to G. Gibson

Bill Alves wrote,

>Anyway, I do not choose tuning systems to make sure that a melody is
>perceived as different if I choose a slightly altered set of intervals.

Well said! Choosing a tuning system with that justification might make
sense in some algorithmic composition contexts but otherwise seems
poorly motivated.

Besides, Turkish musicians are trained in 53-tone and can readily
distinguish between, and perform in, scales where one member is
displaced by a comma (as often happens when comparing Turkish, Persian,
and Hindu scales).

Gregg Gibson wrote,

>>I have used tree diagrams to examine all the possible heptatonic modes
>>of the 19-tone equal that have a modicum of consonant intervals and
>>chords, hence the four new modal genera, which I have previously
>>suggested might profitably be added to the diatonic. In harmony these 28
>>new modes constitute what may be called the chromatic music, whose
>>existence has been hitherto intuited, but never plainly defined.
>
>All of these 28 modes are well-represented in 12-equal. Or, if one accepts
>the 12-equal version of the 7 diatonic modes, one should accept the 12-equal
>version of the other 21 modes as well. I use them all the time on the 12
>guitar and have developed exercises for learning 18 of them. So how are these
>modes a justification for 19-equal?
>
>Let me again suggest that the observed phenomenon of just over a quarter-tone
>being the smallest accepted melodic interval is nothing more that the result
>of a learned 12-equal template (albeit a movable one for those without
>perfect pitch). Intervals from just over a quarter tone to just under
>3/4-tone will be classified in the same group by those whose musical
>experience has been limited to 12-equal. Similarly, intervals less than a
>quarter tone will most closely resemble unisons in their field of experience.
>
>


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From: Gregg Gibson
Subject: Reply to Paul Ehrlich
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Gregg Gibson:
> >}Of
> >}course the centre of 22-tone equal in India is nowadays supposed to be
> >}the South.

Paul Ehrlich:
> >
> >Where did you get this idea?

I am uncertain what you mean. This is a matter of elementary knowledge.
Look in any elementary text on the subject.


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From: Gregg Gibson
Subject: No Disrespect
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Paul Ehrlich:
> It is not difficult to find examples
> >in Wagner, Stravinsky, and jazz where the basic consonances are 7-limit
> >tetrads, by virtue of all other harmonies being even more dissonant. Triads
> >sound incomplete in this sort of context.

By such reasoning one could argue that 15:8 is consonant, because 48:25
is even rougher. Your argument is therefore evidently invalid.

English words such as 'incommensurable' admit of several senses.

Your definition of the syntonic comma is one definition, but it is
irrelevant to a discussion of precisely how the consonant cycles are to
be reconciled in an equal temperament.

Your assertion that you have never denied 22-tone equal to be unfit for
diatonicism comes strangely from someone who has proclaimed this system
the salvation of music, but perhaps you mean to preserve 12-tone equal
for diatonicism. I have heard this sort of thing before - it always
reduces to "something special for ME" and 12-tone equal for everybody
else.


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From: Gregg Gibson
Subject: My Friends the Aliens
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Gregg Gibson said:
> >>The ideal value for the octave itself is of course 1200/19 = 63.16 ;
> >>here are the values for the other six consonances.
> >>3:2 701.96/11 = 63.82
> >>4:3 498.04/8 = 62.26
> >>5:4 386.31/6 = 64.39
> >>5:3 884.36/14 = 63.17
> >>6:5 315.64/5 = 63.13
> >>8:5 813.69/13 = 62.59
> >
> >>If one considers the consonances to be all of equal importance, one can
> >>simply take the average of the above seven figures, which is 63.22, to
> >>arrive at a figure for the octave of 1201.2.

Paul Ehrlich said:
> >Gregg, this is not correct. You are not giving equal importance to the
> >intervals above. Since whatever compromise is made in taking the average will
> >be multiplied 14-fold for the 8:5 but only 5-fold for the 6:5, you are being
> >more permissive for tuning errors in the larger intervals. I am certain
> >Fokker would not have made this mistake, being a brilliant physicist. I know
> >you will probably gloss over this post the way you have dismissed, with total
> >lack of understanding, my other comments. But if you are interested, I will
> >tell you the other things wrong with your calculations here, and how I
> >arrived at an optimal octave of 1202.7 cents for 19-equal.

