Intervals n:m where n+m > 17

The ratio of 19:16 for the minor third came up recently in the list and everyone seemed to take it seriously. This made me want to further my education in this regard.

I feel that (as a rule of thumb for ordinary timbres) we can ignore the accuracy of any ratios whose numerator and denominator (in lowest terms) sum to more than 17. This is a different kind of "limit". It means that even the 11-limit (standard usage) is of limited interest, and 19-limit and above, no interest at all. I mostly even look upon 9-limit as merely something you get for free when you have 7-limit with good fifths.

11:1 or 11:2 and maybe even 11:3 and 11:4 might just sound like something but I suspect it would take a sawtooth wave, a hightly trained ear and an extremely accurate ratio to appreciate 11:7 or 11:8 as anything different from the general dissonance in their vicinity. Sethares theory explains this nicely.

Obviously there is a grey area and it depends on timbre, vibrato etc. I might be persuaded to push the magic number to 19. But even so, 19:16 just looks ridiculous to me (summing to 35). This is merely a 6:5 that's been flattened about 18 cents. Can anyone really tell a 19:16 from a 13:11 (8 cents narrower) or either of these from anything in between.

I believe that some time ago Paul Erlich agreed with me in suggesting that 15:8 is not the slightest bit consonant but just happens to occur in chords with other very consonant intervals. I assume he had in mind the major 7th chord i.e.

C E G B
4 5 6
4 5
4 6
8 15 (really just a dissonance between 9:5 (or 11:6 if you insist) and 2:1)

At 16:42 17/11/98 -0500, Paul Erlich wrote (in private email; hope you don't mind, Paul):
>Actually, I think the modern musician latches onto 16:19:24 as the
>tuning of the minor triad and does not accept 10:12:15. However, when
>presented the intervals in isolation, 6:5 and 5:4 will definitely be
>preferred. There's something about 16:19:24 which give the right "root"
>to the minor triad (since 16 is octave-equivalent to 1) while 10:12:15
>does not. So despite all the theoretical niceties, the real-world
>musician in me can't ignore 19.

I suspect it's just that we're so used to hearing the mistuned thirds of 12-tET that people expect them. 19/16 is essentially identical to (only 2.5 cents flatter than) a 12-tET minor third. I guess it could be that the 'virtual fundamental' and difference-tone effects of the 16:19:24 otonality have a strong enough upward 'push' to overcome the downward 'pull' of the consonance of the just 5:4 and 6:5, only 15.6c away in the 1/6:1/5:1/4 utonality. But in the absence of familiarity with 12-tET I doubt it. Any empirical data?

Paul also wrote:
>The diminished seventh chord is tuned 10:12:14:17, and the dominant
>flat-9 is tuned 8:10:12:14:17, when performed by a barbershop quartet or
>other free-pitched ensemble.

Are you sure the dim7th is not 10:12:14:16.8 i.e.

C Eb F# A
5 6 7
5 6
5 7
3 5 (+13.8c)

Note that I consider it much more meaningful to write 5:3(+13.8c) than 42:25 in this context.

I find it very hard to believe that the singer of the high note could resist the downward pull of the 6:5 and 7:5 only 20.5c away and the 5:3 only 34.3c away. Particularly when you consider that the 13.8c comma, when distributed, leads to only 3.5c errors in the 5:3, 6:5's and 7:6, and only 7c in the 7:5's.

For the dom flat 9th I propose:

C E G Bb Db
4 5 6 7
5 6
5 7
3 5 (+13.8 c)
x y (approx 1280c dissonance)

I believe it's irrelevant whether you call y:x 17:8, 19:9, 21:10, 23:11 or 25:12. It's simply in that highly dissonant region between the octave and the ninth (or the 11:5 if you insist).

Is the 10:12:14:17 barbershop thing unequivocally supported by precise measurements?

How do barbershops sing minor triads? 16:19:24 or 1/6:1/5:1/4?

Is there a document on a website somewhere that gives 'just'-ifications like those above for most chords?

If you've already been over this before (I'm relatively new here) what key words should I search for in the archives?
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

> From: Dave Keenan <d.keenan@uq.net.au>
>
> The ratio of 19:16 for the minor third came up recently in the list and everyone seemed to take it seriously. This made me want to further my education in this regard.
>
> I feel that (as a rule of thumb for ordinary timbres) we can ignore the accuracy of any ratios whose numerator and denominator (in lowest terms) sum to more than 17. This is a different kind of "limit". It means that even the 11-limit (standard usage) is of limited interest, and 19-limit and above, no interest at all. I mostly even look upon 9-limit as merely something you get for free when you have 7-limit with good fifths.

