This is an
**
Opt In Archive
.
**
We would like to hear from you if you want your posts included. For the contact address see
About this archive. All posts are copyright (c).

9000
9050
9100
9150
9200
9250
**9300**
9350
9400
9450
9500
9550
9600
9650
9700
9750
9800
9850
9900
9950

9300 -
**9325 -**

Message: 9325 - Contents - Hide Contents Date: Mon, 19 Jan 2004 02:26:24 Subject: Re: Question for Dave Keenan From: Carl Lumma>> >t might, depending on the value of 's' or hearing resolution assumed >> (this is essentially the only free parameter in harmonic entropy, >> which subsumes considerations of timbre, register,Not register, I thought. The neigborhood around 5/2 probably looks a lot like that around 5/4 but the actual entropy values will be different, right? -Carl

Message: 9326 - Contents - Hide Contents Date: Mon, 19 Jan 2004 05:49:06 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> But remember that the disagreement on things like "beep" >> and "father">>> is not whether they contain consonances but whether those >> consonances>>> have anything to do with their supposed 5-limit mappings. >>>> What would you call a virtual-pitch-based phenomenon where chord >> tones are assigned by the brain to specific partials within a >> subsuming 'harmony' -- to take typical examples, 3:4:5, 4:5:6, 5:6:8, >> or hypothetically, 10:12:15 (or still more hypothetically -- remember >> George Kahrimanis? -- 1/6:1/5:1/4)? >> I suppose you'd like to call it 5-limit consonance, and I would have > no great objection to that. But what I want to know is, how can we > tell whether it's happening or not?A psychological experiment concerning 'roots', perhaps? Ultimately it will come down to perception and the reporting of perception, which as we know, are not amenable to the exact sciences.> Let's assume we agree on what > sounds consonant. How would we tell if the consonance is due to one of > these approximate 5-limit alignments or some other more complex but > more accurate alignment.Well the perceived position of the 'root' would determine that.> e.g. with TOP Beep, you were asking us to believe that the 260 c > generator could be "experienced as" an approximate 5:6 even though it > is only 7 cents away from 6:7, and 55 cents away from 5:6.Not really, because though the optimality of TOP can be seen as concerning *all intervals*, most of which are even more complex than 5:6 and have worse errors, its optimization property still holds for any, say, "product limit" n*d not smaller than the largest prime. So you could use a "product limit" of 5 or even 20 (thus allowing voicings -- favored anyway -- like 2:3:4:5) without running into this difficulty. Also, I'm not convinced what you say would necessarily be impossible in the right context, such as a full 1:2:3:4:5:6:8 chord, or even, perhaps, a 4:5:6:8 chord. After all, Gene and others would have us believe that meantone dominant seventh chords were "experienced as" 4:5:6:7 chords, even though the 6:7 interval would typically be tuned far closer to 5:6. Anyway, when I yielded to you on 'father', the same was pretty much implied for 'beep' . . .

Message: 9327 - Contents - Hide Contents Date: Mon, 19 Jan 2004 05:57:04 Subject: Re: Annotated Dave Keenan file From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I've taken this file > > Good 7-limit generators * [with cont.] (Wayb.) > > and annotated it by adding names when available, and descriptions when > not, of the 7-limit linear temperaments it discusses. While I think > the value judgments are a little eccentric, it surely is worth looking > at the temperaments here which don't have names. Maybe Dave wants to > suggest something?I'd like them to be named "", "", and "" respectively (without the quotes). :-)

Message: 9328 - Contents - Hide Contents Date: Mon, 19 Jan 2004 10:33:16 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> It might, depending on the value of 's' or hearing resolution assumed >>> (this is essentially the only free parameter in harmonic entropy, >>> which subsumes considerations of timbre, register, >> Not register, I thought.Yes register. S is clearly larger lower down.> The neigborhood around 5/2 probably looks > a lot like that around 5/4 but the actual entropy values will be > different, right?What does that have to do with it? The right comparison would be a given interval with that same interval transposed lower down, not between two different intervals. For octave-equivalent harmonic entropy, s still depends on register, but *voicing* and *inversion* of course become irrelevant to the actual entropy values.

