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Message: 1575 - Contents - Hide Contents Date: Fri, 7 Sep 2001 10:09 +01 Subject: Re: Tenney's harmonic distance From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9n9a0e+cngv@xxxxxxx.xxx> In article <9n9a0e+cngv@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () wrote:> How do get Midiconv to input a midi file and output an ascii file of > pitch values which I can edit? You can't.
Message: 1576 - Contents - Hide Contents Date: Sat, 08 Sep 2001 02:41:46 Subject: Re: Distance measures cut to order From: Carl Lumma>> >ne of the most interesting mappings done by Wilson is his mapping >> onto a Penrose tiling, treating it as a two dimensional >> representation of a 5 dimensional space. > >Wowsers!Recall that Erv Wilson is the person who gave us MOS and CS... some examples of his Coxeter-like 2-D projections of tonespace can be seen at... http://www.anaphoria.com/dal.PDF - Type Ok * [with cont.] (Wayb.) ...Unfortunately, they are not very legible in this PDF file. Hard copies of Wilson's articles can be found in back issues of Xenharmonikon, available from Frog Peak music. Lately, Erv has been building 3-D projections of tonespace with Zometool. -Carl
Message: 1577 - Contents - Hide Contents Date: Tue, 11 Sep 2001 08:34:37 Subject: Fwd: Two 21-tone JI scales: detemperings of blackjackk From: Paul Erlich --- In tuning@y..., "Paul Erlich" <paul@s...> wrote: Here's a JI convex detempering of blackjack that seems to give the most possible pure 4:5:6:7 chords (six of them). Is that right? Go to Yahoo groups: /tuning/files/perlich/scales/ * [with cont.] and download blackjack6.gif The JI scale is as follows: ratio note cents 1/1 E[ 0 50/49 Ev 35 15/14 F< 119 320/147 F 147 8/7 Gbv 231 7/6 Gb^ 267 60/49 G 351 5/4 G> 386 21/16 Ab^ 471 4/3 A[ 498 7/5 A> 583 10/7 Bb< 617 3/2 B[ 702 32/21 Bv 729 8/5 C< 814 80/49 C 849 5/3 C> 884 7/4 Db^ 969 25/14 D[ 1004 15/8 D> 1088 40/21 Eb< 1116 The step sizes are 50/49 21/20 64/63 21/20 49/48 360/343 49/48 21/20 64/63 21/20 50/49 21/20 64/63 21/20 50/49 49/48 21/20 50/49 21/20 64/63 21/20 If change the note G by 2400:2401 we have this lattice instead: Go to Yahoo groups: /tuning/files/perlich/scales/ * [with cont.] and download blackjacksup.gif The scale is ratio note cents 1/1 E[ 0 50/49 Ev 35 15/14 F< 119 320/147 F 147 8/7 Gbv 231 7/6 Gb^ 267 49/40 G 351 5/4 G> 386 21/16 Ab^ 471 4/3 A[ 498 7/5 A> 583 10/7 Bb< 617 3/2 B[ 702 32/21 Bv 729 8/5 C< 814 80/49 C 849 5/3 C> 884 7/4 Db^ 969 25/14 D[ 1004 15/8 D> 1088 40/21 Eb< 1116 with step sizes 50/49 21/20 64/63 21/20 49/48 21/20 50/49 21/20 64/63 21/20 50/49 21/20 64/63 21/20 50/49 49/48 21/20 50/49 21/20 64/63 21/20 This could satisfy Kraig Grady, and perhaps even Pierre Lamothe, despite the loss of a tetrad relative to the first scale. Pierre, any comments on this last version? --- End forwarded message ---
Message: 1578 - Contents - Hide Contents Date: Tue, 11 Sep 2001 19:24:47 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjackk From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> This could satisfy Kraig Grady, and perhaps even Pierre Lamothe, > despite the loss of a tetrad relative to the first scale.What do Kraig and Pierre need to make them happy?
Message: 1579 - Contents - Hide Contents Date: Wed, 12 Sep 2001 18:51:22 Subject: Re: Blocks as scales along a line From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:>The notes are > strung out in an approximate line across a diagonal of the > rectangular prism defined by [v_1, ..., v_k], so it is a "scale along > a line" as I called it in the subject header.I might add that given a notation [v_1, ..., v_k] we get C(k,1) = k blocks, C(k,2) = k(k-1)/2 blocks with one comma tempered out, and so forth down to C(k, k-2) = k(k-1)/2 blocks in various linear temperaments, corresponding to faces, and C(k,k-1) = k equal temperaments.
