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Message: 3700 - Contents - Hide Contents Date: Fri, 01 Feb 2002 23:53:39 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:> Gene:>>> As long as 2 is represented, it seems to me any temperament is an >>> octave temperament. The basis I gave was for a fifth and a tritone >>> below a fifth, and I could if I wanted make the fifth a pure fifth, >>> but I could do that, and temper octaves, in the octave basis also. >>> >>> There are three considerations: interval of equivalence of a scale >>> using a given temperament, a basis of generators for the >>> temperament, and the tuning of the temperament. This are independent. > > Paul:>> So why did you say "this was not a temperament"? And isn't it true >> that, if you took it out to, say, 10 notes per approximate octave, >> and tuned the octaves pure, it would _not_ be an octave-repeating >> scale? This seems to be the point Graham is missing. >> The thing he said wasn't a temperament has no notes to an octave,No notes? He said it was generated by a fifth and a fifth-tritone -- so it seems like it could have plenty of notes, up to an infinite number, in fact.
Message: 3701 - Contents - Hide Contents Date: Fri, 1 Feb 2002 00:20:44 Subject: Re: interval of equivalence, unison-vector, period From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 12:07 AM > Subject: [tuning-math] Re: interval of equivalence, unison-vector, period > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> This, I think, corresponds to how Graham thinks of things, >>> and how I _used_ to think of things, before I understood >>> torsion in the period-is-1/2-or-1/9-or-1/N-octave sense. >> >>>> Paul, you're really good at explaining things. >> Please elaborate on this until I understand it. :) >> Oops -- I didn't mean that at all. I meant, before I understood > torsion as it's defined in your dictionary. Thanks for pointing out > my brain fart!Well, OK, you're welcome. But I really thought you were saying you understood something here about which I'm totally clueless.>> >> I don't recall anyone ever responding to the lattice diagram >> I made for the torsion definition: >> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >> >> I thought that showing the pairs of pitches that are separated >> by two unison-vector candidates that are smaller than the >> actual unison-vectors defining the torsional-block might have >> been saying something significant about what a torsional-block >> is, or maybe at least something about this particular example. >> >> Any thoughts? >> Well, you're definitely doing something right in this case, since > 81:80 and 128:125 are definitely intervals that should represent > equivalences hereYES! Good, I'm slowly getting it.> . . . but it won't necessarily be that case that smaller > intervals in the parallelogram than the defining unison vectors > fall into the "equivalent" category for every torsional block.Well, OK, I'll take your word for it. Examples of this would be good.>>> The fog has still not cleared about the three items in the subject >> line. >> Really? OK, first of all, period is specific to MOS scales and the > linear temperaments they come from.Ah! OK, that helps a bit.> Examples: > > meantone temperament > unison vector: 81:80 > interval of equivalence: octave > period: octave > > MIRACLE temperament > unison vectors: 224:225, 385:384, 441:440 > interval of equivalence: octave > period: octave > > diminished/octatonic in 12-tET or 28-tET > unison vector: 648:625 > interval of equivalence: octave > period: 1/4 octave > > 'paultone' > unison vectors: 50:49, 64:63 > interval of equivalence: octave > period: 1/2 octave > > Bohlen-Pierce > unison vectors: 245:243, 3087:3125 > interval of equivalence: tritave (3:1) > period: tritave (3:1)Examples are good, and thanks much for them all. You know what I realize now, upon really seriously studying the tuning-math archives since August? Latticing has almost completely disappeared. That's a big part of the reason why I'm having such a hard time following. Bring back the lattice diagrams! P L E A S E !!!!! (with truckloads of sugar) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3702 - Contents - Hide Contents Date: Fri, 01 Feb 2002 10:05:49 Subject: Re: new cylindrical meantone lattice From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Maybe Gene can help.I haven't been thinking about the spirals, because I don't know what they are for. I have a suggestion for JustMusic lattices, which is to take lattices of higher dimensions, project them onto a plane and give a way of rotating the lattice to look at different projections--some sort of "knobs" you can "turn", giving you orthogonal matricies.
