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Message: 6875 - Contents - Hide Contents Date: Fri, 13 Jun 2003 21:54:52 Subject: Re: Interval Database Experiences From: monz ----- Original Message ----- From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, June 13, 2003 12:24 PM Subject: [tuning-math] Re: Interval Database Experiences> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" > <manuel.op.de.coul@e...> wrote:>>> but I still wonder if that >>> fokker list is complete for this matter, any clues? >>>> I'm not aware of missing important intervals, and open >> to any corrections. >> >> Manuel >> unfortunately, some of the "big names" in tuning history, such as > rameau, helmholtz, etc., have given different, mutually inconsistent > names to different intervals. in the following table, i attempted to > tabulate the most common names for important small 5-limit intervals, > and give the details of the temperaments that result when these > intervals vanish. some of the names of the latter were jovial > creations of this (tuning-math) list. this is meant to replace the > similar table on monz's "equal temperament" dictionary page: > > Yahoo groups: /tuning/database? * [with cont.] > method=reportRows&tbl=10&sortBy=3 hi paul,i thought i replaced the table on my webpage with your database data. didn't i? ...? -monz

Message: 6876 - Contents - Hide Contents Date: Fri, 13 Jun 2003 23:08:50 Subject: Re: Web page From: Carl Lumma Congrats, monz! The row vector's been named after you! -Carl

Message: 6877 - Contents - Hide Contents Date: Fri, 13 Jun 2003 23:11:12 Subject: Re: Baked Alaska From: Carl Lumma>Responding to a question from Carl, I present Baked Alaska, Mmmm... baked...Wait. Maybe what we should be doing here is working on the brat between the fifth and the octave. The Alaska temperaments all have only one size of octave and two sizes of fifths. How hard is this? -Carl

Message: 6878 - Contents - Hide Contents Date: Sat, 14 Jun 2003 23:29:34 Subject: Fried Alaska From: Gene Ward Smith If we define the octave-fifth beat ratio as (2f - 3)/(o - 2) where f is the fifth and o is the octave, and if we use a repeating alaska pattern of scale steps with eight steps of size "a", satisfying 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0 and four steps of size "b" satisfying 256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8- 18432*b^10+4096*b^12 = 0 then we get an alaska-style scale with eight octave-fifth brats equal to 2 and four equal to 1; Fried Alaska: ! alafried.scl Fried alaska, with octave-fifth brats of 1 and 2 12 ! 98.867788 197.735576 299.074366 397.942153 496.809941 598.148731 697.016520 795.884307 897.223099 996.090887 1094.958674 1196.297466

Message: 6879 - Contents - Hide Contents Date: Sat, 14 Jun 2003 18:33:12 Subject: Re: Fried Alaska From: Carl Lumma>If we define the octave-fifth beat ratio as > >(2f - 3)/(o - 2) > >where f is the fifth and o is the octave,f and o are in cents? Could you give the derrivation of this formula?>if we use a repeating alaska pattern of scale steps with eight steps >of size "a", satisfying > > 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0 > >and four steps of size "b" satisfying > >256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8- >18432*b^10+4096*b^12 = 0Honestly, I don't know how you do it, Gene.>then we get an alaska-style scale with eight octave-fifth brats equal >to 2 and four equal to 1; Fried Alaska: > >! alafried.scl >Fried alaska, with octave-fifth brats of 1 and 2 >12 >! >98.867788 >197.735576 >299.074366 >397.942153 >496.809941 >598.148731 >697.016520 >795.884307 >897.223099 >996.090887 >1094.958674 >1196.297466This is fantastic. It's very close to Alaska VI, but with better bad thirds, better fifths, and even slightly better octaves. The good thirds suffer, but as 10ths they'll still be doing quite well. Fantastic! Now, to test to see if the brats make a difference. I'll report back... -Carl

Message: 6880 - Contents - Hide Contents Date: Sat, 14 Jun 2003 05:52:09 Subject: Web page From: Gene Ward Smith I've started webbing again, so there's some new stuff at mictronal.co.uk. Here's the url: . . . . . . . . . . . . . . Music and Mathematics * [with cont.] (Wayb.)