Message: 1580 - Contents - Hide Contents Date: Wed, 12 Sep 2001 10:55:22 Subject: Blocks as scales along a line From: genewardsmith@xxxx.xxx I'm a little insomniaic right now, so I thought I'd explain a neat way to look at and calculate PB scales. The way it's been done might be described as taking a basis for a notation in the p-limit, and setting one basis element aside, finding the block from the rest. The dual point of view says that we select out the corresponding val. Suppose [v_1, ..., v_k] form a notation in the p-limit, where k=pi(p) is the number of primes up to p, and suppose all the vals v_i are positive, so that v_i(q) for q<=p a prime is a positive integer. Suppose also that (b_1, ..., b_k) is a basis for the notation. We create a block of n=v_1(2) scale steps in an octave by setting aside b_1, and replacing the top row representing b_1 with [1 0 0 ... 0] representing 2, which gives us a Graham matrix we can invert and use to define the block. The inverted matrix can be written in terms of our notation as [(1/n)v_1, v_2 - (v_2(2)/n)v_1, ..., v_k - (v_k(2)/n)v_1], which we may verify by applying it to 2 and the basis elements b_i for i>1, obtaining the identity. This means that the condition for a note q being in the block is 0 <= v_1(q) < n, and -1/2 <= v_i(q) - (v_i(2)/n)v_i(q) < 1/2 for each i>1. Thus the m-th scale step expressed in our notation can be calculated from v_1(q) = m and v_i(q) = ceil(-1/2 + (v_i(2)/n)m). The notes are strung out in an approximate line across a diagonal of the rectangular prism defined by [v_1, ..., v_k], so it is a "scale along a line" as I called it in the subject header. I hope this was reasonably coherent.
Message: 1581 - Contents - Hide Contents Date: Thu, 13 Sep 2001 20:48:19 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjackk From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> This could satisfy Kraig Grady, and perhaps even Pierre Lamothe, >> despite the loss of a tetrad relative to the first scale. >> What do Kraig and Pierre need to make them happy?Kraig loves CS scales with all steps as superparticular ratios. As for Pierre . . . I'll let him speak for himself.
Message: 1582 - Contents - Hide Contents Date: Thu, 13 Sep 2001 02:16:19 Subject: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Pierre Lamothe Paul I would believe naïvely that an interval like 64/63 is too small to be seen as a step but I would never use that as an argument since it concerns musical judgment rather than maths and I am searcher in music-math relations, not musician. However I mentionned (post 1000) a problem concerning linear generator picking up unison vectors using convexity. That suggests 64/63 could be a unison vector misinterpreted as a step. Besides you wrote precedently: << Personally, I think Blackjack, let along Canasta, have too many notes to be heard and conceptualized in their entirety, the way diatonic scales and their Middle-Eastern cousins are, and perhaps my decatonic scales can be.>>So rather than comment on receivability in gammier theory of your detempered values, I would suggest a reinterpretation of the Blackjack set using incidentally a decatonic gammier. You have to judge if it has interest from musical viewpoint or even may concern your decatonic scales. If not, it won't be bad for it's not strictly derived from gammier theory. I had to use ad hoc maths to insert anew consistency into a reduced (so desorganized) set of 21 intervals out of a gammier having 41 intervals originally. ----- As alternative to the Blackjack set seen as the scale 0 2 7 9 14 16 21 23 28 30 35 37 42 44 49 51 56 58 63 65 70 72 having alternatively steps 2 and 5, let us work on this approach where the same set would be seen as 9 16 23 30 37 44 51 58 65 0 72 7 14 21 28 35 42 49 56 63 where 2 and 70 are missing for being considered as unison vectors. The values 2 and 70 would keep sense only for the transposition and would be stranger in the modes begining with 0. ----- In appearance, there is no problem to describe the melodic relations here since the steps seem to be simply 7 and 9.>From the tempered viewpoint it would be hard to go overthe following solution. 9---16---23---30---37---44---51---58---65 / / / / / / / / / \ 0 / / / / / / / / 72 \ / / / / / / / / / 7---14---21---28---35---42---49---56---63 However, I recall that the Blackjack set is supposed to be very closed from just values in 11-limit and it's a reduced set (after Canasta 31 and Miracle 41), so the steps are not forcely all retained. So I want to refine the problem seeking to detemper in such way that the intervals would be both of minimal sonance and melodically well-organized. ----- I have a link to images (post 1000) showing that the gammier ib1215 would be the simplest solution using 41 intervals and where it is easy to see that the step relations would be completely desorganized with the Blackjack set. Starting from that it seemed clear that odd 11 couldn't be used to reorganize the melodic relations (even if it keeps sense vertically). Thus, seeking to reorganize the just relations in the 7-limit I used the following relation 12/11 = 35/32 modulo 385/384 to find an optimal solution which requires to use also the step 