RE: hekany dekany dock

Neil, thanks for your interest. Once again, the 22-tone guitar is with
me for the long haul -- my little bits of 22-tone magic so far, some
conforming to my theories, others not, are really just the infancy of
something new, and I don't want to take any shortcuts in the learning
process. Sure, I've jammed on some 22-tone blues, etc., but that's not
what it's all about for me. Right now, I'm too busy with conventionally
tuned music and musicians to do what I will someday need to do -- shut
myself away from the diatonic outside world and really explore musical
space unhindered. I just turned 25 yesterday, and I guess it's keeping
me happy for now to keep surfing the 12-tone wave I'm riding -- there's
nothing like sharing a common musical language with musicians all over
the Western world (or what amounts to the same thing, Cambridge). I'm
not going to rush anything, and when the time is right, the 22-tone
>stuff will come together . . .

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Graham wrote,

>On dissonance of composites in general: it is much easier to form
>consonant chords using composite numbers than prime numbers of
>about the same size. As an octave invariant example, 1:3:5:15
>includes the interval 15/1. Try finding an equally consonant 4
>note octave invariant chord including 13/1. If the consonance of
>an interval reflects its propensity to form consonant chords, then
>composites are more consonant than primes and, in some
>circumstances, the metric should handicap high primes accordingly.

In some circumstances, but not in others (such as when intervals are
sounded alone). Graham, how does your metric address this?

The consonance of a chord depends on all intervals within it, not just
the most dissonant one. 5/6 of the intervals in 1:3:5:15 are consonant
within the 5-limit, which will not be true of any 4-not chord containing
13. That should be enough of a handicap against 13. I admit that there
might not a single metric that will give the consonance of intervals as
well as of chords. The consonance of chords is a combination of
roughness issues (see Plomp & Levelt, Kameoka & Kuriyagawa, Sethares)
and tonalness issues (Parncutt, preliminary model by me (posted once),
upcoming model by me (if I live long enough) based on harmonic entropy
concept).

3:5:9:15 is at least as consonant as 1:3:5:15 although the rectangular
5-limit lattice favors the latter. The triangular lattice seems better
in all circumstances.

>An unrelated (or is it) point. Recently, Gary Morrison wrote:

>> By fibonacci, what I'm referring to is J. Yasser's fibonacci-like
>>sequence of tunings (5 7 12 19 31 50 81...). (By fibonacci-like I mean
>>that the next number in the sequence is the sum of the previous two.) 12

>Firstly, wasn't it Kornerup who first identified this sequence?

In a very different context. Kornerup found a subset of ETs consistent
with meantone notation and composition, and approaching a meantone
tuning (often called golden tuning) where the tone, the minor third, the
perfect fourth, and the minor sixth are all divided into two parts by a
golden section. I don't think he included 5 or 7. Yasser thought that at
any stage in the evolution of tonality, the total number of pitches in
use is one number in the sequence and the number of diatonic pitches is
the previous number. He starts his sequence with 3, 2, 5, 7, . . . I
tend to side with Kraehenbuehl and Schmidt that 12 out of 22 is a more
self-delimiting (i.e., logically closed) scale than 12 out of 19 and a
more logical outcome of 12-tone chromaticism. K&S dealt with JI, so my
agreement with them may be coincidental, although we both rely on the
introduction of the number 7 into consonant frequency ratios. Yasser's
frequency ratios use primes up to 13 and are unbelievably out of tune
with 19-tet.

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Graham wrote,

>On dissonance of composites in general: it is much easier to form
>consonant chords using composite numbers than prime numbers of
>about the same size. As an octave invariant example, 1:3:5:15
>includes the interval 15/1. Try finding an equally consonant 4
>note octave invariant chord including 13/1. If the consonance of
>an interval reflects its propensity to form consonant chords, then
>composites are more consonant than primes and, in some
>circumstances, the metric should handicap high primes accordingly.

In some circumstances, but not in others (such as when intervals are
sounded alone). Graham, how does your metric address this?

The consonance of a chord depends on all intervals within it, not just
the most dissonant one. 5/6 of the intervals in 1:3:5:15 are consonant
within the 5-limit, which will not be true of any 4-not chord containing
13. That should be enough of a handicap against 13. I admit that there
might not a single metric that will give the consonance of intervals as
well as of chords. The consonance of chords is a combination of
roughness issues (see Plomp & Levelt, Kameoka & Kuriyagawa, Sethares)
and tonalness issues (Parncutt, preliminary model by me (posted once),
upcoming model by me (if I live long enough) based on harmonic entropy
concept).

3:5:9:15 is at least as consonant as 1:3:5:15 although the rectangular
5-limit lattice favors the latter. The triangular lattice seems better
in all circumstances.

>An unrelated (or is it) point. Recently, Gary Morrison wrote:

>> By fibonacci, what I'm referring to is J. Yasser's fibonacci-like
>>sequence of tunings (5 7 12 19 31 50 81...). (By fibonacci-like I mean
>>that the next number in the sequence is the sum of the previous two.) 12

>Firstly, wasn't it Kornerup who first identified this sequence?

In a very different context. Kornerup found a subset of ETs consistent
with meantone notation and composition, and approaching a meantone
tuning (often called golden tuning) where the tone, the minor third, the
perfect fourth, and the minor sixth are all divided into two parts by a
golden section. I don't think he included 5 or 7. Yasser thought that at
any stage in the evolution of tonality, the total number of pitches in
use is one number in the sequence and the number of diatonic pitches is
the previous number. He starts his sequence with 3, 2, 5, 7, . . . I
tend to side with Kraehenbuehl and Schmidt that 12 out of 22 is a more
self-delimiting (i.e., logically closed) scale than 12 out of 19 and a
more logical outcome of 12-tone chromaticism. K&S dealt with JI, so my
agreement with them may be coincidental, although we both rely on the
introduction of the number 7 into consonant frequency ratios. Yasser's
frequency ratios use primes up to 13 and are unbelievably out of tune
with 19-tet.

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