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Message: 901 - Contents - Hide Contents Date: Tue, 28 Aug 2001 17:47:25 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma>>> >hat's CS? >>>> The property that every interval in a scale appears in only >> one interval class. For example, 3:2 appears only as a 5th >> in the diatonic scale... but in 12-tET, the tritone appears >> as both a 4th and a 5th, so the diatonic scale in 12-tET is >> non-CS. >>It seems to me that in a 12-et, a tritone would always be 6 steps. >Can you clarify?Scale steps. What recent threads have called "steps" are actually 2nds. Then, there's 3rds (major and minor), etc. Here's what Rothenberg calls a "interval matrix" for the diatonic scale, 1sts 2nds 3rds 4ths 5ths 6ths 7ths 8ths ionian 0 2 4 5 7 9 11 12 dorian 0 2 3 5 7 9 10 12 phrygian 0 1 3 5 7 8 10 12 lydian 0 2 4 [6] 7 9 11 12 mixolydian 0 2 4 5 7 9 10 12 aeolian 0 2 3 5 7 8 10 12 locrian 0 1 3 5 [6] 8 10 12 The tritone is what R. calls an "ambiguous interval". CS is equivalent to no ambiguous intervals. In meantone, the augmented 4th is smaller than the diminished 5th, so the meantone diatonic is CS. In Pythagorean tuning, the aug. 4th is larger than the dim. 5th, so it is also CS. But Rothenberg would call this scale "improper", since its scale steps overlap in interval space. When certain other conditions are met, R. claims proper scales make possible a compositional style in which melodies may be transposed across the modes of a scale without loosing their identity. -Carl
Message: 902 - Contents - Hide Contents Date: Tue, 28 Aug 2001 17:53:55 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma>> >n Just Intonation, these pitches form intervals with C that are >> not equal. >>Your definition said every interval appears in only one class-- >however, these are two different intervals. Naturally, they will be >represented as different in many systems, and will be the same only >in those systems that has their ratio in the kernel.Just for the record, that was Bob, not me. "Interval class" is just a bad way to say "scale step". "Every interval" is just a bad way to say "every acoustic interval". Does that help? See also my previous message in this thread. Gene, was it ever decided if a kernel is equivalent to a set of unison vectors, as we use them? -Carl
Message: 903 - Contents - Hide Contents Date: Tue, 28 Aug 2001 18:38:08 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Carl brought this sort of thing up a while back (actually, he > showed>> that some PBs are not CS, also using very large unison vectors). I >> replied that there needs to be some notion of "good" PBs. What's > the>> weakest such condition we can come up with? If at least one step > size>> is smaller than the chromatic unison vector . . . will that work? >> I don't know what the weakest condition is, and I haven't thought > about the one you propose above. I'm not sure what the point of it is- > -a step of the 7-et is larger than 25/24, after all.You must have missed the message where I corrected this -- I meant "at least one step size is larger than the chromatic unison vector".> > However, the point of what I was sketching out before was that it is > certainly possible to come up with conditions which make sense and > suffice to produce a MOS. > > What's CS?Every specific interval size is always subtended by the same number of steps. Seems to be synonymous with "good" PBs in the untempered JI case.
Message: 904 - Contents - Hide Contents Date: Tue, 28 Aug 2001 18:44:10 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> In 12-tET, six steps can be used "enharmonically" to represent either > an augmented fourth or a diminished fifth. If we think in the key of > C, this implies that the sixth step above C can be used as either an > F# or a Gb. > > In Just Intonation, these pitches form intervals with C that are not > equal. In 12-tET, the irrational approximation of both intervals (sq > root of 2) lies between F# and Gb. On the other hand, in Just > Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and Gb is > 36/25 (1.44)of C.In Just Intonation there are actually several possible ratios for F#, as well as for Gb. So for these purposes, a well-defined tuning such as Pythagorean or meantone would have been better.
