# Re: consonance of MELODIC intervals

```Dear Piet Vos,

I just respond because I see your question related to my opinion that
harmony is based on coincidence detection. I am not aware of any deviation
from Zarlino's simple fraction rule. So I naively read the well known
figure by Buser and Imbert 1992 after Wood 1944 after Helmholtz 1883,
showing hardness of the sound of two violins as a function of frequency
ratios, roughly as follows:

C4-C4:   1:1      (prime)       0/12  1
C4-C5:   2:1      (octave)     12/12  2
C4-G4:   3:2=1.5  (fifth)       7/12: 1.49831 diff. -0.00169
C4-F4:   4:3=1.33 (fourth)      5/12: 1.33484 diff. +0.00151
C4-A4:   5:3=1.66 (sixth)       9/12: 1.68179 diff. +0.01512 very large
C4-E4:   5:4=1.25 (major third) 4/12: 1.25992 diff. +0.00992 large
C4-Dis4: 6:5=1.2  (minor third) 3/12: 1.18921 diff. -0.01079 large
void     7:4=1.75
C4-Fis4: 7:5=1.4                6/12: 1.41421 diff. +0.01421 very large
C4-Gis4: 8:5=1.6                8/12: 1.58740 diff. -0.01260 large
void:    9:5=1.8
void:    7:6=1.166
void:    8:7=1.14857 etc.
C4-D4:   9:8=1.125              2/12: 1.12246 diff. +0.00254
C4-Ais:  (16:9=1.777)          10/12: 1.78180 diff. +0.00402
C4-B4:   (17:9=1.8888)         11/12: 1.88775 diff. -0.00114
C4-Cis4: (17:16=1.0625)         1/12: 1.05946 diff. -0.00304

It is just my guesswork that people may mainly differ with respect to
tolerance against deviation from the matching frequency at equally tempered
scale. In case of a third, that difference amounts about 0.01 times 256 Hz
= 2.56 Hz, it is, in terms of period about 32 microseconds.
This is of the same order as enlargement of perceived octave with respect
to the real one. Has anyone looked for an similar effect on fifth and
third? I realized that octave enlargement equals an first-order ISI
perception error of about 16 microseconds, independent of frequency, that
might correspond to width of action potential.

Regards,
Eckard Blumschein

At 15:36 20.06.00 +0200, you wrote:
>Hi,
>
>Does someone know how to order the 13 within-octave MELODIC intervals
>(prime, minor 2nd,... octave) as a function of consonance. I believe to
>know that there are more than one different possible orderings, in
>dependence of the theory on consonance (psycho-acoustical, mathematical,
>... theory). Thanks in advance for any help!
>
>Piet V.
>
>Piet G. Vos
>section Perception NICI, U. Nijmegen
>P.O.Box 9104
>6500 HE Nijmegen NL
>tel: +31 24 36126 31/20; fax: +31 24 361 60 66; vos@nici.kun.nl
>home-page: http://www.nici.kun.nl/~vos
>
>
>"et altissimus humilissimum facere debet"
>
```