RE: 12-ET Intervals list

>Ciao. I'm working with a friend on a software for guitar and I need a
>list of the intervals in 12-ET from the more consonant to more>
>dissonant. With "consonant" I mean beatless, empty. So I've tried to do
>my own list based only upon my ear:

>octave
>P5
>P4
>6M
>3M
>3m
>6m
>7m
>4+
>2M
>7M
>2m

>This list is istinctive but I'd like to have a more scientific
>justification if possible. Anyone can help me? Maybe with a mathematic
>law on beating and interval spacing (spacing: 10M is more consonant
>than 3M). Ciao.

>-Lorenzo

Hindemith tried to do the same thing, mathematically rather than
instinctively,
and his list was identical to yours through the first seven entries. After
that, he went astray due to his refusal to accept 7-limit interpretations.
So he but the tritone last, although it is a 7:5 out of tune by 17 cents,
which by naive mathematics should be similar in consonance to the minor
sixth, an 8:5 out of tune by 14 cents.

The most common definition of consonance is derived from a psychoacoustical
phenomenon known as roughness. The exact rankings will depend on the
overtone structure of the tones in question; two Japanese (?) workers whose
last names both start with a "K" (anyone?) wrote a paper in JASA giving an
example of how to calculate this. They produced a continuous graph showing
roughness from the unison up to one or two octaves; needless to say, there
were dips in the graph at the first six intervals in your list. The roughest
intervals were out-of-tune octaves, fifths, and especially unisons.

Another component of consonance is tonalness, or how well the tones "fuse"
into a single complex-tone perception. I recently posted an algorithm for
determining the certainty of perceiving any tempered interval as any
interval from the harmonic series. This algorithm only assumes that there
will be some audible harmonic overtones within an optimal (rather high)
frequency range). There is only one free parameter in the algorithm, the
finest pitch resolution of the ear in a harmonic context. Unfortunately,
this algorithm doesn't translate well into an absolute ranking of intervals.
We can, however, look at the worst-case (out of all inversions and
extensions) certainty for the most likely interpretations of the
pitch-classes 0-6:

resolution = 1%:
semitones interpretation certainty
0 1/1 40.1
1 18/17 2.35
2 9/8 4.34
3 6/5 5.31
4 5/4 5.85
5 4/3 13.3
6 7/5 or 10/7 3.42

This basically agrees with the ranking in your list.


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