Message: 6881 - Contents - Hide Contents Date: Sat, 14 Jun 2003 00:21:38 Subject: graph theory glossary From: Carl Lumma Any errors are mine. _________________________________________________________________ For a graph G or a vertex V, define: chromatic number Least number of colors required to color all the adjacent vertices of G differently. circuit Path that begins and ends at the same vertex. clique A complete subgraph of G. Sometimes the largest such subgraph. complete Every pair of vertices in G is connected by exactly one edge. connected On G there exists a path containing all vertices. crossing number The minimum number of crossings with which G can be drawn. degree (valence) Number of branches at V. diameter The longest geodesic in G. dominating set A subgraph S of G such that remaining vertices in G are adjacent to vertices in S. domination number The size of the smallest dominating set in the given graph. eccentricity The longest geodesic involving the given vertex. genus The minimum number of handles that must be added to a plane to embed the given graph without any crossings. geodesic The shortest path between a pair of vertices. loop Circuit of length 1 (an edge that connects a vertex to itself). planar A graph whose edges intersect only at its vertices. radius The shortest geodesic in a graph. tree A simple, undirected, connected, acyclic graph.

Message: 6882 - Contents - Hide Contents Date: Sat, 14 Jun 2003 01:17:15 Subject: Re: graph theory glossary From: Carl Lumma Following Gene's age-old suggestion, I will now attempt to tell if various graph terminology have any music-theoretical importance. Here vertices are pitches, and edges are consonant dyads. Note: This is based on a slightly updated version of the glossary, compared to the last post.>chromatic number >The least number of colors required to color all the adjacent >vertices of G differently.As this goes up, I assume edge connectivity goes up. And edge connectivity is generally Good.>circuit >A path that begins and ends at the same vertex.On the lattice, circuits would constitute scales that are "covered" by dyads. Temperament allows a scale to be covered by a single type of dyad.>circumference >The length of the longest circuit in G.As this goes down, I assume edge connectivity goes up.>clique >A complete subgraph of G. Sometimes the largest such subgraph.Maximal cliques of the n-limit lattice are saturated n-limit chords. In JI they are either otonalities, utonalities, or ASSs.>clique number >The number of vertices in the largest clique of G.Tells you the number of dimensions you need to lattice a scale.>complete >G for which every pair of vertices is connected by exactly one >edge.If G is complete all its dyads are consonant, but it may not be a saturated n-limit chord. Tonal music tends to depend on scales with several disjoint cliques. Though by the 9-limit you pick up an extra 3:2, and the 6:7:9 chord can invoke a new tonality.>connectivity, edge >The minimum number of edges that must be deleted from G to render >it disconnected. Considered beneficial. >connectivity, vertex >The minimum number of vertices that must be deleted from G to >render it disconnected.Also good, but since we're generally more interested in intervals than pitches, edge connectivity seems more natural.>crossing number >The minimum number of crossings with which G can be drawn.The higher the better.>degree (valence) >The number of branches at V.We don't care so much about V.>diameter >The longest geodesic in G.The lower the better. See Paul Hahn's site.>dominating set >A subgraph S of G such that remaining vertices in G are adjacent >to vertices in S.The existence of many dominating sets of similar cardinality seems like a Good Thing for tonal music. Kinda like what I said earlier about cliques, except that here we know we can modulate easily between subgraphs.>domination number >The size of the smallest dominating set in G.Not sure it matters.>eccentricity >The longest geodesic involving V.We don't care so much about V.>genus >The minimum number of handles that must be added to a plane to >embed G in it without any crossings.Higher the better.>loop >A circuit of length 1 (an edge that connects a vertex to itself).Only for octaves!>planar >G whose edges intersect only at vertices.Only for triadic music in JI, we hope.>radius >The shortest geodesic in G.Less important than diameter. -Carl

Message: 6883 - Contents - Hide Contents Date: Sat, 14 Jun 2003 08:52:49 Subject: Re: graph theory glossary From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> crossing number >> The minimum number of crossings with which G can be drawn. >> The higher the better.This connects to the genus, by the way, which is how many holes you need to poke in your donut in order to not have any crossings if you draw the graph on the donut.

Message: 6884 - Contents - Hide Contents Date: Sun, 15 Jun 2003 10:06:57 Subject: Re: Fried Alaska From: Carl Lumma>Now, to test to see if the brats make a difference. I'll report >back...Well, it's hard to test just the brats, since this is slightly different than any non-synched alaska tuning. But after a quick test with a Reed organ patch, I don't prefer this to Alaska VI. I can definitely hear a difference between vallotti and synced vallotti, which are quite close tuning-wise. But I think the best thing would be to render a midi file in Audio Compositor that compares synched and non-synched triads in various inversions. I may do this at some point. -Carl