Message: 9329 - Contents - Hide Contents Date: Mon, 19 Jan 2004 02:59:05 Subject: Re: Question for Dave Keenan From: Carl Lumma>What does that have to do with it? The right comparison would be a >given interval with that same interval transposed lower down, not >between two different intervals.You're right -C. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9330 - Contents - Hide Contents Date: Mon, 19 Jan 2004 06:11:51 Subject: to whoever put up the links From: Paul Erlich Please, User John Starrett on math.cudenver.edu * [with cont.] (Wayb.) should be replaced with User John Starrett on rainbow.nmt.edu * [with cont.] (Wayb.) Thanks! ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9331 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:00:31 Subject: Q: The missing link -- affine geometry & exterior algebra? From: Paul Erlich All right, folks . . . I'm not sure if I missed anything important since I last posted, but before I catch up . . . In the 3-limit, there's only one kind of regular TOP temperament: equal TOP temperament. For any instance of it, the complexity can be assessed by either () Measuring the Tenney harmonic distance of the commatic unison vector 5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047 12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988 () Calculating the number of notes per pure octave or 'tritave': 5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave; .........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave. 12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave; .........TOP tritave = 1901 -> 19.01 notes per pure tritave. The latter results are precisely the former divided by 2: in particular, the base-2 Tenney harmonic distance gives 2 times the number of notes per tritave, and the base-3 Tenney harmonic distance gives 2 times the number of notes per octave. A funny 'switch' but agreement (up to a factor of exactly 2) nonetheless. In some way, both of these methods of course have to correspond to the same mathematical formula . . . In the 5-limit, there are both 'linear' and equal TOP temperaments. For the 'linear' case, we can use the first method above (Tenney harmonic distance) to calculate complexity. For the equal case, two commas are involved; if we delete the entries for prime p in the monzos for each of the commatic unison vectors and calculate the determinant of the remaining 2-by-2 matrix, we get the number of notes per tempered p; then we can use the usual TOP formula to get tempered p in terms of pure p and thus finally, the number of notes per pure p. Note that there was no need to calculate the angle or 'straightness' of the commas; change the angles in your lattice and the number of notes the commas define remains the same, so angles can't really be relevant here. As I understand it, the determinant measures *area* not only in Euclidean geometry, but also in 'affine' geometry, where angles are left undefined . . . Anyhow, since both of these methods could be used to address a 3-limit TOP temperament, in 5-limit could they be still both be expressible in a single form in a general enough framework, say exterior algebra? In the 7-limit, the two methods give us, respectively, the complexity of a 'planar' temperament as a distance, and the cardinality of a 7- limit equal temperament as a volume. But 7-limit 'linear' temperaments get left out in the cold. The appropriate measure would seem to have to be an *area* of some sort -- from what I understand from exterior algebra, this is the area of the *bivector* formed by taking the *wedge product* of any two linearly independent commatic unison vectors (barring torsion). If the generalization I referred to above is attainable, all three of the 7-limit cases could be expressed in a single way. Anyhow, if this is all correct, I want details, details, details. The goal, of course, is to produce complexity vs. TOP error graphs for 7-limit linear temperaments, something I currently don't know how to do. If someone can fill in the missing links on the above, preferably showing the rigorous collapse to a single formula in the 3-limit and with some intuitive guidance on how to visualize the bivector area in affine geometry (or whatever), I'd be extremely grateful.

