Defining "Just" Intonation and "Consonance"

I just figured out the following info which perhaps may interest others.

Just Intonation Ratio Limits [with respect to the denominator]
2 Limit 3/2 5/4 7/4 9/8 11/8 13/8 15/8
Cents 701.96 386.31 968.83 203.91 551.52 840.53 1088.27


3 Limit 4/3 5/3 7/6 11/6 13/12 17/12 19/12
Cents 498.04 884.36 266.87 1049.36 138.57 603.00 795.56

5 Limit 6/5 7/5 8/5 9/5 11/10 13/10 17/10
Cents 315.64 582.51 813.69 1017.60 165.00 454.21 918.64

7 Limit 8/7 9/7 10/7 11/7 12/7 13/7 15/14
Cents 231.17 435.08 617.49 782.49 933.13 1071.70 119.44


Does anyone have any comment as to why the following list of "just" intervals
isn't otherwise accepted for the most "consonant" use insofar as they are based
on the second most consonant interval of a Perfect Fifth besides the Octave.
(The consonance of the unison being obvious.)

Perfect Fifth Derived Ratios for the chromatic scale
(i.e. based on 3/2 and adjusted for octave range)
Int Ratio Cents
P5 3/2 701.96
M2 9/8 203.91
M6 27/16 905.87
M3 81/64 407.82
M7 243/128 1109.78
D5 729/512 611.73
m2 2187/2048 113.69
m6 6561/4096 815.64
m3 19683/1638 317.60
m7 59049/32768 1019.55
P4 177147/131072 521.51
531441/524288 23.46
Octave!!? (1200+23.46)

*D5=diminished fifth, augmented fourth, tritone.

Notably above, the "octave" by the next derived ratio of 531441/524288 equals
23.46 cents! And the Perfect Fourth "commonly" represented by 4/3 = 498.04
cents is opposed to 177147/131072 = 521.51 cents "derived" above. Essentially,
I'd like to know what is the definition of "just" intonation and how "should"
it be applied. I would assume one critical factor would be the limitations of
human hearing in its accuracy of pitch perception but the "Fibonacci spiral" of
the cochlea would also seem significant. Comments on this would be most
appreciated.


Another example similar to the above is the Perfect Fourth derived chromatic
scale:
Int Ratio Cents
P4 4/3 498.04
m7 16/9 996.09
m3 32/27 294.13
m6 128/81 792.18
m2 256/243 90.22
D5 1024/729 588.27
M7 4096/2187 1086.31
M3 8182/6561 384.36
M6 32768/19683 882.40
M2 65536/59049 180.45
P5 262144/177147 678.49
1045876/531441 1176.54
Octave!!? (1176.54-1200) = -23.46

-------------
Benjamin Tubb
AIM: brtubb
ICQ: 650264
brtubb@cybertron.com
http://home.cybertron.com/~brtubb/theory.html

The Music of Stephen Collins Foster (1826-1864)
http://www.geocities.com/Nashville/9958/

>It was not "Ensoniq" the
>company that gave us their implementation -- it was Steven Curtin,

Well, Steve was an Engineer working for Ensoniq at the time, but it's
certainly true that it turned into something of a labor of love for him -
beyond a routine work assignment.