5 with 7 and 9. I don't give more details since it's not a systematic approach. ----- In the following graph the edges (--,\,---) correspond to (5,7,9). (2) seen as unison vector \ 0 --- 9 \ \ 7 --- 16 \ \ 14 --- 23 -- 28 \ \ \ 21 --- 30 -- 35 \ \ 37 -- 42 --- 51 \ \ \ 44 -- 49 --- 58 \ \ 56 --- 65 \ \ 63 --- 72 \ (70) seen as unison vector which is a regular lattice (in sense of "treillis") if the unison vectors are removed. So, detempering as (5,7,9) == (21/20, 16/15 or 15/14, 35/32) we obtain this JI lattice having all low sonance excepted the two using odd 105. | 35/32 | 21/20 | 35/32 | --- ( 1 )----35/32 | | 16/15 | | | | --- 16/15-----7/6 | | 15/14 | | | | --- 8/7------5/4-----21/16 | | | 16/15 | | | | | | --- 128/105----4/3------7/5 | | 15/14 | | | | --- 10/7------3/2----105/64 | | | 16/15 | | | | | | --- 32/21-----8/5------7/4 | | 15/14 | | | | --- 12/7-----15/8 | | 16/15 | | | | --- 64/35----( 2 ) So, rather than that 9---16---23---30---37---44---51---58---65 / / / / / / / / / \ 0 / / / / / / / / 72 \ / / / / / / / / / 7---14---21---28---35---42---49---56---63 we have that 9---16---23---30---37---44 51---58---65 / / / \ / \ \ \ / / / \ 0 / / \ \ \ \ / / 72 \ / / / \ \ \ / \ / / / 7---14---21 28---35---42---49---56---63 ----- I add simply that 9 as a step has to correspond to 35/32 here to give consistency while it may appear as 12/11 in chord. Finally, I could draw images later if this JI interpretation has sense at musical viewpoint. Pierre
Message: 1583 - Contents - Hide Contents Date: Thu, 13 Sep 2001 20:54:59 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Paul Erlich --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> > where 2 and 70 are missing for being considered as unison > vectors. The values 2 and 70 would keep sense only for the > transposition and would be stranger in the modes begining > with 0.Don't assume that the modes beginning with 0 are more important or more special than other modes. Look at my suggestion with superparticular steps. It is not based on the mode beginning with 0. I thought you might like it because for my 22-tET-based decatonic scales, the JI versions with superparticular steps were the ones you found most suitable for your theory.
Message: 1584 - Contents - Hide Contents Date: Thu, 13 Sep 2001 07:08:08 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> I would believe naïvely that an interval like 64/63 is too > small to be seen as a step but I would never use that as an > argument since it concerns musical judgment rather than maths > and I am searcher in music-math relations, not musician.It's 1/44 of an octave, and in some temperings it is larger. It's a single step in the 19,26,31,34 and 41 systems, and certainly a single 19-et step does sound very much like a scale step. I'll probably want to comment further but I need to chase down your reference.
Message: 1585 - Contents - Hide Contents Date: Fri, 14 Sep 2001 21:42:47 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Paul Erlich --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:>As I already said, (even if that concerned true gammier), all >possible melodic modes have not the same importance.The relative importance they may have is related to the musical grammar used, which lies outside the considerations of your model. Before the 17th century, the major mode (what you call "Zarlino") of the diatonic scale was considered one of the _least_ important melodic modes. And it greater importance since the 17th century, I don't believe is accounted for by anything in your theory.>The best >among them are the more compact in the latticeI don't see how one melodic mode can be more compact in the lattice than another. They are all identical lattice configurations.>(in sense of >"réseau").So perhaps by "lattice" you don't mean "harmonic lattice". Can you elucidate the "réseau" concept?
Message: 1586 - Contents - Hide Contents Date: Fri, 14 Sep 2001 07:49:26 Subject: A simple condition for a valid basis From: genewardsmith@xxxx.xxx Recall that the statement of this included a requirement that the block be ordered by the val v dual to the block's basis for the basis to be valid. There is a simple sufficient condition for this. If {b_3, b_5, ..., b_p} is the block's basis, we may form the Graham matrix whose rows are given by the vector corresponding to 2 (in the usual prime notation, that would be [1,0, ..., 0]) and the basis row vectors. If we invert this matrix the columns will be vals, the first of which is the val v/n, where v(2) = n (which we may recall is positive, and gives the number of notes an octave of the block. If the other vals are {u_3, u_5, ..., u_p} we have an expression for any note r in this notation as r = 2^v(r)/n * b_3^u_3(r) * ... * b_p^u_p(r). The note r will be in the block if 0 <= v(r) < n and if -1/2 <= u_q(r) < 1/2 for each odd prime q<=p. A sufficient condition for the notes of the block to be ordered by v is that the product b_3^u_3(r) ... b_p^u_p(r) is not greater than 2^(1/2n) and given the bounds on the exponents u_q(r) a sufficient condition in turn for this is that the product (c_3 c_5 ... c_p)^n, (where c_q = exp(|ln b_q|) is the multiplicative absolute value of b_q) is less than 2.