Message: 9332 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:01:30 Subject: Q: The missing link -- affine geometry & exterior algebra? From: Paul Erlich All right, folks . . . I'm not sure if I missed anything important since I last posted, but before I catch up . . . In the 3-limit, there's only one kind of regular TOP temperament: equal TOP temperament. For any instance of it, the complexity can be assessed by either () Measuring the Tenney harmonic distance of the commatic unison vector 5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047 12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988 () Calculating the number of notes per pure octave or 'tritave': 5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave; .........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave. 12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave; .........TOP tritave = 1901 -> 19.01 notes per pure tritave. The latter results are precisely the former divided by 2: in particular, the base-2 Tenney harmonic distance gives 2 times the number of notes per tritave, and the base-3 Tenney harmonic distance gives 2 times the number of notes per octave. A funny 'switch' but agreement (up to a factor of exactly 2) nonetheless. In some way, both of these methods of course have to correspond to the same mathematical formula . . . In the 5-limit, there are both 'linear' and equal TOP temperaments. For the 'linear' case, we can use the first method above (Tenney harmonic distance) to calculate complexity. For the equal case, two commas are involved; if we delete the entries for prime p in the monzos for each of the commatic unison vectors and calculate the determinant of the remaining 2-by-2 matrix, we get the number of notes per tempered p; then we can use the usual TOP formula to get tempered p in terms of pure p and thus finally, the number of notes per pure p. Note that there was no need to calculate the angle or 'straightness' of the commas; change the angles in your lattice and the number of notes the commas define remains the same, so angles can't really be relevant here. As I understand it, the determinant measures *area* not only in Euclidean geometry, but also in 'affine' geometry, where angles are left undefined . . . Anyhow, since both of these methods could be used to address a 3-limit TOP temperament, in 5-limit could they be still both be expressible in a single form in a general enough framework, say exterior algebra? In the 7-limit, the two methods give us, respectively, the complexity of a 'planar' temperament as a distance, and the cardinality of a 7- limit equal temperament as a volume. But 7-limit 'linear' temperaments get left out in the cold. The appropriate measure would seem to have to be an *area* of some sort -- from what I understand from exterior algebra, this is the area of the *bivector* formed by taking the *wedge product* of any two linearly independent commatic unison vectors (barring torsion). If the generalization I referred to above is attainable, all three of the 7-limit cases could be expressed in a single way. Anyhow, if this is all correct, I want details, details, details. The goal, of course, is to produce complexity vs. TOP error graphs for 7-limit linear temperaments, something I currently don't know how to do. If someone can fill in the missing links on the above, preferably showing the rigorous collapse to a single formula in the 3-limit and with some intuitive guidance on how to visualize the bivector area in affine geometry (or whatever), I'd be extremely grateful.

Message: 9333 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:15:16 Subject: Re: A potentially informative property of tunings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> > wrote:>> Take a generator of 260.76 cents and a period of 1206.55 cents. This >> defines a linear tuning which belongs to a family of related linear >> temperaments. The simplest mapping is the "beep" mapping, which > distributes>> the 27;25 interval: >> >> [(1, 0), (2, -2), (3, -3)] >> >> but after 6 iterations of the generator, there's a better 5:1 at > (1, 6),>> about 15 cents flat (compared with the 51 cent sharp "beep" version > of the>> interval). That means this particular tuning is consistent > with "beep">> temperament only up to a range of 5 generators -- or to coin a > phrase, its>> "consistency range" with respect to "beep" is 5. In comparison, top >> meantone has a "consistency range" of 34: its (17, -35) version of > 5:1 is>> only 2 cents flat, compared with the 4-cent sharp (4, -4). Quarter- > comma>> meantone has a "consistency range" of 29, since it has a better 3:1 > at >> (-11, 30). >> >> First of all, I don't like the term "consistency range", but I > couldn't>> think of anything better. I'd appreciate ideas for what to call this >> property. >> Since you're describing a relationship or comparison between two > temperaments, I would suggest "compatibility range". The > term "consistency" is usually used to describe only relationships > within a single temperament.I don't see any comparison between two temperaments in what Herman is proposing! It all looks "within temperament" to me.

Message: 9334 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:17:54 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> There's also the posssibility that the dominant seventh chord >> functions best when its harmonic entropy (or maybe only the HE of one >> of its dyads) is locally _maximised_. >> What would this look like? The dominant seventh chord is defined > as a local minimum of entropy.Defined?? By whom?

Message: 9335 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:20:28 Subject: Re: Harmonic Entropy and Minkowski's ? function From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It seems to me a Stieltjes integral with respect to the Minkowski ? > function is worth exploring. > > Stieltjes Integral -- from MathWorld * [with cont.] > > Minkowski's Question Mark Function -- from Mat... * [with cont.] > > The ? function has the property that > > ?((p1+p2)/(q1+q2)) = (p1/q1 + p2/q2)/2 > > for adjacent Farey fractions. > > We can ask for Integral exp(-((x-c)/s)^2) d? > > for various values of s, and use it as one definition for the > harmonic entropy of c. ?(x) is continuous and it doesn't look > terribly difficult to compute, though I've never tried.I'll eagerly await further details on the harmonic entropy list.