Message: 3703 - Contents - Hide Contents Date: Fri, 01 Feb 2002 23:56:53 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., graham@m... wrote: >> In-Reply-To: <a3d5gb+ldpj@e...> >> genewardsmith wrote: >> All elements are>>> torsion elements, and we have a finite group, so this is rather >>> different than a block with torsion elements. >>>> But it still involves torsion? >> Certainly, but not in the sense of a torsion block with torsion,since it isn't a block.>>>> This is *not* a temperament, or at least not one >>> I'm interested in hearing, so 2 is not acting as a unison, which is >>> hardly a surprise. >>>> Of course it's a temperament. It's twintone/paultone/pajara. >> Is pajara the new official name? I'd like to get this settled.OK, no more "paultone" or "twintone".>As for this val, which defines only one of two required generator >mappings being a temperament, that's only if you layer on some >interpretation and perform the extra calculations to find a good >choice for the second generator; taken by itself, it isn't one. It's >telling us to send the octave to a unison, and 5 and 7 both to 1/9; >it's only after you stick in half-octaves and send 7 to some tuning >of 64/9 and 5 to a half-octave below that that pajara emerges. Read >literally as a temperament, it sends 2 to 1 and 5 and 7 to 1/9, and >I don't think that qualifies. >>> The octave is acting as a unison, but it's more complicated than that. As >> it has torsion, it's actually half an octave that's acting as a commatic >> unison vector. >> I would say it's acting as a generator, but if you make 2 a unisonit becomes a torsion element, since its square is an octave. This, along with my message to Monzo this morning, seems to show the very real problems with considering 2 a unison!
Message: 3704 - Contents - Hide Contents Date: Fri, 01 Feb 2002 08:23:23 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Well, you're definitely doing something right in this case, since >> 81:80 and 128:125 are definitely intervals that should represent >> equivalences here > >> YES! Good, I'm slowly getting it. > >>> . . . but it won't necessarily be that case that smaller >> intervals in the parallelogram than the defining unison vectors >> fall into the "equivalent" category for every torsional block. > >> Well, OK, I'll take your word for it. Examples of this would > be good.For example, in the Blackjack block (see for example the JI blackjack block I just made for you), one of the three defining unison vectors of the 3-dimensional parallelepiped is 36:35, while 64:63 appears a lot _within_ the block -- yet there is no torsion.
Message: 3705 - Contents - Hide Contents Date: Fri, 01 Feb 2002 10:06:56 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Well . . . I revised the question in the next message, which I hope > you get to see.Meaning, the next message after the one where the question appeared, which is back 11 messages: Yahoo groups: /tuning-math/message/3109 * [with cont.]
