Tuning-Math Archive Section 11: 10700

Message: 10700 - Contents - Hide Contents Date: Mon, 29 Mar 2004 17:32:11 Subject: Re: Some 13-limit microtemperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Gene Ward Smith wrote: > I take it you must have started with 6 commas, then. Could you tell us > what they are? Then I could try duplicating the result.

2401/2400, 4375/4374, 3025/3024, 9801/9800, 2080/2079, 4096/4095, 4225/4224, 6656/6655, 10648/10647, 123201/123200; a total of ten commas.

> Oh yes, the standout temperament was already top of my 13- and 15- limit > microtemperament lists.

It's come up before, I know. You want to name it?

Message: 10702 - Contents - Hide Contents Date: Mon, 29 Mar 2004 20:06:42 Subject: Haluska commas From: Gene Ward Smith Haluska defines a (p-limit) comma as an interval such that any smaller p-limit interval has a greater height. He uses the numerator as a height function, but nothing essential is changed by taking Tenney height instead, and I think that will actually make it easier to compute a comma list. I propose we call a p-limit interval m/n>1 a "Haluska comma" if for any p-limit interval a/b>1, a/b<m/n ==> m*n<a*b. The first Haluska comma for any odd p will be 2, followed always by 3/2 and 4/3. If p>3 we then get 5/4 and 6/5, etc.

Message: 10703 - Contents - Hide Contents Date: Mon, 29 Mar 2004 21:04:37 Subject: 5-limit Haluska commas From: Gene Ward Smith Here is how the (infinite) list of 5-limit Haluska commas less than 81/80 starts out: diaschisma 2048/2025 |11 -4 -2> kleisma 15625/15552 |-6 -5 6> schisma 32805/32768 |-15 8 1> semithirds comma 274877906944/274658203125 |38 -2 -15> ennealimma 7629394531250/7625597484987 |1 -27 18> kwazma 9010162353515625/9007199254740992 |-53 10 16> monzisma 450359962737049600/450283905890997363 |54 -37 2> senior comma 381520424476945831628649898809/381469726562500000000000000000 |-17 62 -35> piratic comma 17763568394002504646778106689453125/17763086495282268024161967871623168 |-90 -15 49> Taking them in adjacent pairs gives 34, 53, 118, 441, 612, 612, 1171, 2513...

Message: 10705 - Contents - Hide Contents Date: Mon, 29 Mar 2004 22:36:29 Subject: 7-limit Haluska commas From: Gene Ward Smith If we start the list from 225/224, the 7-limit Haluska commas begin 225/224 [-5, 2, 2, -1] 2401/2400 [-5, -1, -2, 4] 4375/4374 [-1, -7, 4, 1] 250047/250000 [-4, 6, -6, 3] 78125000/78121827 [3, -13, 10, -2] 205891132094649/205885750000000 [-7, 30, -9, -7] Taking the commas in pairs gives us miracle, ennealimmal, ennealimmal, ennealimmal and finally something which breaks out of the ennealimmal gravitational field: |88 151 183 35 43 1> [<1 -19 -33 -40|, <0 88 151 183|] Generator: 731/3125, representing an interval of 147/125. Even for me, this is getting over the top, but it does have a rather low badness rating, comparable to meantone or miracle if you believe in log-flatness.