Message: 1587 - Contents - Hide Contents Date: Fri, 14 Sep 2001 23:50:52 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Pierre Lamothe Paul, In writing << all possible melodic modes have not the same importance >> I was at hundred miles to think you could take that in the sense of historical importance. In the Zarlino gammier 1 5 2 6 6 3 0 4 1 1 5 2 6 there exist 16 modes. One of them is the non-convex 1 16/15 6/5 4/3 3/2 5/3 15/8 2 . 5 . 6 . 3 0 4 . 1 . 2 . I wanted to say uniquely that a such mode is less important, for instance, than the doristi mode 1 16/15 6/5 4/3 3/2 8/5 16/9 2 . . . . 6 3 0 4 . 1 5 2 . for reason which appears clearly in the lattice. Like the Zarlino mode itself, the second implies for instance a) convexity and minimality (about the space obtained by rotation), being . o o o o . . o o o o o . . o o o o . rather than the following (periodic but non-convex) for the first one . . . o o o o . o o o o o . o o o o . . . b) the maximality for the number of triads (5 rather than 2 in the first case). ----- I never thought that the reality was reduced at my poor little considerations on a simple aspect. Besides, I remember a time where I didn't resist to start a discussion after your comments. Now, I could'nt. ----- You wrote: << So perhaps by "lattice" you don't mean "harmonic lattice". Can you elucidate the "réseau" concept? >> In question for Gene I wrote: << We don't have this problem in French for we use "treillis" in the first sense of partial ordering where two elements have always inferior and superior "bornes", and "réseau" in the sense of discrete Z-module. >> A module is like a vector space, If a vector V exists, then any kV exists, where k is an element of - a field like R (reals) for the vector space - a ring like Z (integers) for the module A Z-module is a module where the ring is Z. A "réseau" (discrete Z-module) is discrete i.e. has all its elements separate in R (at center of a ball where there not exist another element). What you name "harmonic lattice" is based on a represention of a such lattice in which you represent some edges like, for instance, edges 6/5, 5/4, 3/2 which are differences between two vectors. Z-module and vector space are suitable for interval space while a pitch space is an "espace affine" or an "espace affine muni d'une origine" when a tonic is defined. The unison element don't exist in these "espaces affines" but it's the neutral element of a "réseau" and a vector space. Pierre
Message: 1588 - Contents - Hide Contents Date: Fri, 14 Sep 2001 00:58:17 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Pierre Lamothe Paul wrote: << Don't assume that the modes beginning with 0 are more important or more special than other modes. Look at my suggestion with superparticular steps. It is not based on the mode beginning with 0. >> I don't assume that. I wrote my remark (that 2 and 70 keep sense for the transposition) precisely for that. << The values 2 and 70 would keep sense only for the transposition and would be stranger in the modes beginning with 0. >> In your case the modes are the 21 rotations and there exist only one mode beginning by 0. In my case there exist 60 decatonic modes beginning with 0. By transposition, (now in "pitch space"), these modes may begin with 0, 7, 14, 21, etc. When a mode begin with 7n distinct from 0 then 7n+2 is now stranger to the mode while 2 is used. The "note" 7n+2 is stranger for being in the same class as the the "tonic" 7n. As I already said, (even if that concerned true gammier), all possible melodic modes have not the same importance. The best among them are the more compact in the lattice (in sense of "réseau"). So the interval space (obtained by rotation) is minimal and chords are maximal. ----- Since I don't have time to write now on the last point I could quote a french post to Antti Kartunnen in june 2001 about the criteria used with modal classes. That contains figures helping to understand my point about inequal importance of modes. << - modalité - transposabilité - minimalité - harmonicité - simplicité - connexité La modalité réfère à la dualité majeure mineure. La transposabilité réfère au nombre de modes possibles par changement de tonique (ça définit la classe modale) et renversement. La minimalité réfère à l'espace des intervalles générés par le mode. L'harmonicité réfère aux accords de base possible. Bien que l'harmonie relève d'abord de la microtonalité, il y a ici un aspect macrotonal. Voici comment caractériser l'harmonicité et la transposabilité des 4 classes modales dans ib1051. o - o | \ | \ 10 modes (majeurs/mineurs) - 3 accords o - o - o o | \ o - o - o 3 modes (symétriques) - 2 accords \ | o o | \ o - o - o 3 modes (symétriques) - 2 accords \ | o o o \ | o - o 4 modes (majeurs/mineurs) - 1 accord \ | o La simplicité réfère à la sonance d'un intervalle et la connexitéau nombre de connexions (conjointes) sur cet intervalle. Mathématiquement, c'est liée à la symétrie (modalité) la convexité (transposabilité, minimalité, harmonicité) la centralité (simplicité, connexité) Pierre
Message: 1589 - Contents - Hide Contents Date: Sat, 15 Sep 2001 00:08:24 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> (in sense of >> "réseau").> So perhaps by "lattice" you don't mean "harmonic lattice". Can you > elucidate the "réseau" concept?"Réseau" is French for "lattice" in the sense we've been using it; "lattice" has another meaning in English-language mathematics which French sensibly gives another name to.