Even though you can have ratios that approx. others closely, many times the difference tone will be wide enough to change its perception. I think it is much to soon to jump the gun and start making rules.
Lets take the 13/11 and the 19/16 as the basis of a minor chord. With the first we could have (transposed up an octave) 22/26/33 and the second 16/19/24 now the difference tones generated by the first are 4, 7, and 11 with the second 3,5,8. We can see that the difference of 19/16 will generate the third tone of our chord while in the first , the four will clash with the 33. The 16/19/24 is more consonant to my ear. Limits are mathematical constructs that get us into new areas, but we are no where close to drawing any
conclusions as to how are musical mind perceives or processes information. Also the work of LaMonte Young continues to pioneer music is all the area omitted by such rules. Our perception of processing harmonics is probably no more limited that our perception of colors!
On the other hand, just as LaMonte uses no 5s you as an aesthetic decision you might wish to explore the musical area below the 19th! v
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Dave: I suggest you do your own listening experiments, with a few
different timbres, to understand the pull of 17 in 8:10:12:14:17. It's
stronger than an interval-by-interval analysis would suggest, as
evidenced by the instability of its utonal counterpart (which has the
same intervals). Increasing the number of notes in a chord increases the
strength of combination tones in a disproportionate way (due to the
nonlinearity of difference tones and to the multiplicity of new
intervals between various combinations of "real" notes and existing
combination tones). Also, once there are more than two or three notes,
the ear tries really hard to reduce the whole thing into a single
complex tone, making harmonic series fragments advantageous. But timbre
affects all of this to a huge extent.

>Obviously there is a grey area and it depends on timbre, vibrato etc. I
might be persuaded to >push the magic number to 19. But even so, 19:16
just looks ridiculous to me (summing to 35). >This is merely a 6:5
that's been flattened about 18 cents. Can anyone really tell a 19:16
from a >13:11 (8 cents narrower) or either of these from anything in
between.

As a lone interval, 6:5 certainly dominates and 19:16 is nearly
impossible to pick out. As the top interval of a major triad, 6:5 is
unquestionably what we want to hear. But as the bottom interval of a
minor triad, every trained musician I've asked definitely prefers 19:16.
It may be partially familiarity but that can't be all because then the
same effect would show up in the major triad and the lone interval.

>At 16:42 17/11/98 -0500, Paul Erlich wrote (in private email; hope you
don't mind, Paul):
>>Actually, I think the modern musician latches onto 16:19:24 as the
>>tuning of the minor triad and does not accept 10:12:15. However, when
>>presented the intervals in isolation, 6:5 and 5:4 will definitely be
>>preferred. There's something about 16:19:24 which give the right
"root"
>>to the minor triad (since 16 is octave-equivalent to 1) while 10:12:15
>>does not. So despite all the theoretical niceties, the real-world
>>musician in me can't ignore 19.

>I suspect it's just that we're so used to hearing the mistuned thirds
of 12-tET that people expect >them. 19/16 is essentially identical to
(only 2.5 cents flatter than) a 12-tET minor third. I guess it >could be
that the 'virtual fundamental' and difference-tone effects of the
16:19:24 otonality have a >strong enough upward 'push' to overcome the
downward 'pull' of the consonance of the just 5:4 >and 6:5, only 15.6c
away in the 1/6:1/5:1/4 utonality. But in the absence of familiarity
with 12->tET I doubt it. Any empirical data?

I've had my piano in meantone for a few months now, and the minor triads
definitely sounded quite wrong at first. They sound quite right now.
Familiarity is an important factor, even on the timescale as short as a
few bars of a single piece of music.