Message: 9336 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:23:48 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> There's also the posssibility that the dominant seventh chord >>> functions best when its harmonic entropy (or maybe only the HE of one >>> of its dyads) is locally _maximised_. >>>> What would this look like? >> Pretty much like a dominant seventh chord in 12-tET. > > Using noble-mediants to estimate them, a max entropy minor seventh > should be around 1002 cents, a max entropy diminished fifth should be > around 607 cents, and a max entropy minor third should be around 284 > cents.Why look at these intervals and not the major third, etc.?

Message: 9337 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:25:01 Subject: Re: Octacot? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Given dicot and tetracot,Don't forget tricot. shoudn't octafifths really be octacot?

Message: 9338 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:26:26 Subject: Re: A potentially informative property of tunings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Herman, > > I quite agree that it would be very useful to know, for any n-limit > temperament, the max number of generators before we obtain a better > approximation for some n-limit consonance, than that given by the > temperament's mapping. > > I'd love to see this figure for all our old favourites. The only thing > that bothers me is that I assume it will vary according to which > particular optimum generator we use, and if so, then it isn't entirely > a property of the temperament (i.e. the map). > > But otherwise, it works fine for me to say that temperament X has a > _consistency_limit_ of Y generators.I'd avoid the term 'limit' since it's already so loaded. But I agree with Dave, and disagree with George, about the appropriateness of 'consistency' here.

Message: 9339 - Contents - Hide Contents Date: Tue, 20 Jan 2004 05:10:00 Subject: Harmonic Entropy and Minkowski's ? function From: Gene Ward Smith It seems to me a Stieltjes integral with respect to the Minkowski ? function is worth exploring. Stieltjes Integral -- from MathWorld * [with cont.] Minkowski's Question Mark Function -- from Mat... * [with cont.] The ? function has the property that ?((p1+p2)/(q1+q2)) = (p1/q1 + p2/q2)/2 for adjacent Farey fractions. We can ask for Integral exp(-((x-c)/s)^2) d? for various values of s, and use it as one definition for the harmonic entropy of c. ?(x) is continuous and it doesn't look terribly difficult to compute, though I've never tried.

Message: 9340 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:34:31 Subject: Poor man's ?tropy From: Gene Ward Smith If G_s(x) = sqrt(exp(-(x/2s)^2)/(sqrt(2 pi)s) is the Gaussian distibution with standard deviation s, and ?' is the distribution/generalized function which is the derivative of the Minkowski ? function (for all reals, not just [0,1]), and if "*" as usual denotes the convolution product, then the ? harmonic entropy with parameter s is defined as Ent_s = ?' * G_s This is very nice, but the convolution product can be much faster to compute if we use something other than G_s. In particular, the uniform distibuition of width s, which is U_s(x) = 1/s when -s/2 <= x <= s/2, and zero otherwise. We have ?' * U_s = (1/s) Integral_{y-s/2...y+s/2} d? = ?(x+s/2)-?(x-s/2)/(2s), which is a central difference operator (in place of a derivative) applied to ?(x) and easy to compute, since ? is easy to compute. If you convolve U_s with itself, you get a triangular distribution, and if we use ?' * U_s * U_s we get something closer to Gaussian smoothing. If Q(x) is the integral of ?(x), this is like taking two successive central difference operators on Q, so a question arises as to how difficult it is to compute Q.

Message: 9341 - Contents - Hide Contents Date: Tue, 20 Jan 2004 05:37:14 Subject: Re: Harmonic Entropy and Minkowski's ? function From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> We can ask for Integral exp(-((x-c)/s)^2) d? > > for various values of s, and use it as one definition for the > harmonic entropy of c. ?(x) is continuous and it doesn't look > terribly difficult to compute, though I've never tried.It may be worth noting that the above is a convolution of ? with the Gaussian distribution. Convolution -- from MathWorld * [with cont.] This is a smoothing operation, and while the derivaitve ?'(x) does not exist as a function, it is a well-defined generalized function like the Dirac delta. The above definition can be thought of as Gaussian smoothings of ?' into actual functions.

Message: 9342 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:37:51 Subject: Re: Harmonic Entropy and Minkowski's ? function From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I'll eagerly await further details on the harmonic entropy list.Must it be moved there? That list has been incredibly counterproductive, at least to me, since it's served to keep me out of the conversation. Why not just repost relevant articles?