Message: 6885 - Contents - Hide Contents Date: Sun, 15 Jun 2003 14:50:05 Subject: Re: Interval Database Experiences From: monz> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, June 15, 2003 1:38 AM > Subject: [tuning-math] Re: Interval Database Experiences > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > >> hi paul, >> >>>> i thought i replaced the table on my webpage with >> your database data. didn't i? ...? >> >> >> >> >> -monz >> you never did, and more recently i'm waiting for the harmonic entropy > definition page to get updated as well (as per our e-mail) . . .sorry about falling behind like this. i've been really busy with other stuff. i'll try to chat with you on Yahoo messenger at some point and we'll get these done. -monz

Message: 6886 - Contents - Hide Contents Date: Sun, 15 Jun 2003 08:38:10 Subject: Re: Interval Database Experiences From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > i thought i replaced the table on my webpage with > your database data. didn't i? ...? > > > > > -monzyou never did, and more recently i'm waiting for the harmonic entropy definition page to get updated as well (as per our e-mail) . . .

Message: 6887 - Contents - Hide Contents Date: Sun, 15 Jun 2003 22:18:45 Subject: Re: Fried Alaska From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> If we define the octave-fifth beat ratio as >> >> (2f - 3)/(o - 2) >> >> where f is the fifth and o is the octave, >> f and o are in cents? Could you give the derrivation > of this formula?They are frequency ratios; if we have a "trine" of 1, f, o, then the beats between f and 1 are 2f-3 and between o and 1 o-2, the above is therefore a beat ratio. There are five others, since we also have beats from f to o; if r is the above ratio the others are given by r, 2r - 3, 3/r - 2 and their reciprocals.>> if we use a repeating alaska pattern of scale steps with eight steps >> of size "a", satisfying >> >> 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0 >> >> and four steps of size "b" satisfying >> >> 256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8- >> 18432*b^10+4096*b^12 = 0 >> Honestly, I don't know how you do it, Gene.Since you have Maple, you could do it too, using the resultant function.

Message: 6888 - Contents - Hide Contents Date: Sun, 15 Jun 2003 22:20:54 Subject: Re: Fried Alaska From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> I can definitely hear a difference between vallotti and synced > vallotti, which are quite close tuning-wise.If you can tell the difference between werckmeister and sync-werckmeister, which are nearly identical, it would definately be a plus for the theory that syncing the beats matters.

Message: 6889 - Contents - Hide Contents Date: Sun, 15 Jun 2003 22:22:37 Subject: Re: Interval Database Experiences From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> you never did, and more recently i'm waiting for the harmonic entropy >> definition page to get updated as well (as per our e-mail) . . .> sorry about falling behind like this. i've been > really busy with other stuff. i'll try to chat with > you on Yahoo messenger at some point and we'll get > these done.When you do that, could you also look at my question about the correct defintion of "microtemperament"?

Message: 6890 - Contents - Hide Contents Date: Sun, 15 Jun 2003 17:02:28 Subject: Re: Interval Database Experiences From: monz hi Gene,> From: "Gene Ward Smith" <gwsmith@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, June 15, 2003 3:22 PM > Subject: [tuning-math] Re: Interval Database Experiences > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>> you never did, and more recently i'm waiting for the harmonic entropy >>> definition page to get updated as well (as per our e-mail) . . . >>> sorry about falling behind like this. i've been >> really busy with other stuff. i'll try to chat with >> you on Yahoo messenger at some point and we'll get >> these done. >> When you do that, could you also look at my question about the correct > defintion of "microtemperament"?you were never very specific about what's wrong with it. the most foolproof way to get me to actually update the page would be to write a commentary that i could simply copy-and-paste and add as an addendum at the bottom of the page. -monz

Message: 6891 - Contents - Hide Contents Date: Sun, 15 Jun 2003 20:40:45 Subject: "microtemperament" definition (was: Interval Database Experiences) From: monz hi Gene,> From: "Gene Ward Smith" <gwsmith@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, June 15, 2003 7:11 PM > Subject: [tuning-math] Re: Interval Database Experiences > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> you were never very specific about what's wrong >> with it. >> What I mean by a "microtemperament" is something like this: one where > all of the relevant consonant intervals are within a cent of being > pure. What you've defined is a regular temperament of dimension > greater than or equal to one, so I think it should not be used as a > definition of microtemperament. > > I'd first define "regular temperament" as the expression of musical > intervals by means of a set of generators g1, ..., gn such that each > interval is represented as a number of the form g1^e1 ... gn^en, then > the dimension of the regular temperament as the number of generators > minus one. An equal temperament, with one generator, would be regular > of dimension zero; p-limit just intonation would be regular with > dimension pi(p), which is the number of primes <= p. A linear > temperament, or temperament of dimension one, has two generators.OK, thanks. umm ... i'm still real busy with "regular life" (meaning that i have to do other work to make money), so i'll still need more help with this. so, should i put your second paragraph in my "temperament" definition, and then the first paragraph in "microtemperament"? please advise. -monz