Message: 1590 - Contents - Hide Contents Date: Sat, 15 Sep 2001 06:14:25 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> A Z-module is a module where the ring is Z.To keep things from getting too confusing around here, I'll mention again that a Z-module is the same as an abelian group.> A "réseau" (discrete Z-module) is discrete i.e. has all its > elements separate in R (at center of a ball where there not > exist another element).I think again we could fix our terminology around here and call this a lattice. If we want the other sense of lattice, why not call it "treillis"? If we are really lucky, the Anglo-Saxons will start doing this too and we all will be happier.> Z-module and vector space are suitable for interval space > while a pitch space is an "espace affine" or an "espace > affine muni d'une origine" when a tonic is defined.I don't see that you need the concept, but if you think you do why not just call it "affine space", since you are writing in English? If a tonic is defined then it seems to me you are back with groups. What is something acted on by a lattice, I wonder--it's a sort of symmetric space?
Message: 1591 - Contents - Hide Contents Date: Sat, 15 Sep 2001 08:40:25 Subject: An interesting class of notations From: genewardsmith@xxxx.xxx Recall that "jacks", the ratios between adjacent elements in the half (octave) of the Farey sequence from 1/2 to 1, plus those derived from approaching 1/2 from below, began as superparticular ratios, or epimorios, whose denomiators are either triangular or square numbers. If t(n) = n(n+1)/2 is the nth triangular number, we may denote these by T(n) = t(n)/(t(n)-1) = and S(n) = n^2/(n^2-1). We then have T(n) = S(n)S(n+1), S(n) = T(2n-1)T(2n), as we may verify by a simple calculation. Let us began from T(2)S(2) = (3/2)(4/3) = 2. If we take the ratio between them, 9/8, and add to it 4/3 then anything which is a product of powers of T(2) and S(2) (which is the full 3-limit) will likewise be a product of powers of 4/3 and 9/8; this is therefore a note-basis for a notation for the 3-limit. If we write the matrix as [ 2 -1] B(1) = [-3 2] then the inverse matrix [2 1] N(1) = [3 2] gives us the notation itself. We now apply the splitting rule for the larger element, T(2), and get T(2) = S(2)S(3) = (4/3)(9/8). If we take the interval from 9/8 to 4/3 we have 32/27, and we get [9/8, 32/27] or [-3 2] [3 2] B(2) = [5 -3], B(2)^(-1) = N(2) = [5 3] Continuing on in this way we get <T(3),T(4),S(4)> leading to [10/9,16/15,81/80], so that [ 1 -2 1] B(3) = [ 4 -1 -1], [ 5 2 3] [-4 4 -1] N(3) = [ 8 3 5] [12 4 7] Since (16/15)^5(81/80)^5 < 2, we can compute a pentatonic scale whose nth note is given by (10/9)^n * (16/15)^ceil(-1/2+2n/5) * (81/80)^ceil(-1/2+3n/5), giving us 1-9/8-4/3-3/2-16/9-(2) as our PB scale. Continuing on in this way, we get <S(3),T(4),S(4)> leading to [ 4 -1 -1] [ 7 5 3] B(4) = [-3 -1 2] N(4) = [11 8 5] [-4 4 -1] [16 12 7] and scale 1-10/9-4/3-3/2-5/3-9/5-(2); then <T(4),T(5),S(4),T(6)> and [-2 1 -1 1] [10 2 7 5] B(5) = [ 2 -3 0 1] N(5) = [16 3 11 8] [ 6 -2 0 -1] [23 5 16 12] [-5 2 2 -1] [28 6 19 14] with scale 1-16/15-9/8-56/45-4/3-7/5-3/2-8/5-16/9-28/15-(2), then <T(5),S(4),T(6),S(5)> and [-3 -1 2 0] [12 7 10 3] B(6) = [ 6 -2 0 -1] N(6) = [19 11 16 5] [ 1 2 -3 1] [28 16 23 7] [-5 2 2 -1] [34 19 28 8] with scale 1-16/15-9/8-56/45-4/3-3/2-8/5-16/9-28/15-(2). People may not think much of these PBs, but there is actually a lot of flexibility in these ratios and we could even build scales with only epimoric steps out of them for those who are fans of that sort of thing. We do finally run into two factors which make the process no longer automatic--36/35, which is simultaneously T(8) and S(6), and so which breaks down both as S(9)S(10) and as T(11)T(12), and 50/49, the first jack appearing with a denominator which is neither triangular nor square.
Message: 1592 - Contents - Hide Contents Date: Sat, 15 Sep 2001 08:57:00 Subject: Re: An interesting class of notations From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> [-3 -1 2 0] [12 7 10 3] > B(6) = [ 6 -2 0 -1] N(6) = [19 11 16 5] > [ 1 2 -3 1] [28 16 23 7] > [-5 2 2 -1] [34 19 28 8]I wrote the same scale down twice, the one which goes with the above notation is 1-16/15-28/25-25/21-5/4-4/3-7/5-3/2-8/5-5/3-16/9-15/8-(2).