Message: 905 - Contents - Hide Contents Date: Tue, 28 Aug 2001 19:18:26 Subject: Re: More microtemperaments From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:> The departure is from a JI odd limit. That is, all odd numbers up to the > one you pick are involved in ratios, and then you octave reduce. The > "mapping by steps" is your homomorphism.I think the best way to answer my questions may be to do some detective work and then ask about anything which remains unclear. I'm going therefore to take a look at the first 5-limit example, and add commentary. My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a is how sharp our "3" is, and b is how sharp the "5" is, then we want a and b in the hexagonal region determined by |a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8 We need further conditions to determine the tuning, so let's look at what Graham does. "3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28" I don't know what these are. "5/19, 317.0 cent generator basis: (1.0, 0.26416041678685936)" If we set r = 0.26416041678685936, then 5/19 is a convergent for r. It's not clear why it is singled out; convergents for r are 1/3, 1/4, 4/15, 5/19, 9/34,14/53, ... "mapping by period and generator: [(1, 0), (0, 6), (1, 5)]" This seems to explain where r came from: if we send (a, b) to a + b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns out to be log_2(3), and (1,5) goes to 1+5*r which is the approximation of log_2(5) we get when both octaves and fifths are exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's basic condition that all primes up to the last will be exactly represented? "mapping by steps: [(15, 4), (24, 6), (35, 9)]" It seems as if this may have something to do with the convergents to r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et [15, 24, 35] from the convergent 4/15. We may then proceed to the others: [19, 30, 44] = [ 4, 6, 9] + [15, 24, 35] [34, 54, 79] = [15, 24, 35] + [19, 30, 44] [53, 84, 123] = [19, 30, 44] + [34, 54, 79] after which a slew of semiconvergents come in. Graham says this is "my homomorphism", but I'm getting a whole collection. "unison vectors: [[-6, -5, 6]]" 2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything using the matrix M = [0 1] [6 5] to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so that [1, r']M = [6r', 1+5r'] and so 6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the column vector V = [ 1 ] [ 6 r'] [1+5r'] has unison vectors generated by [-6, -5, 6]. "highest interval width: 6" How did we get to intervals and scales? "complexity measure: 6 (7 for smallest MOS)" How is this defined? "highest error: 0.001126 (1.351 cents)" 5/3 is off by this amount. "unique" What is unique? This system is so close to the 53-et that it would seem to make sense to adjust the fifth, the octave or both and make it exactly the 53-et.
Message: 906 - Contents - Hide Contents Date: Tue, 28 Aug 2001 19:44:34 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> Gene, was it ever decided if a kernel is equivalent to a set > of unison vectors, as we use them?Here are some questions: (1) Is the unison a unison vector? (2) If q is a unison vector, is q^2 a unison vector? (3) Are products of unison vectors unique--that is, if we have unison vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the exponents ei determined?
Message: 907 - Contents - Hide Contents Date: Tue, 28 Aug 2001 19:57:31 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma>> >ene, was it ever decided if a kernel is equivalent to a set >> of unison vectors, as we use them? >> Here are some questions: > > (1) Is the unison a unison vector? No. > (2) If q is a unison vector, is q^2 a unison vector?No, but it does point to a unison.> (3) Are products of unison vectors unique--that is, if we have > unison vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the > exponents ei determined?I don't know. -Carl
Message: 908 - Contents - Hide Contents Date: Tue, 28 Aug 2001 20:20:42 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma Sorry Bob, I did not see your message until after I had posted mine (we both answered in the same way). Paul also makes a good point -- you're assuming the 'classical' 12-tone, 5-limit scale here (such as Ellis' "duodene"), which is common practice in many music theory text books, but often leads to trouble here, where we take nothing for granted when it comes to JI! -Carl> In Just Intonation, these pitches form intervals with C that are > not equal. In 12-tET, the irrational approximation of both > intervals (sq root of 2) lies between F# and Gb. On the other hand, > in Just Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and > Gb is 36/25 (1.44)of C. So it becomes clear that F# is lower than > Gb, and the 12-tET interval of 1.414... is an irrational > approximation in between them.
Message: 909 - Contents - Hide Contents Date: Tue, 28 Aug 2001 20:22:52 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote: >>> Gene, was it ever decided if a kernel is equivalent to a set >> of unison vectors, as we use them? >> Here are some questions: > > (1) Is the unison a unison vector? Yes. > > (2) If q is a unison vector, is q^2 a unison vector? Yes.Don't know about (3). "Unison vector" sometimes means any element of the kernel, but sometimes "the set of unison vectors of G" means "the generators of the kernel for G" . . . and in the case of chromatic unison vectors, we're pointing to an _altered_ equivalence, not a true equivalence.