David
any prime above 1 is significant
and though these intervals are queer or bad thirds whatever at first they cannot be ignored as they are fundamental wrungs on the spiral. Nineteen is significant because it is known as a twin prime its (twin) being 17 --seperated by only one interval the composite 18. My current tuning is limited to 17 but when I begin to use nineteen I will be able to answer deeper concerns but 31 seems like the next nugget.
Love and Tets
Pat Pagano, Director
South East Just Intonation Society

Dave Keenan wrote:

> From: Dave Keenan <d.keenan@uq.net.au>
>
> The ratio of 19:16 for the minor third came up recently in the list and everyone seemed to take it seriously. This made me want to further my education in this regard.
>
> I feel that (as a rule of thumb for ordinary timbres) we can ignore the accuracy of any ratios whose numerator and denominator (in lowest terms) sum to more than 17. This is a different kind of "limit". It means that even the 11-limit (standard usage) is of limited interest, and 19-limit and above, no interest at all. I mostly even look upon 9-limit as merely something you get for free when you have 7-limit with good fifths.
>
> 11:1 or 11:2 and maybe even 11:3 and 11:4 might just sound like something but I suspect it would take a sawtooth wave, a hightly trained ear and an extremely accurate ratio to appreciate 11:7 or 11:8 as anything different from the general dissonance in their vicinity. Sethares theory explains this nicely.
>
> Obviously there is a grey area and it depends on timbre, vibrato etc. I might be persuaded to push the magic number to 19. But even so, 19:16 just looks ridiculous to me (summing to 35). This is merely a 6:5 that's been flattened about 18 cents. Can anyone really tell a 19:16 from a 13:11 (8 cents narrower) or either of these from anything in between.
>
> I believe that some time ago Paul Erlich agreed with me in suggesting that 15:8 is not the slightest bit consonant but just happens to occur in chords with other very consonant intervals. I assume he had in mind the major 7th chord i.e.
>
> C E G B
> 4 5 6
> 4 5
> 4 6
> 8 15 (really just a dissonance between 9:5 (or 11:6 if you insist) and 2:1)
>
> At 16:42 17/11/98 -0500, Paul Erlich wrote (in private email; hope you don't mind, Paul):
> >Actually, I think the modern musician latches onto 16:19:24 as the
> >tuning of the minor triad and does not accept 10:12:15. However, when
> >presented the intervals in isolation, 6:5 and 5:4 will definitely be
> >preferred. There's something about 16:19:24 which give the right "root"
> >to the minor triad (since 16 is octave-equivalent to 1) while 10:12:15
> >does not. So despite all the theoretical niceties, the real-world
> >musician in me can't ignore 19.
>
> I suspect it's just that we're so used to hearing the mistuned thirds of 12-tET that people expect them. 19/16 is essentially identical to (only 2.5 cents flatter than) a 12-tET minor third. I guess it could be that the 'virtual fundamental' and difference-tone effects of the 16:19:24 otonality have a strong enough upward 'push' to overcome the downward 'pull' of the consonance of the just 5:4 and 6:5, only 15.6c away in the 1/6:1/5:1/4 utonality. But in the absence of familiarity with 12-tET I doubt it. Any empirical data?
>
> Paul also wrote:
> >The diminished seventh chord is tuned 10:12:14:17, and the dominant
> >flat-9 is tuned 8:10:12:14:17, when performed by a barbershop quartet or
> >other free-pitched ensemble.
>
> Are you sure the dim7th is not 10:12:14:16.8 i.e.
>
> C Eb F# A
> 5 6 7
> 5 6
> 5 7
> 3 5 (+13.8c)
>
> Note that I consider it much more meaningful to write 5:3(+13.8c) than 42:25 in this context.
>
> I find it very hard to believe that the singer of the high note could resist the downward pull of the 6:5 and 7:5 only 20.5c away and the 5:3 only 34.3c away. Particularly when you consider that the 13.8c comma, when distributed, leads to only 3.5c errors in the 5:3, 6:5's and 7:6, and only 7c in the 7:5's.
>
> For the dom flat 9th I propose:
>
> C E G Bb Db
> 4 5 6 7
> 5 6
> 5 7
> 3 5 (+13.8 c)
> x y (approx 1280c dissonance)
>
> I believe it's irrelevant whether you call y:x 17:8, 19:9, 21:10, 23:11 or 25:12. It's simply in that highly dissonant region between the octave and the ninth (or the 11:5 if you insist).
>
> Is the 10:12:14:17 barbershop thing unequivocally supported by precise measurements?
>
> How do barbershops sing minor triads? 16:19:24 or 1/6:1/5:1/4?
>
> Is there a document on a website somewhere that gives 'just'-ifications like those above for most chords?
>
> If you've already been over this before (I'm relatively new here) what key words should I search for in the archives?
> -- Dave Keenan
> http://dkeenan.com
>
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On Thu, 21 Jan 1999 15:37:47 +1000, Dave Keenan <d.keenan@uq.net.au> wrote:

>Obviously there is a grey area and it depends on timbre, vibrato etc. I might be persuaded to push the magic number to 19. But even so, 19:16 just looks ridiculous to me (summing to 35). This is merely a 6:5 that's been flattened about 18 cents. Can anyone really tell a 19:16 from a 13:11 (8 cents narrower) or either of these from anything in between.

