Re:JI as fractions of an octave.

This is a beginner question, I just thought of it, but I'm sure
others already have.

I'm thinking that the interval '3/2' must be half of the distance
between 2/2 and 4/2 (or 1/1 and 2/1 respectively)? If so, then
wouldn't cents be inconsistent (because the interval 3/2, ~702
cents, is not half of an octave, 1200 cents)? Also it seems like my
explanation would explain why a 7 chord sounds good, it's the octave
divided into fourths with the intervals 4/4,5/4,6/4,& 7/4 creating
the chord 1:1.25:1.5:1.75.

But, I've read many times that the tritone (600 cents) is half an
octave. So am I missing something?

Evan

hi Evan,

--- In tuning@yahoogroups.com, "o2b_bambooguy" <send2evan@h...> wrote:

> This is a beginner question, I just thought of it, but
> I'm sure others already have.
>
> I'm thinking that the interval '3/2' must be half of the
> distance between 2/2 and 4/2 (or 1/1 and 2/1 respectively)?
> If so, then wouldn't cents be inconsistent (because the
> interval 3/2, ~702 cents, is not half of an octave,
> 1200 cents)? Also it seems like my explanation would
> explain why a 7 chord sounds good, it's the octave
> divided into fourths with the intervals 4/4,5/4,6/4,& 7/4
> creating the chord 1:1.25:1.5:1.75.
>
> But, I've read many times that the tritone (600 cents)
> is half an octave. So am I missing something?

yes. you're missing the difference between the arithmetic
of musical ratios, and the human perception of pitch --
which is logarithmic.

it has to do with what we perceive as "octave equivalence",
which is the "affect" of prime-factor 2, since the most
basic rational interpretation of the "octave" is the ratio
2:1.

because doubling or halving a frequency results in the
perception of "octave equivalence" -- i.e., all pitches
which are related by a 2:1 ratio are considered in some
sense to be "the same" -- the actual *frequency* measurement
of octaves keeps changing, always becoming double the
one below or half the one above.

for example, the octave below A-440 Hz is A-220 Hz,
which is 220 Hz less. but the octave above A-440 is
A-880 Hz, which is 440 Hz more.

because of this, the "arithmetic mean" between two
frequencies or two ratios is never the *logarithmic* mean.

thus, while 2:3:4 makes a nice arithmetic proportion
which has a "root" at the bottom, a 3:2 "perfect-5th"
in the middle, and 2:1 (i.e., 4:2) "octave" at the top,
the 3 will be slightly higher than the logarithmic mean.

as you observed, the logarithmic mean of the octave is
the equal-tempered "tritone" of 600 cents. the arithmetic
mean is the 3:2 "perfect-5th" of about 702 cents.

your explanation of "why the 7 chord sounds good" mirrors
Harry Partch's concept of the "numerary nexus". those
four ratios form a nice proportion of 4:5:6:7, which
is the most consonant chord that can be produced with
four different pitches which do not exhibit any octave
equivalence.

i hope that helps.

if you're a newbie to tuning theory, you might want
to take a look at my Encyclopaedia of Tuning.
try these for starters, and follow the links from there:

http://tonalsoft.com/enc/affect.htm
http://tonalsoft.com/enc/prime.htm
http://tonalsoft.com/enc/octave.htm
http://tonalsoft.com/enc/nexus.htm

-monz