Message: 910 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:04:59 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote: Paul's answers were yes, yes, don't know and Carl's answers were no, no, don't know. I've remarked that I don't know if a unison vector is an element of the kernel or a generator of the kernel, and apparently that has not been decided. If your answer to (1) is "no", then your answer to (3) should probably be "yes", since otherwise a product of unison vectors will equal 1. Let's assume Paul's answer to (3) is also yes, then we have two types of definition: (1) Carl type: Unison vectors are defined to be generators of the kernel of some homomorphism. (2) Paul type: Unison vectors are defined to be members of the kernel of some homomorphism. To pin this down further, here is another question: (4) If we are considering octaves to be equivalent, is 2 a unison vector?
Message: 911 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:06:14 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma>>> >ene, was it ever decided if a kernel is equivalent to a set >>> of unison vectors, as we use them? >>>> Here are some questions: >> >> (1) Is the unison a unison vector? > > Yes.Really? Why would it be? And how do you define 'unison vector', then?>> (2) If q is a unison vector, is q^2 a unison vector? > > Yes.I think I get a different PB if I use 5:4 instead of 25:16...> "Unison vector" sometimes means any element of the kernel, but > sometimes "the set of unison vectors of G" means "the generators > of the kernel for G" . . . Whew. -Carl
Message: 912 - Contents - Hide Contents Date: Tue, 28 Aug 2001 22:14 +0 Subject: Re: More microtemperaments From: graham@m... genewardsmith@j... () wrote:> My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a > is how sharp our "3" is, and b is how sharp the "5" is, then we want > a and b in the hexagonal region determined by > > |a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8 Yes. > We need further conditions to determine the tuning, so let's look at > what Graham does. > > "3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28" > > I don't know what these are.They're 5-limit consistent equal temperaments, being used to calculate the linear temperaments.> "5/19, 317.0 cent generator > > basis: (1.0, 0.26416041678685936)" > > If we set r = 0.26416041678685936, then 5/19 is a convergent for r. > It's not clear why it is singled out; convergents for r are 1/3, 1/4, > 4/15, 5/19, 9/34,14/53, ...Because 19=4+15 is the simplest sum of numbers from the above list that fits this temperament.> "mapping by period and generator: > [(1, 0), (0, 6), (1, 5)]" > > This seems to explain where r came from: if we send (a, b) to a + > b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns > out to be log_2(3), and (1,5) goes to 1+5*r which is the > approximation of log_2(5) we get when both octaves and fifths are > exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's > basic condition that all primes up to the last will be exactly > represented?The condition is that the worst error is as low as possible.> "mapping by steps: > [(15, 4), (24, 6), (35, 9)]" > > It seems as if this may have something to do with the convergents to > r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et > [15, 24, 35] from the convergent 4/15. We may then proceed to the > others: > > [19, 30, 44] = [ 4, 6, 9] + [15, 24, 35] > [34, 54, 79] = [15, 24, 35] + [19, 30, 44] > [53, 84, 123] = [19, 30, 44] + [34, 54, 79] > > after which a slew of semiconvergents come in. Graham says this > is "my homomorphism", but I'm getting a whole collection.Then this is a subtlety of "homomorphism" I wasn't aware of. I remember you showing how a linear (2-D) temperament can be described using the mappings of two equal temperaments.> "unison vectors: > [[-6, -5, 6]]" > > 2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything > using the matrix M = > > [0 1] > [6 5] > > to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so > that [1, r']M = [6r', 1+5r'] and so > 6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the > column vector V = > > [ 1 ] > [ 6 r'] > [1+5r'] > > has unison vectors generated by [-6, -5, 6].Not sure about this bit.> "highest interval width: 6" > > How did we get to intervals and scales?From the set of 5-limit intervals: 1:1, 5:4, 6:5, 3:2 and equivalents and inversions. The highest number of generators you need to describe all these intervals is 6.> "complexity measure: 6 (7 for smallest MOS)" > > How is this defined?The number before times the number of periods to an octave. The number of complete otonalities you can play is the number of notes in the generated scale minus this.> "highest error: 0.001126 (1.351 cents)" > > 5/3 is off by this amount.That'll be it then.> "unique" > > What is unique?It means each interval being approximated has a unique mapping to the temperament. For example, meantone fails to be unique in the 9-limit because 9:8 and 10:9 map the same way.> This system is so close to the 53-et that it would seem to make sense > to adjust the fifth, the octave or both and make it exactly the 53-et.Maybe, but it could be a useful way of choosing subsets of 53-et. Graham
Message: 913 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:14:42 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote: > > Paul's answers were yes, yes, don't know and Carl's answers were no, > no, don't know. I've remarked that I don't know if a unison vector is > an element of the kernel or a generator of the kernel, and apparently > that has not been decided. If your answer to (1) is "no", then your > answer to (3) should probably be "yes", since otherwise a product of > unison vectors will equal 1. Let's assume Paul's answer to (3) is > also yes, then we have two types of definition: > > (1) Carl type: Unison vectors are defined to be generators of the > kernel of some homomorphism. > > (2) Paul type: Unison vectors are defined to be members of the kernel > of some homomorphism.Gene, did you read the rest of my message???
