero even if the numbers are large?
>

If you are tuning by eliminating audible low freq beating, then you
are most likely approaching an interval within say the 9 or 11 limit,
or whatever your talented ear can hear, for instance, tuning just 3:2
on the piano. If your JI scale includes 964:771, then its your call.
If your scale includes irrational roots, or ratios designed to
impersonate irrationality, then calling it Just is out of the
question, IMHO.
Beat per second is frequency dependent. Small whole number ratios
are related to harmonic series, but scales do not necessarily derive
from either pure harmonics nor overtone series.
When you say the limitation around zero, I guess you mean how close
to just do your unisons or 3:2 have to get to just? On a piano, some
notes or ranges you want to zero out to 1/10 bps, some you are pleased
to leave at 1/2 bps. Especially if just one unfortunate string has a
beating overtone, you will need to "feather" in beats surrounding that
tone.

--- In tuning@yahoogroups.com, "Richard Eldon Barber"
<bassooner42@y...> wrote:
> --- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...>
wrote:
> > Thanks friends.
> >
> > But Richard , what do you mean by small ? if we choose
denominator as
> > 8,10 , 11 , 15 , the ratio is just?
> >
> > May be it relates to harmonic numbers in two sound of ratio ? may
be
> > relate to beat number ? the ratio like 964:771 is non-just or
> > pseudo-just or even just (as carl said n*d=743244>100 so not
just)? If
> > beat per second is a tool for identifying ratio as just what is
its
> > limitation around zero even if the numbers are large?
> >
>
>
>
> If you are tuning by eliminating audible low freq beating, then
you
> are most likely approaching an interval within say the 9 or 11
limit,
> or whatever your talented ear can hear,

Playing with sawtooth waves with headphones, I could tune ratios as
complex as 17:13 by eliminating beats by ear. But this is quite a bit
beyond what I could hear in a practical musical situation.