Since you presume to assert that I have "glossed over" and dismissed
your posts with "total lack of understanding" and since you compound
your folly by seeking to assert that the great Fokker would have fallen
into the gross error into which you have fallen, sir, I am compelled to
explain to you that:

1. Fokker's method is as I have stated. Read the references for
yourself. If you do not have access to Fokker's original texts, use
Mandelbaum.

2. Fokker could have used _no_ method to achieve equal weighting of
deviations of the consonances. The most _basic_ principle of
temperament, which is taught to all _beginners_ in this discipline, is
that there can be _no_ temperament which compromises the consonances
equally. This arises from the mathematics of the case. _Learn_
mathematics before you make such statements, which again, only show your
ignorance.

Now that I have administered the bitter pill, let me add a large
tablespoon of sugar. You are quite bright. You were led into this faulty
conclusion by assuming that the consonances are independent, and so,
vary independently when tempered. I use non-mathematical language here
deliberately. But in fact the consonances consist of three pairs, 5:4 &
8:5, 6:5 & 5:3, and 3:2 & 4:3, each of which pair varies as if it were a
single interval, so far as temperaments are concerned. Therefore, to
seek some system which shall weight deviations from the consonances
equally, even in the case in which the octave is itself tempered, is the
classic pons asinorum of temperament, for no such system can exist. Were
however, you to confine yourself to the narrower (but unfortunately,
perfectly irrelevant) objective of finding a tempered octave which
should deteriorate both of the members of each _pair_ of the consonances
equally, either from just or from their 19-tone equal values,
independent of the other two pairs, then, and then only, would your
objection be valid.

I advise you to study the cycle of 22-tone equal consonant fifths. See
if you can find a consonant third, four fifths above the tonic. If you
cannot (and you cannot) your arguments fall to the ground. I repeat that
when I was a beginner, I made the same mistake you are making.

Your assertion that the chromatic modes of 19-tone equal are just as
well expressible in 12-tone equal, could never be made by anyone who has
heard them, and extensively used them in music with any harmonic
element. Even in melody, and even insofar as the diatonic modes are
concerned, the 12-tone equal modes so deviate from the tonal fabric of
just intonation that they are very evidently deteriorated. But in the
chromatic modes, where the augmented tone is so much used, the
difference is enormous.

Finally, let me observe that the value of 1202.7 cents for the octave,
while apparently very precise, and possibly indicative that one person
may have copied the other without acknowledgement, actually follows from
the decision to round the 19-tone equal tuning degree, 63.16 cents,
upward by the nearest tenth of a cent. 63.2 cents gives an octave of
63.2 x 19 = 1200.8 cents. 63.3 cents gives an octave of 63.3 x 19 =
1202.7 cents. 63.4 cents gives an octave of 1204.6 cents. Back in the
early 90's I settled on the value of 63.3 cents as the value which seems
best when taken to the nearest tenth of a cent. Almost anyone who
thought of tempering the 19-tone octave would be led to 1202.7 cents as
the most eligible of the three values. I repeat that I have no interest
in disputing priority of this discovery. The true credit, insofar as it
belongs to anyone, belongs partly to Fokker, who to my knowledge at
least, first suggested tempering the octave with the purpose of
improving the other consonances. He therefore put the germ of this idea
in my own mind, and it is my habit to make occasional reference to this
as a kind of tribute. This is a very important discovery, for by it the
19-tone equal becomes virtually as smooth harmonically as the 31-tone
equal.

I bear you _no_ personal ill will. I myself _do_ sometimes make bad,
terribly embarassing mistakes. We all do. The best defense against
despising ourselves is to adopt an elaborately civil tone in all public
discourse.


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From: Gregg Gibson
Subject: Mandelbaum's Use of 31-tone Equal
PostedDate: 20-12-97 05:51:47
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