Message: 914 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:15:56 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> "Unison vector" sometimes means any element of the kernel, but > sometimes "the set of unison vectors of G" means "the generators of > the kernel for G" . . . and in the case of chromatic unison vectors, > we're pointing to an _altered_ equivalence, not a true equivalence.It seems to me we should decide which way it's going to be. As for your last point, the chromatic unison vector is in the kernel of one homomorphism but not of another.
Message: 915 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:17:20 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>>>> Gene, was it ever decided if a kernel is equivalent to a set >>>> of unison vectors, as we use them? >>>>>> Here are some questions: >>> >>> (1) Is the unison a unison vector? >> >> Yes. >> Really? Why would it be?It's mapped to a unison.> >> (2) If q is a unison vector, is q^2 a unison vector?>> >> Yes. >> I think I get a different PB if I use 5:4 instead of 25:16...Just because it's also a unison vector, doesn't mean the resulting PB is the same!>>> "Unison vector" sometimes means any element of the kernel, but >> sometimes "the set of unison vectors of G" means "the generators >> of the kernel for G" . . . > > Whew.I hoped Gene would understand this, but apparently he skipped over this and came to the same conclusion independently (based on your answer and the first part of mine).
Message: 916 - Contents - Hide Contents Date: Tue, 28 Aug 2001 21:20:18 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> "Unison vector" sometimes means any element of the kernel, but >> sometimes "the set of unison vectors of G" means "the generators of >> the kernel for G" . . . and in the case of chromatic unison > vectors,>> we're pointing to an _altered_ equivalence, not a true equivalence. >> It seems to me we should decide which way it's going to be.Too late -- it's already been used both ways. Why don't we just drop the "unison vector" terminology on this list and use "kernel" terminology instead, as I suggested before?
Message: 917 - Contents - Hide Contents Date: Wed, 29 Aug 2001 00:27:30 Subject: Re: Now I think "the hypothesis" is true :) From: Carl Lumma>> >Unison vector" sometimes means any element of the kernel, but >> sometimes "the set of unison vectors of G" means "the generators >> of the kernel for G" . . .Ah, so my answers were based on the latter, and yours on the former?>>>> (2) If q is a unison vector, is q^2 a unison vector? >>>> >>> Yes. >>>> I think I get a different PB if I use 5:4 instead of 25:16... >>Just because it's also a unison vector, doesn't mean the resulting >PB is the same!And your reasoning here an example of the former?>>>> (1) Is the unison a unison vector? >>> >>> Yes. >>>> Really? Why would it be? >> It's mapped to a unison.Sounds like a tautology to me. -Carl
Message: 918 - Contents - Hide Contents Date: Wed, 29 Aug 2001 01:24:06 Subject: Defining CS and propriety for newbies (was: Now I think "the hypothesis" is tru) From: Dave Keenan --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> "Interval class" is just a bad way to say "scale step".Eek! No it isn't. I remember you had this problem before. A scale step is the distance between two _consecutive_ scale degrees. For example, in western diatonic scales we have whole-tone steps and half-tone steps. We don't have minor third steps or perfect fifth steps.> "Every > interval" is just a bad way to say "every acoustic interval". > Does that help?I don't think it helps. "Acoustic" simply means "relating to sound". Everything we deal with here is acoustic. I don't think that any the above terminology is very good, when trying to define CS or propriety for newbies. Instead of your "interval class" I use "number of scale steps", and instead of your "interval" I use "size (in cents)". In these definitions I use "interval" to mean "the distance between a specific pair of notes of the scale". Now the definitions: A scale is proper if all intervals spanning the same number of scale steps, have a range of sizes (in cents) that does not overlap but may meet, the range of sizes for any other number of scale steps. A scale is strictly-proper if all intervals spanning the same number of scale steps, have a range of sizes (in cents) that is disjoint from (does not meet or overlap), the range of sizes for any other number of scale steps. Examples. 