Message: 10706 - Contents - Hide Contents Date: Mon, 29 Mar 2004 08:48:15 Subject: Some 13-limit microtemperaments From: Gene Ward Smith I took all the 13-limit superparticular commas with numerator greater than 2000 four at a time, giving 15 linear temperaments, listed below in TOP/Graham badness order. The first temperament on the list seems to be a standout, it has TM basis {1716/1715, 2080/2079, 3025/2024, 4096/4095}, a period of 1/2 octave and a 44/39 generator. Ets are 224, 270, and 494. [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] 173.898312 [72, 108, 72, 36, -54, 4, -88, -192, -352, -136, -290, -525, -148, -418, -320] [[18, 28, 41, 50, 62, 67], [0, 4, 6, 4, 2, -3]] 330.983530 [36, 54, 36, 18, 108, 2, -44, -96, 38, -68, -145, 51, -74, 170, 307] [[18, 28, 41, 50, 62, 65], [0, 2, 3, 2, 1, 6]] 353.702340 [16, 84, 46, 98, 108, 96, 28, 100, 112, -129, -63, -60, 116, 133, 11] [[2, 4, 9, 8, 12, 13], [0, -8, -42, -23, -49, -54]] 366.221040 [18, 27, 18, -126, -81, 1, -22, -262, -195, -34, -386, -288, -416, -294, 186] [[9, 15, 22, 26, 26, 30], [0, -2, -3, -2, 14, 9]] 366.529508 [2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110] [[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] 376.528284 [34, 111, 64, -28, 27, 97, 6, -162, -83, -163, -449, -348, -300, -161, 197] [[1, 7, 20, 13, -1, 8], [0, -34, -111, -64, 28, -27]] 388.395969 [92, 78, 62, -44, -54, -90, -160, -388, -426, -75, -372, -414, -338, -381, -24] [[2, 26, 24, 21, -4, -6], [0, -46, -39, -31, 22, 27]] 402.962307 [20, -30, -10, -80, 0, -94, -72, -196, -74, 61, -82, 111, -190, 37, 296] [[10, 16, 23, 28, 34, 37], [0, -2, 3, 1, 8, 0]] 442.288853 [90, 135, 90, -90, -135, 5, -110, -454, -547, -170, -676, -813, -564, -712, -134] [[45, 71, 104, 126, 156, 167], [0, 2, 3, 2, -2, -3]] 568.498130 [38, -3, 8, 64, -81, -93, -94, -30, -269, 27, 159, -177, 152, -257, -517] [[1, -7, 3, 1, -11, 22], [0, 38, -3, 8, 64, -81]] 576.552838 [18, 27, 18, 144, 189, 1, -22, 166, 233, -34, 241, 339, 342, 464, 121] [[9, 15, 22, 26, 37, 41], [0, -2, -3, -2, -16, -21]] 623.344226 [36, 54, 36, 18, -162, 2, -44, -96, -390, -68, -145, -576, -74, -588, -627] [[18, 28, 41, 50, 62, 69], [0, 2, 3, 2, 1, -9]] 650.602169 [56, 24, 26, -62, -162, -92, -116, -292, -464, -7, -227, -465, -264, -551, -331] [[2, 4, 5, 6, 6, 5], [0, -28, -12, -13, 31, 81]] 715.033013 [54, 81, 54, -108, -243, 3, -66, -358, -585, -102, -531, -864, -490, -882, -441] [[27, 43, 63, 76, 93, 99], [0, -2, -3, -2, 4, 9]] 872.189210

Message: 10707 - Contents - Hide Contents Date: Mon, 29 Mar 2004 22:38:48 Subject: Re: 5-limit Haluska commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>

>> Taking them in adjacent pairs gives >> 34, 53, 118, 441, 612, 612, 1171, 2513... >

> Back to the good old zoom diagrams! (Sorry for a dumb question): > Can the temperament for adjacent pairs of commas be found from > the wedgie? Or is it done from [0 0 1], [comma], [comma] by > inverting the matrix...

You can get it from the wedge product, which in this case is equivalent to saying you can get it from the 3D vector cross product.

Message: 10708 - Contents - Hide Contents Date: Mon, 29 Mar 2004 14:55:17 Subject: Re: Some 13-limit microtemperaments From: Graham Breed Gene Ward Smith wrote:

> I took all the 13-limit superparticular commas with numerator greater > than 2000 four at a time, giving 15 linear temperaments, listed below > in TOP/Graham badness order. The first temperament on the list seems > to be a standout, it has TM basis {1716/1715, 2080/2079, 3025/2024, > 4096/4095}, a period of 1/2 octave and a 44/39 generator. Ets are 224, > 270, and 494.

I take it you must have started with 6 commas, then. Could you tell us what they are? Then I could try duplicating the result. Or you could even past them straight into the box at Temperament Finder * [with cont.] (Wayb.) It'd help, anyway, if you were to paste commas into your messages instead of typing them. It took me a while to work out that 3025/2024 was supposed to be 3025/3024. Oh yes, the standout temperament was already top of my 13- and 15-limit microtemperament lists. Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10711 - Contents - Hide Contents Date: Tue, 30 Mar 2004 09:51:23 Subject: Re: 98 out of 99 From: Carl Lumma