I guess it's mainly due to my familiarity with Ben Johnston's and Wendy
Carlos's use of harmonic scales containing a series of the upper harmonics
that I think of it as a 19:16 rather than a 13:11 or a 25:21. It would also
depend on the context: in an 8-11-13 chord, it would make sense to call it
a 13:11. But whatever you call them, the roughly 300-cent intervals aren't
quite as smooth as a 6/5 or a 19-tet minor third. I said it seemed
dissonant, but I think maybe "dissonant" is too strong a word. It's
certainly "darker" than a just minor third.

Thanks everyone for your intelligent responses on the n+m > 17 thing. Two things are clear.
1. I need to do more listening experiments.
2. Sometimes people use ratios like 19:16 only to mean "a dissonance in this general vicinity", or "about this far from the nearby consonance" unfortunately others then mistakenly assume the precise ratio to be significant, as if there is a significant local dissonance minimum there, and start talking 19-limit or even higher numbered nonsense. It might be better to use cent values or invent some kind of "approximate ratio" symbol. 19~:16, 19;16 ?

Regards,
-- Dave Keenan
http://dkeenan.com

>From: Dave Keenan <d.keenan@uq.net.au>

>I feel that (as a rule of thumb for ordinary timbres) we can ignore the
>>accuracy of any ratios whose numerator and denominator (in lowest terms)
>sum to >more than 17....
>
>11:1 or 11:2 and maybe even 11:3 and 11:4 might just sound like something
>but I >suspect it would take a sawtooth wave, a hightly trained ear and an
>extremely >accurate ratio to appreciate 11:7 or 11:8 as anything different
>from the >general dissonance in their vicinity. Sethares theory explains
>this nicely.
>
>Obviously there is a grey area and it depends on timbre, vibrato etc. I
>might >be persuaded to push the magic number to 19. But even so, 19:16
>just looks >ridiculous to me (summing to 35). This is merely a 6:5 that's
>been flattened >about 18 cents.

First of all, as some have already said, everything depends on musical
context. (I think that principle is also sometimes lost in discussions on
this forum about which scale is better than which.) I can think of musical
contexts in which 19/16 would be much better than 6/5. The question is not
necessarily which can be distinguished and identified, but which works
better in the music.

Secondly, I think that adding the numerators and denominators is not a
useful measure in most cases. There are many details that rules of thumb
ignore. For example, the special case of the harmonic series (especially
given an appropriate musical context). Ratios with powers of two in the
denominator have a special identity to me, and in such a context, 11/8 and
19/16 (also 13/8 and 17/16) have particular sound that "locks in,"
especially with harmonic timbres.

11/8 is especially a favorite of mine, and I've gotten feedback from
musically naive listeners that have commented on its distinctive sound in a
composition. 11/8 and 13/8 in particular is unlikely to be confused with
12TET pitches or their JI counterparts (4/3 and 45/32 in the first case and
8/5 and 5/3 in the second) since they lie close to the midpoint between
those pitches. I've come to hear 11/8 as a consonance in many musical
contexts, a wonderfully spicy one, as in the beautiful 8/11/14 chord.

So I certainly don't think we can "ignore the accuracy" of these ratios.
It's possible that 19/16 doesn't have a special identity to you distinct
from 6/5, but that may not mean that the musical usefulness would be
synonymous or, indeed, that other listeners agree.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Dave Keenan wrote:

>I suspect it's just that we're so used to hearing the mistuned thirds
of 12-tET that people expect >them. 19/16 is essentially identical to
(only 2.5 cents flatter than) a 12-tET minor third. I guess it >could be
that the 'virtual fundamental' and difference-tone effects of the
16:19:24 otonality have a >strong enough upward 'push' to overcome the
downward 'pull' of the consonance of the just 5:4 >and 6:5, only 15.6c
away in the 1/6:1/5:1/4 utonality. But in the absence of familiarity
with 12->tET I doubt it. Any empirical data?