1. Improper Number of steps in interval 4 1 2 3 <----------> Ranges <--------> <--------> <----------> | | | | | | | | 0 100 200 300 400 500 600 700 etc. Interval size (cents) 2. Proper Number of steps in interval 1 2 3 4 Ranges <--------> <--------> <--------x--------> | | | | | | | | 0 100 200 300 400 500 600 700 etc. Interval size (cents) 3. Strictly proper Number of steps in interval 1 2 3 4 Ranges <--------> <--------> <-------> <-------> | | | | | | | | 0 100 200 300 400 500 600 700 etc. Interval size (cents) The following is supposedly Erv Wilson's definition of CS, as conveyed by Kraig Grady. A scale is CS if all intervals of the same size (in cents), span the same number of scale steps. CS is supposed to be a useful property for a scale to have, but notice that, by this definition, any random scale that has no-two-intervals-the-same-size is trivially CS, even if it has two intervals that differ by only 0.00001 cent spanning different numbers of scale steps! A more meaningful definition for CS would be of the form: A scale is CS if all intervals in the same range of sizes (in cents), (with all ranges defined so as to be disjoint), span the same number of scale steps. Notice that this is almost equivalent to strict-propriety, written conversely. However a scale which is not strictly proper (i.e. it has number-of-step ranges that meet or overlap) might be able to have these ranges split into sub-ranges in such a way thay they no longer overlap and it is thereby CS. 4. CS? Number of steps in interval 4 4 1 2 3 <-> 3 <---> Ranges <--------> <--------> <---> <-> | | | | | | | | 0 100 200 300 400 500 600 700 etc. Interval size (cents) But clearly, this division into non-overlapping sub-ranges must be musically meaningful and in particular the sub-ranges must not be allowed to be too narrow, or else we are back to the trivial case where every subrange can consist of a single size. Various ways of defining allowable ranges for CS, have been proposed, but none universally agreed upon. I ask their authors to explain what they are, should they be so inclined. -- Dave Keenan
Message: 919 - Contents - Hide Contents Date: Wed, 29 Aug 2001 01:27:25 Subject: Re: More microtemperaments From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> I've altered my temperament finding program to accept only temperaments > with a worst error of less than 2.8 cents. I think this is the cutoff for > a microtemperament. Results are at > > <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 2... * [with cont.] (Wayb.)> > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> > <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> > <22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> > <26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.)> > <29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.)> > > Some of them don't have as many as 10 results.Oh Graham, you're wonderful! -- Dave Keenan
Message: 920 - Contents - Hide Contents Date: Wed, 29 Aug 2001 01:35:34 Subject: Re: More microtemperaments From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> I've altered my temperament finding program to accept only temperaments > with a worst error of less than 2.8 cents. I think this is the cutoff for > a microtemperament. Results are at > > <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 2... * [with cont.] (Wayb.)> > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> > <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> > <22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> > <26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.)> > <29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.)> > > Some of them don't have as many as 10 results.Something must be wrong. How come schismic didn't make it into 5-limit? Couldn't you be missing some by not taking your consistent ET's out far enough. But there's definitely no need to go past 215-tET (within 2.8 cents of anything). -- Dave Keenan
Message: 921 - Contents - Hide Contents Date: Wed, 29 Aug 2001 01:55:00 Subject: Re: Now I think "the hypothesis" is true :) From: Dave Keenan --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> What recent threads have called "steps" are > actually 2nds.Yes. "Seconds" are the _only_ things that are called steps. Lest ye doubt, please see p63 of 404 Not Found * [with cont.] Search for http://depts.washington.edu/pnm/CLAMPITT.pdf in Wayback Machine "A step interval is an interval whose two boundary pitches are adjacent pitches of a scale." -- Dave Keenan