>> >f you take a 5x5x5 chord cube in the 9-limit lattice of quintads, >> you get 153 notes to an octave. Reducing this by 99-et leads to >> 98 out of the 99 possible notes, the odd note out being 40, which >> represents 250/189 (its TM reduced representative.) We may harmonize >> this by adding the chord [1 -3 1], which is >> 25/21-250/189-125/84-250/147-125/63. This is the 1-6/5-4/3-3/2-12/7 >> utonal quintad over 125/126. >> >> All quintads [i j k] with absolute values less of i,j, and k less >> than 3, with the addition of [1 -3 1], is therefore one way to >> harmonize everything in 99-et. This is a set of 126=5^3+1 quintads, >> 63 otonal and 63 utonal, each of which is distinct in 99-et. I may >> try this for my next piece. >

>Could you remind me how [a b c] represents a quintad in a 9-limit >lattice? (I know, I should look in the archives...)

Ok, I'll take a stab at this, and maybe Gene can step in later. This is the lattice *of* quintads -- Z3, I believe. Gene has usually used done this with the dual to A3 in the 7-limit. I think there may have been a post at some point about how to get it to work in the 9-limit.... one can extend the chords indefinitely and still use Z3, but not usually without leaving out certain modulations. For example, I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc... -Carl

Message: 10712 - Contents - Hide Contents Date: Tue, 30 Mar 2004 18:45:25 Subject: Re: 98 out of 99 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Ok, I'll take a stab at this, and maybe Gene can step in later. > This is the lattice *of* quintads -- Z3, I believe. Gene has usually > used done this with the dual to A3 in the 7-limit. I think there may > have been a post at some point about how to get it to work in the > 9-limit.... one can extend the chords indefinitely and still use Z3, > but not usually without leaving out certain modulations. For example, > I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc...

9/5 and 9/7 are simply 3*3/5 and 3*3/7; in other words I'm not treating 9 any differently for this purpose, only using it when constructing chords. This works fine for the 9-limit but obviously not beyond.

Message: 10713 - Contents - Hide Contents Date: Tue, 30 Mar 2004 11:06:17 Subject: Re: 98 out of 99 From: Carl Lumma

>9/5 and 9/7 are simply 3*3/5 and 3*3/7; in other words I'm not >treating 9 any differently for this purpose, only using it when >constructing chords. This works fine for the 9-limit but obviously >not beyond.

I don't follow. I can extend the scheme to the 11-limit and not have modulations by any ratios of 11, even though the chords contain 11-identities. Why are ratios of 9 any different? -Carl

Message: 10714 - Contents - Hide Contents Date: Tue, 30 Mar 2004 22:22:32 Subject: Re: Stoermer From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>

>>> Great. Xenharmonikon readers might like to know: are there any 23- >>> limit superparticulars above 10,000,000? >> >> Nope. >

> I spoke too soon, after only looking at the strict 23 limit table. For > some reason probably connected to the fact that 19 is the larger of a > twin prime pair, there are two 19-limit commas smaller than any > strictly 23-limit comma. They are: > > 5909761/5909760 > |-8 -5 -1 0 2 2 2 -1> > > 11859211/11859210 > |-1 -4 -1 1 -4 1 0 4>

As expected, the former appears in XH17; the latter doesn't. So John Chalmers could inform the readership that if the 19-limit list is supplemented with 11859211/11859210, the "less than 10^7" qualification can be dropped, and the lists are complete. The total number of superparticulars in the 23-prime-limit would then be 241.

Message: 10715 - Contents - Hide Contents Date: Tue, 30 Mar 2004 02:09:23 Subject: 98 out of 99 From: Gene Ward Smith If you take a 5x5x5 chord cube in the 9-limit lattice of quintads, you get 153 notes to an octave. Reducing this by 99-et leads to 98 out of the 99 possible notes, the odd note out being 40, which represents 250/189 (its TM reduced representative.) We may harmonize this by adding the chord [1 -3 1], which is 25/21-250/189-125/84-250/147-125/63 This is the 1-6/5-4/3-3/2-12/7 utonal quintad over 125/126. All quintads [i j k] with absolute values less of i,j, and k less than 3, with the addition of [1 -3 1], is therefore one way to harmonize everything in 99-et. This is a set of 126=5^3+1 quintads, 63 otonal and 63 utonal, each of which is distinct in 99-et. I may try this for my next piece. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10716 - Contents - Hide Contents Date: Tue, 30 Mar 2004 18:57:48 Subject: Re: Musical harmony a fuzzy entropic characterization From: Carl Lumma

>Vidyamurthy and Murty, Musical harmony a fuzzy entropic >characterization, Fuzzy Sets and Systems 48 (1992) #2, 195-200

Boy does this look familiar, but I can't find the reference... -Carl

Message: 10719 - Contents - Hide Contents Date: Wed, 31 Mar 2004 10:40:59 Subject: Re: Questions for Carl From: Carl Lumma

>I was wondering if you could bring me up to speed regarding some >gaps I have in my understanding of things that are posted on this >list.