Message: 1593 - Contents - Hide Contents Date: Sun, 16 Sep 2001 06:24:19 Subject: Epimoric epilogue From: genewardsmith@xxxx.xxx I previously derived the PB scale 1-10/9-6/5-4/3-3/2-5/3-9/5-(2) (and once again wrote it down incorrectly) from the notation I called N(4), with basis <16/15, 25/24, 81/80>, which in turn was dervied from the basis <S(3), T(4), S(4)> = <9/8, 10/9, 16/15> by dividing adjacent elements. This second basis, consisting of epimoric elements, is well-suited to producing scales with epimoric steps, and so one might wonder if we can obtain such scales as PBs. The scale above was calculated as the one whose nth degree is (16/15)^n * (25/24)^ceil(b1 + 5n/7) * (81/80)^ceil(b2 + 3n/7), where the base points were b1 = b2 = -1/2. However, we may pick any base points we like in the square region -1 < b1, b2 <= 0 and obtain a PB, and this region is chopped up by lines into subregions which produce a variety of scales and scale modes. If we consider the JI diatonic scale written in the notation, defined by [h_7, h_5, h_3] we get 9/8 --> [1, 1, 1] 5/4 --> [2, 1, 1] 4/3 --> [3, 2, 1] 3/2 --> [4, 3, 2] 5/3 --> [5, 4, 2] 15/8--> [6, 5, 3] If we add (or on the left, multiply) we get [21, 17, 10]; subtracting 3*[7, 5, 3] we get [0, 2, 1]; if therefore we adjust the base point by setting b1 = -1/2 + 2/7 = -3/14 and b2 = -1/2 + 1/7 = -5/14, we get something which should be in the middle of the region producing the diatonic JI if there is one. Calculating (16/15)^n * (25/24)^ceil(-3/14 + 5n/7) * (81/80)^ceil(-5/14 + 3n/7) we find that we do, in fact, obtain the diatonic JI as a PB scale. If we want to obtain an et suitable to the nature of the scale, we can look at the group dual to the smallest basis element, 81/80, giving us p h_7 + q h_5; for p and q both positive we obtain h_7 + h_5 = h_12, 2h_7 + h_5 = h_19 and so forth. It should be noted that we are speaking of vals, not simply ets here; we have [17] h_7 + 2h_5 = [27] [40] in null(81/80), but [17] 2h_7 + h_3 = [27] [39] in null(25/24). The same sort of considerations apply to the other notations discussed, giving rise to many PBs with epimoric scale steps.
Message: 1594 - Contents - Hide Contents Date: Sun, 16 Sep 2001 19:07:33 Subject: Lattices From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> Now, I understand you want to avoid confusion precising that > a Z-module is simply an abelian group, but since Paul ask I > precise "my" (which is not mine) concept of lattice, I think > it would have been irrelevant to simply identify lattice with > abelian group.I agree we should not identify lattice with abelian group, we should also require it to be a discrete subgroup of R^n. Two examples are: (1) Note classes in the p-limit, symmetrically arranged. For example we can define Q(3^a 5^b 7^c) = a^2+b^2+c^2+ab+ac+bc, and thereby put a metric on the 7-limit note classes, making it into a lattice. (2) Notes under an L1 (taxicab) norm; if we set ||2^a 3^b 5^c 7^d|| = ln(2)|a| + ln(3)|b| + ln(5)|c| + ln(7)|d| we exhibit it as a discrete subgroup of R^4, and hence a lattice.> I wanted to say here that, mathematically speaking, interval > space (representable by module and vector space) is a more > fundamental object than a pitch space (representable by affine > space with origin or not). Sure, we can identify space of > bipoints in affine space with vector space but we absolutely > not need to refer at a such pitch space to treat fundamentally > the interval space.I agree, and I haven't seen anyone get any milage out of affine concepts, with or without a distinguished point.