Message: 922 - Contents - Hide Contents Date: Wed, 29 Aug 2001 03:23:23 Subject: Re: Defining CS and propriety for newbies (was: Now I think "the hypothesis" is tru) From: Carl Lumma>> >Interval class" is just a bad way to say "scale step". >> Eek! No it isn't. I remember you had this problem before. > > A scale step is the distance between two _consecutive_ scale > degrees. For example, in western diatonic scales we have > whole-tone steps and half-tone steps. We don't have minor third > steps or perfect fifth steps.You're right. I usually use "scale interval" here. I think my past problem was "scale degrees", which of course are pitches, not intervals.>> "Every interval" is just a bad way to say "every acoustic >> interval". Does that help? >> I don't think it helps. "Acoustic" simply means "relating to > sound". Everything we deal with here is acoustic.Not really. Propriety has only to do with the relative sizes of a scale's intervals, not with the actual sizes -- in fact, Rothenberg discards the interval matrix in favor of the rank- order matrix very early on. Now, "acoustic" may not be the best way to get this across, I'll agree. How would you say it?> //propriety stuff// Great job! > A more meaningful definition for CS would be of the form: > > A scale is CS if all intervals in the same range of sizes (in > cents), (with all ranges defined so as to be disjoint), span > the same number of scale steps.Now you've got me confused.> Notice that this is almost equivalent to strict-propriety, written > conversely. However a scale which is not strictly proper (i.e. it > has number-of-step ranges that meet or overlap) might be able to > have these ranges split into sub-ranges in such a way thay they no > longer overlap and it is thereby CS. > > 4. CS? > Number of steps in interval 4 4 > 1 2 3 <-> 3 <---> > Ranges <--------> <--------> <---> <-> > | | | | | | | | > 0 100 200 300 400 500 600 700 > > Interval size (cents) > > But clearly, this division into non-overlapping sub-ranges must be > musically meaningful and in particular the sub-ranges must not be > allowed to be too narrow, or else we are back to the trivial case > where every subrange can consist of a single size.There are many scales which have overlapping intervals subtending the same number of scale steps, which also lack ambiguous intervals. Therefore they are CS but not proper. These cases don't represent a failure of either concept, though. Propriety is based on the idea that listeners order scale intervals by size. CS is based on the idea that listeners can recognize particular intervals. Actually, both properties can result in what I'd call "convenience items", such as transpositional coherence (cough!), and handy things for PBs. I think these are primarily what motivated Wilson with the concept. But on perceptual grounds, I'd say CS doesn't have much of a leg to stand on (Paul's consonance-only CS may be another matter), but if it does, it would still have the leg for improper CS's. -Carl
Message: 923 - Contents - Hide Contents Date: Wed, 29 Aug 2001 06:02:29 Subject: Some definitions From: genewardsmith@j... DEFINITIONS (1) A *tone group* T is a finitely generated subgroup of the group (R+, *) of the positive real numbers under multiplication. A *tone* is any element of (R+, *), that is, any positive real number considered multiplicatively. Hence, T is generated by n tones {t1, ..., tn}, which have the property that t1^e1 * ... * tn^en != 1 so long as the integer exponents e1...en are not all zero. The number n is the *rank* of T, and any element t \in T is a tone of T. The tones {t1, ..., tn} are the *generators* of T. The canonical examples are the p-limit groups, so we also define the *p-limit group* T_p to be the tone group generated by the primes less than or equal to the prime p. (2) A *note group* N is a finitely generated free abelian group which we may identify with row vectors with integer entries. It is a sort of generalized musical notation, since the canonical example makes [a, b] represents "a" octaves and "b" fifths in a meantone system. If we equate A 440 to [0,0], then any note in ordinary musical notation may be translated into or out of this system--ordinary musical notation can be thought of as representing this note group. Here of course we do *not* equate B# with C, etc. The elements of N we call notes, or N-notes if we need to be specific. (3) A *tuning map* or *tuning* for the note group N is a homomorphism "tune" of N into (R+, *); it is defined by its values tune([1, 0, .. , 0]) = t1, tune ([0, 1, 0, ...,0]) = t2, ... tune([0, 