You may have me confused with someone who understands the things posted on this list. :)

>1. I notice that (sometimes)generators-to-primes "line up" with >temperament values (like 12 19 28) when you invert a matrix of >commas, and then multiply by the determinant. Is there a rule >for this?

This sounds vaguely like Paul E.'s procedure to find the number of notes in a periodicity block, which would correspond to the number of pitches in an ET based on those commas (barring torsion). In general, there are recurrence relations that give the number of pitches in 'good' temperaments. Some of them can be picked off the Stern-Brocot tree. Gene knows more about this.

>2. I still am not clear on how period values are calculated. (Using >matrices). Using wedge products I see how they are calculated >(from the wedgie) but I still am having trouble convincing myself >why wedging period ^ generators leads to the same wedgie as you >would obtain from monzo ^ monzo or value ^ value. (I know that >monzo wedgie is backwards from value wedgie, etc)

I have yet to understand the techniques based on wedge products. Maybe you can help explain them to me once you've mastered them!

>3. I get how periods may be part of an octave, when gcd(values) >is not 1. Once again, is there a rule where these generators >and periods "line up" with temperament values?

This sounds like your first question. What exactly do you mean by "temperament values"? And "line up"? -Carl

Message: 10720 - Contents - Hide Contents Date: Wed, 31 Mar 2004 10:47:36 Subject: Re: 98 out of 99 From: Carl Lumma

>>> >ould you remind me how [a b c] represents a quintad in a 9-limit >>> lattice? (I know, I should look in the archives...) >>

>> Ok, I'll take a stab at this, and maybe Gene can step in later. >> This is the lattice *of* quintads -- Z3, I believe. Gene has >> usually used done this with the dual to A3 in the 7-limit. I >> think there may have been a post at some point about how to get >> it to work in the 9-limit.... one can extend the chords >> indefinitely and still use Z3, but not usually without leaving >> out certain modulations. For example, I'm not sure how Z3 can >> hold modulations by 9:5, 9:7, etc... >

>Do you mean the Z3 group?

I mean the cubic lattice. Gene once told me it was called Z3. Maybe it's the lattice that you get when you assume the symmetries of the Z3 group??

>What's A3?

The FCC (usually 7-limit) lattice.

>Still don't see how the [i j k] >represents a quintad...Thanks

Actually, if you follow the thread, you'll see I'm still waiting for Gene to explain how he can get modulations of 9:5, 9:7, etc, by moving only distance 1 on the lattice. I thought the whole point of the observation that the dual of the 7-limit lattice is also a lattice was that you can represent all the possible modulations as a single step (there are 6 possible modulations in the 7-limit, and every point in the cubic lattice is connected to 6 others). -Carl

Message: 10721 - Contents - Hide Contents Date: Wed, 31 Mar 2004 19:55:20 Subject: Re: Questions for Carl From: Graham Breed Paul G Hjelmstad wrote:

> 1. I notice that (sometimes)generators-to-primes "line up" with > temperament values (like 12 19 28) when you invert a matrix of > commas, and then multiply by the determinant. Is there a rule > for this?

"Invert a matrix ... and then multiply by the determinant" gives the adjoint. Once you know the word, it's easier to use it, because the inverse as such isn't that interesting. Yes, the column of the adjoint corresponding to the row of the original matrix that represents the octave gives you a representative ET mapping/val/constant structure. If you want to temper out all the commas, that's the equal temperament you were looking for. If you don't temper out any columns, the constant structure refers to the periodicity block. Fokker only worked with octave-equivalent matrices, so his determinant only told him how many notes there were. The octave-specific method gives you the mappings for the other primes, and also helps you find and remove torsion. If you temper out some but not all commas, this is one equal temperament that's a special case of whatever dimensioned temperament you end up with.