Message: 1595 - Contents - Hide Contents Date: Sun, 16 Sep 2001 03:33:26 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Pierre Lamothe Gene, You give me a precious feedback about English and mathematics. I will use simply the word lattice (to translate "réseau") and the word treillis for the partial ordering structure I use so abundantly. ----- Now, I understand you want to avoid confusion precising that a Z-module is simply an abelian group, but since Paul ask I precise "my" (which is not mine) concept of lattice, I think it would have been irrelevant to simply identify lattice with abelian group. Paul is also a scientific and I believe he has not only interest for simplified notion. If I would have asked myself this question I would have been confused with a such identification. OK, a lattice is an abelian group, however how could we use matrices if it was only a group? We use here clearly two composition laws. What seemed to me important here was the linearization process of the group and the lack of continuity in the action group. It's why I talked about vectors and use of integers rather than reals. What would be interessant would be to show the pertinence of the lattice representation (depending of the linear independance of primes) which is remarkably suitable to focus on the local part around unison, this only part concerning (in a modelization viewpoint) the music, where distinction of sonance quality has more sense than a simple aspect of the sensation. ----- Now, about the affine space. I agree we have no need for that. I mentioned that only in relation with a precedent discussion where Paul had difficulty to understand I talked always about intervals without reference to pitch space. I wanted to say here that, mathematically speaking, interval space (representable by module and vector space) is a more fundamental object than a pitch space (representable by affine space with origin or not). Sure, we can identify space of bipoints in affine space with vector space but we absolutely not need to refer at a such pitch space to treat fundamentally the interval space. ----- As you may know, I'm not a mathematician (and decidely not a fan of calculation) so it may seem paradoxal I could appear more fussy than you about mathematical definitions. It's simple. That is the principal matter of my thoughts under the mathematical angle. Besides, it's very hard for me to speak about other angles in English. I never try to apply mathematical tools a priori without a deep motivation in term of modelization. Moreover I learn only the parts I need. In my research, I approach always with instinctive representation, formulating first some intuitive notions, and when I need to refine, then I find generally in the mathematics the pretty concepts filling well my need. Now I begin to look at relation between topologic, metric and normologic concepts to precisely describe what I'm discovering and exploring in the musical interval space. ----- I'm closing here my actual short session of posts. Pierre
Message: 1596 - Contents - Hide Contents Date: Mon, 17 Sep 2001 19:47:33 Subject: Re: Ten notes to the octave From: paul@xxxxxxxxxxxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> Starting from a basis <T(4),T(5),S(4),T(6)> = > <10/9,15/14,16/15,21/20> we defined another basis > <21/20,28/27,64/63,225/224>, the corresponding notation of which is > N(5), defined by the matrix > > [10 2 7 5] > [16 3 11 8] > [23 5 16 12] > [28 6 19 14] > > which we may write as [h10,h2,g7,h5], where we write g7 and not h7 > since we have defined h7 so that h7(7) = 20, the nearest integer to > 7 log_2(7). By choosing various base points, we may produce a variety > of PBs, for instance > > (21/20)^n * (28/27)^ceil(-.3+ 2.N) * (64/63)^ceil(-.65 +.7 n) * > (225/224)^ceil(-.25+.5n) > > leading to > > 1 – 15/14 – 7/6 – 5/4 – 4/3 – 45/32 – 3/2 – 5/3 – 7/4 –15/8 – (2). > > Since we have already shown the JI diatonic scale to be a PB, we > might wonder if we can obtain a PB containing it using this basis, > and perhaps also one with epimoric intervals. For instance, we might > consider > > 1 – 21/20 – 9/8 – 5/4 – 4/3 – 7/5 – 3/2 – 5/3 – 7/4 – 15/8 – (2). > > The question of whether this is a PB turns out to be a matter of > definitions. If instead of partitioning into half-open and non- > intersecting regions to define blocks, we could use closed regions > which intersect on their boundaries.I'm unclear about the distinction here. Can you be more explicit? I don't see any grounds for considering this _not_ a PB under any reasonable definitions. Anyway, please read this: Yahoo groups: /tuning/message/20642 * [with cont.] Yahoo groups: /tuning/message/20694 * [with cont.]> We then can have more than one > candidate for a given scale degree; if we allow ourselves to select > any such candidate we choose then the above can be seen as a block, > but not otherwise. We might call such a thing a periodic semiblock, > or PS. Its scale steps in any case are rather marginal, given the > fact that (21/20)^2 < 10/9.I don't get this . . . scale steps rather marginal?> > In the first of these ten-note scales we had the interval 45/32, > which is approximately 7/5; in fact 45/32 = (7/5)(225/224), > illustrative of the fact that we might want to temper 225/224 out for > this scale. Ets doing this can be expressed as combinations > p h10 + q h2 + r g7, and ets n with cons(7, n) < 1.1 and 225/224~1 > are 12, 19, 22, 31, 41, 53, and 72. Also of interest with these > scales are ets with 64/63~1, if we again look for examples with cons > (7,n)<1.1 we obtain 12, 15, 22, and 27. > > The 22 et, which appears on both lists, would seem especially well > suited to these scales. If we look at our two scales in the 22et, the > first has steps 2232223222, and the second 2322223222; we see that > these are in fact distinct rather than modal variants. If Paul will > send me his paper or put it into a folder on tuning-math, I'll try to > collect on a bet I made with myself that these scales are both > discussed in it.You win! They are the main topic of the paper. But I already mentioned both of these scales to you here on this list, as "my" decatonic scales. I came up with them in 1991, age 19 -- and that's why I'm interested in microtonality today!