0, ..., 1]) = tn. The image T = tune(N) under this map is the *corresponding tone group*; if T is also of rank n then tune is a *tuning isomorphism* and {t1, ..., tn} are generators for T. (4) If there are n primes less than or equal to a prime p, we define the note group N_p to be the rank n free group, and the just tuning map to be the tuning just([1, 0, ..., 0]) = 2, just([0, 1, ... , 0]) = 3, ..., just([0, 0, ..., 1]) = p. "Just" is therefore a tuning isomorphism from N_p to T_p. (5) The dual N` to a note group N is the group N` = Hom(N, Z) of homomorphisms from N into the integers. The elements of N` we call ets, or N-ets if we need to be specific. If we take N concretely as consisting of row vectors with integer entries, then we may take N` to be column vectors with integer entries. If h and g are any two ets and v is a note, then h+g is an et defined by "h+g"(v) = h(v)+g(v); the 0-division et "0" which sends all notes v to 0 is the identity; this defines the group structure on N`. Concretely, it is represented by adding the two column vectors, just as the group N is defined by adding row vectors. If n is any positive integer, we define the et h_n in the dual to the p-limit group N_p` to be the column vector with entries round(n log_2 (p_i)), so that h_n = [ n ] [round(n log_2(3)] [round(n log_2(5)] . . . [round(n log_2(p)]. Here "round(x)" is the real number x rounded to the nearest integer, so that round(x)-1/2<x<=round(x)+1/2 We should note that it is *not* always the case that h_n + h_m = h_{n+m}. (6) If M is any subgroup of the note group N, then null(M) is the subgroup of N` consisting of all elements h \in N such that h(m) = 0 for every m \in M. (7) If M is any subgroup of the dual to a note group N`, then null(M) is the subgroup of N consisting of all v \in N such that h(v) = 0 for every h \in M. (8) If h \in N` is any nonzero et, then kernel(h) or null(h), the *kernel* of h is the subgroup of rank n-1 of N defined as null(H), where H is the group generated by h. In other words, the kernel is the set of all notes v such that h(v)=0. (9) A set of n-1 notes {u1, ..., u_{n-1}}in kernel(h) is a *generating set* if any element u of the kernel is a Z-linear sum u = j_1 u_1 + ... + j_{n-1}, where the j_i are integers. (10) A nonzero et h is *reduced* if the coordinates of h are relatively prime; that is, if no integer greater than one divides all the v_i where v_i is the number in the ith row of v. (11) If {u_i} is a generating set for h, then we may divide it into "a" *commas* and "b" *chromas*, where a+b = n-1. The subset of kernel(h) generated by the commas is the *commatic kernel* K, and the quotient group N/K is the *commatic note group*. If all the generators are commas, the commatic note group is simply Z and if h is reduced we may identify it with the homomorphism from N to N/K. If all of the generators are chromas, the commatic kernel is {0} and the commatic note group is N. In general, the commatic note group is of rank b.
Message: 924 - Contents - Hide Contents Date: Wed, 29 Aug 2001 07:39:32 Subject: An example From: genewardsmith@j... Let's put the definitions I just gave to use by looking at how the Blackjack scales might be derived. We may start from either ets or commas, but ets are easier to find and so it probably makes the most sense to began there. If we look at 7-limit ets, we find 10,12,15,19,22,27,31,41,68,72,99 as ets h_n with n between 10 and 100 and cons(7,n)<1. If we pick h_{31} and h_{41}, we generate a rank 2 group M which also contains h_{10} = h_{41}-h_{31}, h_{72}=h_{41}+h_{31}, etc. Then K=null(M) is generated by the notes [-5, 2, 2, -1] and [-5,-1,-2,4], which correspond to the tones 225/224 and 2401/2400. If we look for ets contained in M we find h_{10}, h_{11}, h_{20}, h_{21}, ... and so forth. If we select h_{21}, we find ker(h_{21}) is generated by [2,2,-1,-1] (corresponding to 35/35) and K. If we make [2,2,-1,-1] a chroma and {[-5,2,2,-1], [-5,-1,-2,4]} commas then K is the commatic kernel and L=N_7/K is a note group of rank 2. If we choose a tuning for L in a reasonable way we now should have a good tone system for the 7-limit. "Reasonable" might for instance mean tuning octaves pure and picking a good value for the remaining generator. A particularly practical form of "reasonable" is to tune another et in M; thus we could have 21 notes out of 31 with 36/35 one step, 21 notes out of 41 with 36/35 two steps, or 21 notes out of 72 with 36/35 three steps.
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