Actually, yes. There was an experiment (prompted by the Bohlen-Pierce
scale) that tested untrained listeners' responses to three chords:
4:5:6, 3:5:7, and 10:12:15 (=1/6:1/5:1/4). The listeners were asked to
rate 15 different chords, presented in random order. The chords were the
three just ones, and versions where the middle note was detuned -30
cents, -15 cents, +15 cents, and +30 cents. The results found the
listeners dividing into two very clear groups. One group, dubbed "pure",
clearly preferred the just versions of 4:5:6 and 3:5:7, and gave the
lowest rating to the versions with 30-cent errors. The other group,
dubbed "rich," liked the +15- and -15-cent-detuned versions of 4:5:6 and
3:5:7 better than the other three versions. Both groups of reponses for
4:5:6 and 3:5:7 were roughly symmerical around zero mistuning. Both
groups, though, preferred the version of 10:12:15 where the middle note
was detuned -15 cents (thus very close to 16:19:24). The write-up
attributes this to the relatively high numbers used in writing 10:12:15,
but goes no further.

This article appears in the book _Harmony and Tonality_ edited by J.
Sundberg, which I believe was the 1987 proceedings of the Royal Swedish
Academy of Music, Stockholm. I forget the name of the article itself,
but most of the articles in this book are well worth reading.

> any prime above 1 is significant

Just as an aside, 1 is not considered prime. The official mathematical definition explicitly excludes 1, presumably to ensure that prime factorizations are unique.

Gary Morrison wrote:

> From: Gary Morrison <mr88cet@texas.net>
>
> > any prime above 1 is significant
>
> Just as an aside, 1 is not considered prime. The official mathematical definition explicitly excludes 1, presumably to ensure that prime factorizations are unique.

As the musical and mathematical world share parallels it is good to get in mind that a musical "prime" is different than a mathematical prime. Something I think Paul
Erlich hints at with his understanding of odd as opposed to primes. Didn't Euler use a term I can't remember that was "Tolent" or something like that. Can't find it in
the dictionary but its meaning where one number had a "prime" relation to the other and both had no common denominator. In this case, looking at the 45/32 in the just
major scale it limit would not be 45 because of the relation between 45/32 and 5/4 for example. I know I not being completely clear here but if we can find this word we
can say that it would be limit would be determined by the lowest odd number within the context of an uninterrupted chain of "tolent". You would have to have this because
if you allowed the chain to be interrupted you could say that even though you have a 7/6 and an 7/4 in a scale they are a 3/2 apart (this being a partial chain) but
sooner or later you have to examine it relation to a 5/4 or 1/1. It is if the simplest latticing of a scale would determine a limit. Wilson has suggested the term "Cap"
opposed to "limit" as being less confusing cause with limit you are never sure whether it is inclusive or not!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Gary Morrison is right, of course, when he writes:

"Just as an aside, 1 is not considered prime. The official mathematical
definition explicitly excludes 1, presumably to ensure that prime
factorizations are unique."

This does not detract from the fact that it can be musically useful to
treat 1 as an independent factor, e.g. in diamonds and combination-product
sets. Although the contrast here is certainly that between prime
factorization and factorization in general, I'm not confident enough to
characterize mathematically the ontological status of equalities in
alternative factorings which may arise from this and related factorings,
but the musical utility is clear. For example, Wilson's most-favored
Eikosany, based on the set where each tone is represented by three factors
out of the set of factors (1,3,7,9,11,15), takes advantage of the presence
of 1 and two composite numbers as independent factors to produce a set with
what I would characterize as a maximum of local melodic resources. By
contrast, all prime factor sets are minimally redundant melodically and
thus less useful in more traditional musical settings.

Wilson did some interesting work using the vertices of Penrose tilings as
pitches on graphs of tonal space. When 1 and composite numbers were used as
factors, many surprising equalities appeared, creating opportunities for
random walks across the lattice to spontaneously introduce pitch class
repetitions. I recommend the exercise highly...

I must argue that one (1) is thee prime. I do not care what official anything calls it.