Message: 1597 - Contents - Hide Contents Date: Mon, 17 Sep 2001 05:06:36 Subject: Ten notes to the octave From: genewardsmith@xxxx.xxx Starting from a basis <T(4),T(5),S(4),T(6)> = <10/9,15/14,16/15,21/20> we defined another basis <21/20,28/27,64/63,225/224>, the corresponding notation of which is N(5), defined by the matrix [10 2 7 5] [16 3 11 8] [23 5 16 12] [28 6 19 14] which we may write as [h10,h2,g7,h5], where we write g7 and not h7 since we have defined h7 so that h7(7) = 20, the nearest integer to 7 log_2(7). By choosing various base points, we may produce a variety of PBs, for instance (21/20)^n * (28/27)^ceil(-.3+ 2.N) * (64/63)^ceil(-.65 +.7 n) * (225/224)^ceil(-.25+.5n) leading to 1 – 15/14 – 7/6 – 5/4 – 4/3 – 45/32 – 3/2 – 5/3 – 7/4 – 15/8 – (2). Since we have already shown the JI diatonic scale to be a PB, we might wonder if we can obtain a PB containing it using this basis, and perhaps also one with epimoric intervals. For instance, we might consider 1 – 21/20 – 9/8 – 5/4 – 4/3 – 7/5 – 3/2 – 5/3 – 7/4 – 15/8 – (2). The question of whether this is a PB turns out to be a matter of definitions. If instead of partitioning into half-open and non- intersecting regions to define blocks, we could use closed regions which intersect on their boundaries. We then can have more than one candidate for a given scale degree; if we allow ourselves to select any such candidate we choose then the above can be seen as a block, but not otherwise. We might call such a thing a periodic semiblock, or PS. Its scale steps in any case are rather marginal, given the fact that (21/20)^2 < 10/9. In the first of these ten-note scales we had the interval 45/32, which is approximately 7/5; in fact 45/32 = (7/5)(225/224), illustrative of the fact that we might want to temper 225/224 out for this scale. Ets doing this can be expressed as combinations p h10 + q h2 + r g7, and ets n with cons(7, n) < 1.1 and 225/224~1 are 12, 19, 22, 31, 41, 53, and 72. Also of interest with these scales are ets with 64/63~1, if we again look for examples with cons (7,n)<1.1 we obtain 12, 15, 22, and 27. The 22 et, which appears on both lists, would seem especially well suited to these scales. If we look at our two scales in the 22et, the first has steps 2232223222, and the second 2322223222; we see that these are in fact distinct rather than modal variants. If Paul will send me his paper or put it into a folder on tuning-math, I'll try to collect on a bet I made with myself that these scales are both discussed in it.
Message: 1598 - Contents - Hide Contents Date: Mon, 17 Sep 2001 19:52:57 Subject: Re: Miracle theory From: paul@xxxxxxxxxxxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> The "miracle" generator splits the difference between 15/14 and > 16/15, and hence is associated to the scales with the jumping jack > 225/224 in the kernel. If we seek other generators with miracle > properties, and obvious place to began is with other jumping jacks. > If we look instead at 81/80, we get the meantone miracle, which is > well-known.The meantone miracle? If we look higher, we find 2401/2400 = (49/48)(49/50),> 3025/3024 = (55/54)(55/56), and so forth. We therefore might expect a > miracle generator somewhere in the interval between 50/49 and 49/48, > which is to say somewhere between 35 and 35.7 cents.What do you mean by miracle generator here? Miracle generator for us has meant specifically 116.7 cents . . .> > Checking ets with 2401/2400 in the kernel and with good properties, > we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which > gives us a pretty good notion of where the next miracle is hiding.You need to define your conception of "miracle" for us. On> the high side we have 8/270 = 35.555 cents, and on the low side 5/171 > = 10/342 = 35.08 cents. > > The meantone miracle is really better viewed as a flat fifth miracle; > we have 9/8 = (3/2)^2 1/2, so that by extending the circle of fifths > and reducing by octaves we obtain the true miracle of the meantone. > It seems to me it would make sense to do likewise with these other > miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle > fourths in a scale whose interval of repetition is the major sixth > would seem to be the way to take full advantage of the miracle > generator.Would the octave no longer be an interval of repetition in your view? Can you clarify at all? What do you meant by "take full advantage" that may be different from our previous focus on number of consonant chords?
Message: 1599 - Contents - Hide Contents Date: Mon, 17 Sep 2001 09:05:39 Subject: Miracle theory From: genewardsmith@xxxx.xxx The "miracle" generator splits the difference between 15/14 and 16/15, and hence is associated to the scales with the jumping jack 225/224 in the kernel. If we seek other generators with miracle properties, and obvious place to began is with other jumping jacks. If we look instead at 81/80, we get the meantone miracle, which is well-known. If we look higher, we find 2401/2400 = (49/48)(49/50), 3025/3024 = (55/54)(55/56), and so forth. We therefore might expect a miracle generator somewhere in the interval between 50/49 and 49/48, which is to say somewhere between 35 and 35.7 cents. Checking ets with 2401/2400 in the kernel and with good properties, we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which gives us a pretty good notion of where the next miracle is hiding. On the high side we have 8/270 = 35.555 cents, and on the low side 5/171 = 10/342 = 35.08 cents. The meantone miracle is really better viewed as a flat fifth miracle; we have 9/8 = (3/2)^2 1/2, so that by extending the circle of fifths and reducing by octaves we obtain the true miracle of the meantone. It seems to me it would make sense to do likewise with these other miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle fourths in a scale whose interval of repetition is the major sixth would seem to be the way to take full advantage of the miracle generator. In the same way, for the 35-cent miracle, we have 49/48 = (7/4)^2 1/3, so a circle of 7/4's with an interval of repetition of 3 would make for an interesting collection of scales.
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