Message: 9343 - Contents - Hide Contents Date: Tue, 20 Jan 2004 06:39:11 Subject: A two-dimensional ? function From: Gene Ward Smith I found this on the web; it could well be useful in defining harmonic entropy for triads. 403 Forbidden * [with cont.] (Wayb.)

Message: 9344 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:47:02 Subject: Re: Harmonic Entropy and Minkowski's ? function From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I'll eagerly await further details on the harmonic entropy list. >> Must it be moved there? That list has been incredibly > counterproductive, at least to me, since it's served to keep me out > of the conversation.I don't understand.> Why not just repost relevant articles?I could do that if necessary.

Message: 9345 - Contents - Hide Contents Date: Tue, 20 Jan 2004 00:34:15 Subject: Re: A potentially informative property of tunings From: Herman Miller On Mon, 19 Jan 2004 00:52:19 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: >>> [(1, 0), (2, -2), (1, 6)] superpelog >>Why are you calling this "superpelog"? It's a Dave Keenan style >duplicated pelog, isn't it?It approximates two different versions of pelog, which turned up in the Scala archive as pelog_pa.scl and pelog_pb.scl, credited to "von Hornbostel" (presumably Erich von Hornbostel). The "type a" pelog is what we're calling pelogic, but I'm actually more interested right now in the "type b" pelog, which sounds to my ears a bit more authentic. In any rate, there's so much variability among pelog scales that both versions are useful. The nine-note MOS of this tuning is (maybe coincidentally) similar to the tuning of the giant bamboo flutes on the CD _Music of the Gambuh Theater_, which isn't exactly 9-ET. This tuning also has some useful resources in its own right, beyond the pelog approximations, such as a pretty good 3:5:7:9:11 chord. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9346 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:47:04 Subject: Re: Octacot? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Given dicot and tetracot, >> Don't forget tricot.Too late, I already did. Not only that, googling restricted to Yahoo groups turns up mostly French language groups, plus Bisexual Pagan Teens. Can you brief me?

Message: 9347 - Contents - Hide Contents Date: Tue, 20 Jan 2004 07:41:35 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> There's also the posssibility that the dominant seventh chord >> functions best when its harmonic entropy (or maybe only the HE of one >> of its dyads) is locally _maximised_. >> What would this look like?Pretty much like a dominant seventh chord in 12-tET. Using noble-mediants to estimate them, a max entropy minor seventh should be around 1002 cents, a max entropy diminished fifth should be around 607 cents, and a max entropy minor third should be around 284 cents.> The dominant seventh chord is defined > as a local minimum of entropy.Who defined it as such, when, and why? As far as I know, the only widely accepted definition of the dominant seventh chord is a chord containing the diatonic scale degrees V, VII, II, IV. It may or may not be a local harmonic entropy minimum depending how the scale is tuned. It seems to me that the more dissonant it is, the more relief is likely to be felt when it "resolves" to a consonance.

Message: 9348 - Contents - Hide Contents Date: Tue, 20 Jan 2004 19:51:08 Subject: Re: Octacot? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote:>>> Given dicot and tetracot, >>>> Don't forget tricot. >> Too late, I already did. Not only that, googling restricted to Yahoo > groups turns up mostly French language groups, plus Bisexual Pagan > Teens. Can you brief me?As usual, you need only look here: Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=6 or here: Tonalsoft Encyclopaedia of Tuning - equal-temp... * [with cont.] (Wayb.)

Message: 9349 - Contents - Hide Contents Date: Tue, 20 Jan 2004 08:31:42 Subject: Octacot? From: Gene Ward Smith Given dicot and tetracot, shoudn't octafifths really be octacot? "Octacot" can be described as nonoctave 88tet with octaves, or as tetracot sliced in half. If we are using 5/34 as a generator for tetracot, then we get octacot by using 5/68 instead. Octacot has TM basis {245/243, 2400}, and the same TOP tuning as tetracot. The latter fact shows we can't rely just on TOP tuning for naming things--the fact that octacot has a generator half as large surely amounts to something. We would now have: dicot 5/17 generator tetractot 5/34 generator octacot 5/68 generator for the whole cot family.

9000
9050
9100
9150
9200
9250
**9300**
9350
9400
9450
9500
9550
9600
9650
9700
9750
9800
9850
9900
9950

9300 -
**9325 -**