> 2. I still am not clear on how period values are calculated. (Using > matrices). Using wedge products I see how they are calculated > (from the wedgie) but I still am having trouble convincing myself > why wedging period ^ generators leads to the same wedgie as you > would obtain from monzo ^ monzo or value ^ value. (I know that > monzo wedgie is backwards from value wedgie, etc)

Do you mean the whole column mapping intervals to periods? It's a fiddly calculation, and I'd need to check with my source code, which you have anyway. But do it once and you can forget about it. It happens that the period and generator mappings are both vals. The generator mapping is special (and unique) in that it refers to an imaginary equal temperament with zero notes to the octave. It must be a val, because you get it from the adjoint of the matrix of commas. It's what you get when one of the "commas" you temper out is an octave. You could get it by repeatedly subtracting more sensible vals, because the difference between vals is always a val (even if it isn't an equal temperament). For temperaments like mystery that divide the octave into a middling number of steps, it's more obvious the period mapping is a val as well.

> 3. I get how periods may be part of an octave, when gcd(values) > is not 1. Once again, is there a rule where these generators > and periods "line up" with temperament values?

I don't see what you mean by this.

> 4. Geometry. I've got some questions... I'll discuss in the > relevant post > > Thanks! Anyone else, feel free to chime in...

Yes, well, I've done so. I don't know if you've given up on me yet. Graham

Message: 10722 - Contents - Hide Contents Date: Wed, 31 Mar 2004 00:28:41 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> As expected, the former appears in XH17; the latter doesn't. So John > Chalmers could inform the readership that if the 19-limit list is > supplemented with 11859211/11859210, the "less than 10^7" > qualification can be dropped, and the lists are complete. The total > number of superparticulars in the 23-prime-limit would then be 241.

That and a cite of Dick Lehmer's article would require only a short paragraph in the next edition of XH.

Message: 10723 - Contents - Hide Contents Date: Wed, 31 Mar 2004 20:51:02 Subject: Re: 98 out of 99 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>> Do you mean the Z3 group? >

> I mean the cubic lattice. Gene once told me it was called Z3. > Maybe it's the lattice that you get when you assume the symmetries > of the Z3 group??

The cyclic group of order 3, the cubic lattice, and the ring of 3adic integers are all called Z3; we just have to live with it.

> Actually, if you follow the thread, you'll see I'm still waiting > for Gene to explain how he can get modulations of 9:5, 9:7, etc, > by moving only distance 1 on the lattice.

You can't, but then you can't get any other modulations this way either. Moving a distance of 1 always exchanges otonal and utonal tetrads/quintads. I thought the whole

> point of the observation that the dual of the 7-limit lattice is > also a lattice was that you can represent all the possible > modulations as a single step (there are 6 possible modulations > in the 7-limit, and every point in the cubic lattice is connected > to 6 others).

The six tetrads you get are the six tetrads sharing an interval. To do the 9-limit more intrinsically, we would use a packing, not a lattice, and we would have ten quintads, not six. We could take all points [a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for minor quintands, and link them when they share an interval. I don't know what that looks like, but it's three-dimensional.

Message: 10724 - Contents - Hide Contents Date: Wed, 31 Mar 2004 00:34:14 Subject: 11 and 13 limit Haluska commas From: Gene Ward Smith Here they are: 11 limit 2401/2400 |-5 -1 -2 4 0> 3025/3024 |-4 -3 2 -1 2> 4375/4374 |-1 -7 4 1 0> 9801/9800 |-3 4 -2 -2 2> 151263/151250 |-1 2 -4 5 -2> 1771561/1771470 |-1 -11 -1 0 6> 3294225/3294172 |-2 2 2 -7 4> 781258401/781250000 |-4 2 -11 2 6> 4882812500/4882786447 |2 0 13 -9 -2> 2541867610898/2541865828329 |1 -26 0 2 10> 13 limit 10648/10647 |3 -2 0 -1 3 -2> 123201/123200 |-6 6 -2 -1 -1 2> 5767168/5767125 |19 -1 -3 -1 1 -3> 192914176/192913083 |8 -13 0 3 -2 3> 4060088955/4060086272 |-25 7 1 0 -2 5> 14869765625/14869757952 |-15 -3 7 -5 4 1> 13051691536000/13051688172831 |7 -5 3 -9 -3 8> 28344980104623/28344976000000 |-10 10 -6 5 -6 4>