repeating patterns

Here's something I've wanted to try, maybe someone already has. Suppose I
took two 12TET scales separated by 80/81. Then one would have both nice
sounding thirds and a wide open modulation horizon.

One idea I had for a keyboard setup was to use the left hand on one 12TET and
the right hand on the other. Then to play a simple diatonic scale one might
have a pattern like RRLRRLLR.

Equal tempered scales have the property that each note has the same set of
intervals available. With other scales, notes will fall into some set of
classes based on what precise intervals are available. Most non-ET scales
repeat on an octave cycle, so a typical 12 note per octave scale would have
12 distinct classes of notes.

The scale I propose above would have 24 notes per octave, but break into just
two classes.

Conventional ET is not the only system with interval symmetry. One can have
multiple generating intervals instead of just one. For example if one takes
the set of pitches (2**n)(3**m) then every note has the same intervals
available.

If I understand how the "k-limit" term is used, it might be useful to make a
finer distinction. Suppose I take a set of notes (2**n)(3**m)(5**p) where n
and m are unbounded but p is constrained to be 0 or 1. My understanding is
that this scale would be called 5-limit. The scale where p is also allowed
unbounded values would also be 5-limit. Perhaps the case of unbounded
exponents is not of sufficient interest to merit a new term?

Jim


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From: "Paul H. Erlich"
Subject: Reply to Gregg Gibson
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>>Nevertheless
>>most researchers have found Arab singers quite unable to reliably
>>reproduce these very narrow intervals, much less anything so esoteric as
>>the neutral third, which usually amounts to nothing more than a slightly
>>inaccurate minor third.
>
>That is pure hogwash. You obviously have not listened to much Arabic music
>with a sensitive ear. You should be more respectful of the world's musical
>cultures that to toss out anything that does not fit your preconceptions.
>There are Arabic scales whose aesthetic effect lies in the division of the
>octave into 3 large and 4 small intervals, unlike the usual 5 large and 2
>small. The performance of these scales by singers and other instrumentalists
>is no less accurate than the typical Western classical performance. I'd love
>to see your reaction to Gamelan scales!
>
>>The
>>second defect of the 22-tone equal involves its failure to close the
>>cycle of fifths.
>
>19-tone closes the cycle of fifths after 19 fifths. 22-tone closes after 22
>fifths.
>
>Of course, you probably don't mean the failure to close the cycle of fifths
>at all, you probably mean the failure of three fifths (minus an octave) to
>approximate a 5:3 major sixth, and of four fifths (minus two octaves) to
>approximate a 5:4 major third. In other words, you mean the non-vanishing of
>the syntonic comma.
>
>> I am aware that some will say to themselves - so what?
>>But a musical system that does not close the cycle of fifths has at a
>>stroke isolated itself from 99% of the music not merely of the western
>>19th century, but from virtually the whole of the western tradition,
>
>If we substitute "the non-vanishing of the syntonic comma" for "non-closing
>of the cycle of fifths" above, you are correct. However, 19th century
>composers such as Schubert rely on other properties of 12-tone tuning, such
>as the "closing of the cycle of major thirds" (really the vanishing of the
>diesis) that make 19-tone unusable for this music.
>
>>and
>>from many other musical traditions as well.
>
>Care to give an example?
>
>>Such musical systems, like
>>just intonation, are mere curiosities, and are far more impoverished in
>>usable, aurally distinct resources even than the 12-tone equal.
>
>You are calling Turkish, Hindu, and many other musics "impoverished." They
>often sound so to Western ears, but the riches lie deeply buried and require
>an open mind to begin to appreciate.
>
>>The four scales I gave have consonant triads (major or minor) available
>>on four of their seven degrees. For example:
>
>>C D E F G Ab B
>
>>has consonant triads on C, E, F & G, but not on D, Ab or B. I hope this
>>clarifies the matter.
>
>This is an important fact that I use all the time in 12-tone. What does
>19-tone add to these scales besides a bit of smoothness to the harmony?
>
>>Your refer to the 7 and 11 limits. The latter is a mere fantasm of the
>>just intonationists, and is not audible to the ear as anything other
>>than dissonance.
>
>How insulting! Your ear, perhaps. You are negating the testimony of Partch
>and others, including my own experience.
>
>>The septimal limit is more interesting. But it is
>>dissonant, however this may trouble those who aspire to forever extend
>>the boundaries of consonance, until someday I presume, we shall find
>>everything consonant, and music need trouble itself no longer with any
>>rules or constraints whatever...
>
>That day has already passed, in case you didn't notice...
>
>>Your reference to individuals being trained to reliably distinguish
>>melodic intervals as close as 10 cents is the most fantastic piece of
>>information I have ever been privileged to encounter.
>
>A tenth of a tone is 20 cents, but no matter. Intervals of this order of
>magnitude are used in speech, are reliably detected and impart meaning.
>


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From: "Paul H. Erlich"
Subject: Escaping the diatonic hegemony
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> A 19-tone unequal tuning would result in
> appreciably more intervallic confusion, given the fact that the 19-tone
> equal tuning degree is close to the limen.