Message: 6892 - Contents - Hide Contents Date: Mon, 16 Jun 2003 10:45:57 Subject: Re: Fried Alaska From: Carl Lumma>>> >f we define the octave-fifth beat ratio as >>> >>> (2f - 3)/(o - 2) >>> >>> where f is the fifth and o is the octave, >>>> f and o are in cents? Could you give the derrivation >> of this formula? >>They are frequency ratios; if we have a "trine" of 1, f, o, then >the beats between f and 1 are 2f-3 and between o and 1 o-2, the above >is therefore a beat ratio.Ah! That's sooo simple. Thanks!>>> if we use a repeating alaska pattern of scale steps with eight steps >>> of size "a", satisfying >>> >>> 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0 >>> >>> and four steps of size "b" satisfying >>> >>> 256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8- >>> 18432*b^10+4096*b^12 = 0 >>>> Honestly, I don't know how you do it, Gene. >>Since you have Maple, you could do it too, using the resultant function.Yeah, but how'd you get those polynomials? -Carl

Message: 6893 - Contents - Hide Contents Date: Mon, 16 Jun 2003 10:54:43 Subject: Re: Fried Alaska From: Carl Lumma>> > can definitely hear a difference between vallotti and synced >> vallotti, which are quite close tuning-wise. >>If you can tell the difference between werckmeister and >sync-werckmeister, which are nearly identical, it would definately be >a plus for the theory that syncing the beats matters.These temperaments are besides the point. We need a series of triads tuned as near as they can be, with a selection of brats. If you provide them in cents in root position, I'll make an audio sequence and post it. -Carl

Message: 6894 - Contents - Hide Contents Date: Mon, 16 Jun 2003 10:56:19 Subject: Re: Fried Alaska From: Carl Lumma>These temperaments are besides the point. We need a series of >triads tuned as near as they can be, with a selection of brats. >If you provide them in cents in root position, I'll make an >audio sequence and post it.Oh, and don't publish the brats. Just number the triads. -Carl

Message: 6895 - Contents - Hide Contents Date: Mon, 16 Jun 2003 21:01:54 Subject: Re: Fried Alaska From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> These temperaments are besides the point. We need a series of > triads tuned as near as they can be, with a selection of brats. > If you provide them in cents in root position, I'll make an > audio sequence and post it.I['m not sure what you mean--different tunings can have the same brat, and for any given brat, you can get arbitrarily close to JI.

Message: 6896 - Contents - Hide Contents Date: Mon, 16 Jun 2003 14:26:53 Subject: Re: Fried Alaska From: Carl Lumma>> >hese temperaments are besides the point. We need a series of >> triads tuned as near as they can be, with a selection of brats. >> If you provide them in cents in root position, I'll make an >> audio sequence and post it. >>I'm not sure what you mean--different tunings can have the same >brat,Of course. And they should be included in the mix.>and for any given brat, you can get arbitrarily close to JI.Oh yeah? Then brats are nonsense, I say. I want two chords, with brats of 2 and 13/8, each 1 cent RMS from 3:4:5 to the nearest .1 cent RMS. -Carl

Message: 6897 - Contents - Hide Contents Date: Mon, 16 Jun 2003 21:34:17 Subject: Re: Fried Alaska From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> and for any given brat, you can get arbitrarily close to JI. >

Message: 6898 - Contents - Hide Contents Date: Mon, 16 Jun 2003 02:11:08 Subject: Re: Interval Database Experiences From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> you were never very specific about what's wrong > with it.What I mean by a "microtemperament" is something like this: one where all of the relevant consonant intervals are within a cent of being pure. What you've defined is a regular temperament of dimension greater than or equal to one, so I think it should not be used as a definition of microtemperament. I'd first define "regular temperament" as the expression of musical intervals by means of a set of generators g1, ..., gn such that each interval is represented as a number of the form g1^e1 ... gn^en, then the dimension of the regular temperament as the number of generators minus one. An equal temperament, with one generator, would be regular of dimension zero; p-limit just intonation would be regular with dimension pi(p), which is the number of primes <= p. A linear temperament, or temperament of dimension one, has two generators.

Message: 6899 - Contents - Hide Contents Date: Mon, 16 Jun 2003 14:38:51 Subject: Re: Fried Alaska From: Carl Lumma>>> >nd for any given brat, you can get arbitrarily close to JI. >>>> Oh yeah? Then brats are nonsense, I say. > >wha???At the least it means we'd need to consider the error as well as the brat -- brats alone would not be reliable. -Carl

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