Kraig Grady wrote:

> From: Kraig Grady <kraiggrady@anaphoria.com>
>
> Gary Morrison wrote:
>
> > From: Gary Morrison <mr88cet@texas.net>
> >
> > > any prime above 1 is significant
> >
> > Just as an aside, 1 is not considered prime. The official mathematical definition explicitly excludes 1, presumably to ensure that prime factorizations are unique.
>
> As the musical and mathematical world share parallels it is good to get in mind that a musical "prime" is different than a mathematical prime. Something I think Paul
> Erlich hints at with his understanding of odd as opposed to primes. Didn't Euler use a term I can't remember that was "Tolent" or something like that. Can't find it in
> the dictionary but its meaning where one number had a "prime" relation to the other and both had no common denominator. In this case, looking at the 45/32 in the just
> major scale it limit would not be 45 because of the relation between 45/32 and 5/4 for example. I know I not being completely clear here but if we can find this word we
> can say that it would be limit would be determined by the lowest odd number within the context of an uninterrupted chain of "tolent". You would have to have this because
> if you allowed the chain to be interrupted you could say that even though you have a 7/6 and an 7/4 in a scale they are a 3/2 apart (this being a partial chain) but
> sooner or later you have to examine it relation to a 5/4 or 1/1. It is if the simplest latticing of a scale would determine a limit. Wilson has suggested the term "Cap"
> opposed to "limit" as being less confusing cause with limit you are never sure whether it is inclusive or not!
> -- Kraig Grady
> North American Embassy of Anaphoria Island
> www.anaphoria.com
>
> ------------------------------------------------------------------------
> To unsubscribe from this mailing list, or to change your subscription
> to digest, go to the ONElist web site, at http://www.onelist.com and
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> ------------------------------------------------------------------------
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Patrick Pagano wrote:

"I must argue that one (1) is thee prime. I do not care what official
anything calls it."

Mr. Pagano:

It's not a matter of whether something is 'official' or not, but that terms
are precisely defined so that all that follows -- in this case, the entire
body of number theory -- from the definitions is consistant with and
consequent to those definitions. For musical purposes, the set containing 1
and all the prime numbers may indeed be relevant and useful, but it does
not make 1 a prime number.

Take a few minutes and consider the standard definition of a prime number:

An integer p is called a prime number if p is larger than 1 and has no
positive divisors other than 1 and itself.

Mr. Wolf
I totally agree that the definition you are stating is true. yet I am inclined
to think of prime as something that initiates or that is required for a system
to work ,in a sense that prime to myself an perhaps myself only defines as
progenator or first because 1,3,5,7,11,13,17, need one to exist(in my opinion).
So I apologize for any misunderstanding and maybe I should not have commented
on your reply as I came in late. But using the Lambdoma of the Timaeus 1
generates everthing else so to me is (prime) and "theoretically" 0/0 is
outside even that system(according to Kayser) and is in a sense a quasi-prime
relationship.I had always been taught that prime is a # divisible only by
itself and one,so one being included in that definition--even if serves to
qualify a system- is essential to that system. But yet again by the book you
are 100% correct. So mine is not an argument just a theoretical deviation for
my own purposes.I just view the fundamental as a prime. False yes in number
theory,not in personal taste.

Sorry for the confusion
Resonate and Extenuate
Pat Pagano, Director
South East Just Intonation Society

Daniel Wolf wrote:

> From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>
>
> Patrick Pagano wrote:
>
> "I must argue that one (1) is thee prime. I do not care what official
> anything calls it."
>
> Mr. Pagano:
>
> It's not a matter of whether something is 'official' or not, but that terms
> are precisely defined so that all that follows -- in this case, the entire
> body of number theory -- from the definitions is consistant with and
> consequent to those definitions. For musical purposes, the set containing 1
> and all the prime numbers may indeed be relevant and useful, but it does
> not make 1 a prime number.
>
> Take a few minutes and consider the standard definition of a prime number:
>
> An integer p is called a prime number if p is larger than 1 and has no
> positive divisors other than 1 and itself.
>
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Kraig Grady wrote:

> Didn't Euler use a term I can't remember that was "Tolent" or something
like that. Can't find it in
> the dictionary but its meaning where one number had a "prime" relation to
the other and both had
> no common denominator.

It's Euler's "totient" function. The operand is a natural number. The value
is the number of integers
between 1 and the number itself which are coprime to it, i.e. have a
greatest common divisor of 1.

Manuel Op de Coul coul@ezh.nl