The semitones in LucyTuning are different enough to both be useful.
The chromatic semitone had a mysterious sound that is quite audible.
Whether it can be sung accurately is another matter. It is right on
the borderline as regards the melodic limen, but I reckon a _good_
singer, with the right training, could manage it. I don't place
much importance on this, however, because temperament is only an
issue for fixed pitch intruments. The vocals can take care of
themselves -- witness rock with 12 equal guitars.

> Modulation would also be made
> more difficult, and certain consonances would be too deteriorated for
> use.

No consonant chords are deteriorated in LucyTuning relative to 19
equal. 19 notes will get you a long way, modulation wise.

>> Gregg's dislike of non-meantone temperaments is well established. As
>> an existentialist, I cannot agree, because I use and will continue to
>> use such temperaments.
>
> What do you mean by existentialist? Do you have a favorite tuning? If
> so, why do you value it especially? If you tell me, I will refrain from
> sending my Gestapo to stop you from using the temperament or system you
> prefer.

An existentialist is someone who believes existence precedes essence.
Is that clear? Oh, um, well ... In this case, I simply mean that
the fact I use such temperaments means that I must approve of them.
The justifications I give may not be my real reasons, but it is
beyond doubt that such reasons exist.

The two temperaments I use most often -- and therefore the ones I
must value most highly -- are 31 equal and a schismic fourth mapping.
31 equal needs no introduction, of course. I use a 12 note mapping
-to a conventional keyboard, usually with Ab rather than G#. I
sometimes use other meantone temperaments with the same mapping.
I am fully aware of the simplifications that result with meantone.
I prefer 31 equal because of its good septimal approximations.
Actually, the second scale (after 12 equal) I ever tuned up is a
subset of 19 equal. It is a good way of cutting your meantone
teeth, because of the high contrast between different semitones.

Now, to the schismic fourth. I outlined this on the list before.
To summarise, it uses schismic temperament (a major third is
associated with -8 rather than +4 fifths) with 12 steps to a fourth.
This is consistent with 29 equal. The actual tuning I use is near
enough Pythagorean. This mapping is harder to use than the
meantone one, but the advantage is in purer intervals and more
melodic sublety. The more I use it, the more I get to like it.
Although the harmonies are good, I haven't worked out any chord
sequences yet, other than approximations of standard diatonic ones,
which require comma shifts.

Being able to use such small intervals means you can add an
emotional edge not usually possible on a fixed pitch instrument.
And, when you want a pure interval, it's really ease to find.
If I want, I can get a mistuned affect by hitting the "wrong"
keys.

Originally, I was tuned to 53 equal. As well as the relatively poor
septimal approximations, I didn't like the commas being so small.
They have a sort of distressing sound as a result of being barely
audible, and can't function melodically. Switching to 41 equal
solved both problems. After some theoretical work, though, I found
the better approximation. The commas are smaller -- slightly larger
than Pythagorean -- but just large enough to be audible.

I'm also starting to play with 22 equal, now that I've got a
symmetrical keyboard. I have previously used Paul Erlich's 22 note
mapping with 46 equal. I gave up on this because I decided the
schismic mapping works better, although it has even more notes.

I have made some use of 7 and 5 equal, extended just intonations,
and various "ethnic" tunings.


If acoustic instruments are to be liberated from 12 equal, I
suggest they be designed to play in the widest possible variety
of tunings, rather than simply adopting a new temperament. The
choice of tuning is as personal as orchestration and would, in an
ideal world, be available to all composers.

I can't see it's at all reasonable to expect the world in general to
switch to 19 equal. It may work for wind instruments, as a template
from which to bend to 7-limit harmony, but it is too extreme for
people to accept it otherwise. By "extreme" I mean relative to 12
equal which has a special place as all other temperaments are, in a
sense derived from it. 12 equal is a meantone, schismic and
diaschismic scale. As Gregg observed, other scales tend to be
melodically confounded with 12 pitch classes. Orchestral instruments
incompatible with the repertoire of -- like it or not -- 12 biased
music aren't going to catch on.


SMTPOriginator: tuning@eartha.mills.edu
From: Steven Rezsutek
Subject: Re: 19tet vs. meantone?
PostedDate: 17-12-97 20:57:18
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