Message: 3706 - Contents - Hide Contents Date: Fri, 1 Feb 2002 00:30:37 Subject: Re: interval of equivalence, unison-vector, period From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 12:23 AM > Subject: [tuning-math] Re: interval of equivalence, unison-vector, period > >>>> . . . but it won't necessarily be that case that smaller >>> intervals in the parallelogram than the defining unison vectors >>> fall into the "equivalent" category for every torsional block. >> >>>> Well, OK, I'll take your word for it. Examples of this would >> be good. >> For example, in the Blackjack block (see for example the JI blackjack > block I just made for you), one of the three defining unison vectors > of the 3-dimensional parallelepiped is 36:35, while 64:63 appears a > lot _within_ the block -- yet there is no torsion.Hmmm ... so by "smaller", you mean in interval size. But what about in taxicab-metric size? It seems to me that a unison-vector within a torsional block must *always* be smaller in taxicab-size than the unison-vectors which define the torsional-block. Yes? Might I be onto something here that's obvious, but that might have some special meaning that hasn't been noticed? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3707 - Contents - Hide Contents Date: Fri, 1 Feb 2002 02:09:31 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 1:42 AM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> you seem to concur that perhaps any meantone would do >> what 2/9-comma did even for the block >> in question. >> I mean even for the parallelogram in question.Well, it makes sense. But as I said, I think I'd like to make some different lattices and compare them. Still, even if you feel that this "choice of meantone" thing is a moot point, I still think it's fascinating to lattice the meantones this way and *see* their deviation from JI. Another thing I like is that since 12-EDO is ~= 1/11-comma meantone, it too can be plotted and examined in this way. That's something I'd like to study for a bit. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3708 - Contents - Hide Contents Date: Fri, 01 Feb 2002 08:35:32 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Friday, February 01, 2002 12:23 AM >> Subject: [tuning-math] Re: interval of equivalence, unison- vector, period >> >>>>>> . . . but it won't necessarily be that case that smaller >>>> intervals in the parallelogram than the defining unison vectors >>>> fall into the "equivalent" category for every torsional block. >>> >>>>>> Well, OK, I'll take your word for it. Examples of this would >>> be good. >>>> For example, in the Blackjack block (see for example the JI blackjack >> block I just made for you), one of the three defining unison vectors >> of the 3-dimensional parallelepiped is 36:35, while 64:63 appears a >> lot _within_ the block -- yet there is no torsion. > >> Hmmm ... so by "smaller", you mean in interval size. > > But what about in taxicab-metric size? > It seems to me that a > unison-vector within a torsional block must *always* be smaller > in taxicab-size than the unison-vectors which define the > torsional-block. Yes?Usually, but not necessarily -- for example, pitch two pitches near the pair of corners of the parallelogram that are furthest from one another -- this will tend to be longer than at least the shortest unison vector. However, with the hexagonal periodicity blocks we have a better attempt to do just what you describe -- especially if the taxicab distance is evaluated with respect to a triangular lattice like the ones I tend to use.
Message: 3709 - Contents - Hide Contents Date: Fri, 01 Feb 2002 10:09:58 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Maybe Gene can help. >> I haven't been thinking about the spirals, because I don't know >what they are for. I have a suggestion for JustMusic lattices, which >is to take lattices of higher dimensions, project them onto a plane >and give a way of rotating the lattice to look at different >projections--some sort of "knobs" you can "turn",Note the 5 "knobs" you can "turn" in Dave Keenan's 4-dimensional Tumbling Dekany: ÐÏà ±á * [with cont.] (Wayb.) So you mean like this?
Message: 3710 - Contents - Hide Contents Date: Fri, 1 Feb 2002 00:43:30 Subject: Re: Fwd: Re: interval of equivalence, unison-vector, periodd From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 1:02 PM > Re: interval of equivalence, unison-vector, period > >>> But what *is* the period? I mean, not what interval >> or size, but what is it? What significance does it have? >> It's the smallest interval at which a scale can be > transposed without changing the scale at all. For example, > the diminished (octatonic) scale in 12-tET has a period > of 1/4-octave. For another, my symmetrical decatonic scale > in 22-tET has a period of 1/2-octave.Ah ... OK, I can grasp that. But then what makes the "interval of equivalence" different from that?> The way Gene does things, unison vectors are all > _small intervals_ defined with specific ratios, for > example 81:80 but not 81:40, and then Gene can construct > temperaments or whatever, and then octave-equivalence > can be stuck back in at the end, if desired. If you don't > do it this way, you won't be able to deal with torsion > properly.Can you explain why not? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3711 - Contents - Hide Contents Date: Fri, 1 Feb 2002 02:11:27 Subject: Re: new cylindrical meantone lattice From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 2:05 AM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Maybe Gene can help. >> I haven't been thinking about the spirals, because I don't > know what they are for. I have a suggestion for JustMusic > lattices, which is to take lattices of higher dimensions, > project them onto a plane and give a way of rotating the > lattice to look at different projections--some sort of > "knobs" you can "turn", giving you orthogonal matricies.Thanks, Gene. This has been part of the desiderata ever since I figured out how to lattice >3 dimensions, c. 1998. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3712 - Contents - Hide Contents Date: Fri, 1 Feb 2002 00:53:07 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 1:05 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > >>>>> because of the disappearance >>>> of the syntonic comma unison-vector, this flat lattice >>>> should be imagined to wrap around as a cylinder, so that >>>> the right and left edges connect. Thus the centered >>>> meantone may imply either of any pair of pitches which >>>> would be separated by a comma on the flat lattice, and >>>> each of those pairs of points on the flat lattice map to >>>> the same point on a cylinder. >> This is true also for an infinite number of pitches on the > flat lattice that are even further from the line. So the > centering of the meantone line between 1/1 and 3/2 is > irrelevant. So is the use of 2/9-comma meantone as opposed > to any other meantone, as far as I can tell.I don't know about that, Paul. My intuition tells me that if composers choose particular flavors of meantone based on criteria such as the amount-of-error-from-JI relationships among the basic consonant intervals (M3/m6, m3/M6, p4/p5), then they *do* intend to emphasize/deemphasize specific JI intervals, because the meantone they choose will do that. The centering of the meantone line -- which I prefer to call a spiral since it belongs on a cylindrical lattice -- within the PB, distributes the amount of error from JI as evenly as possible among the intervals closest to the 1/1. That seems to me to be something with an actual musical application. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3713 - Contents - Hide Contents Date: Fri, 1 Feb 2002 02:31:15 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 1:21 AM > Subject: [tuning-math] Re: new cylindrical meantone lattice > >>>>> Now about your other two objections: >>>> >>>>>>>>> (a) the density of points along the line, which doesn't >>>>> appear to be meaningful; >>>> >>>>>>>> I'm hoping that the post I just sent before this one, >>>> about composer choosing particular flavors of meantone, >>>> addresses this one. >>>>>> Not at all -- I was referring to the fact that, for example, >>> in 5/18-comma meantone, the points on the spiral are rather >>> far apart from one another -- that doesn't seem particularly >>> meaningful. >> >>>> OK, the only way I can respond to this properly is to go ahead >> and create a 5/18-comma lattice and examine it. That's not going >> to happen until tomorrow. >> It's already on your meantone webpage applet!!Ah, OK ... I really need to clean this page up. You're confusing an older idea I had, which I should probably clarify or maybe even delete, with what I'm talking about now. That applet was an idea that I had, against which you argued quite strongly, and I pretty much agree that there's not a whole lot of meaning in it. The significant thing is *does* show is the *angle* of the meantone, which corresponds to the angle of spiral on the cylinder. The *insignificant* part of it is the distribution of points along those lines. Those are merely the chains of JI pitches that the meantones give exactly, which have no significance at all on a flat lattice like this ... other than that one plotted step in each direction (positive and negative) along the meantone axis shows the single pair of complementary JI intervals which are given exactly and really *are* audibly significant in the meantone. The distribution of points that really matters is that of the chain of meantone generators, which can be seen on the cylindrical lattice at the bottom of the "meantone" webpage, and here: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) And those don't vary a lot ... only a tiny bit of variation in lattice taxicab step-size from one meantone to the next, within a fairly narrow overall range. Since the meantone cylinder is exactly perpendicular to the syntonic-comma metric, the circles which ring the cylinder and represent those commas for each pair of JI pitches are equidistant. The meantone generators are all plotted at some point along each of these circles, the exact point being determined by the intersection of the meantone spiral with the syntonic-comma circle. Thus, the only difference in the plotting of the chains of generators (along the spirals) between various meantones is the slight change in spacing of those points due to the varying angle of the meantone spiral. So in fact, on my newer meantone cylinder lattices, the density of points along the spiral really doesn't change all that much. Does that clear up your first objection, and thus all three? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3714 - Contents - Hide Contents Date: Fri, 1 Feb 2002 01:01:18 Subject: Re: new cylindrical meantone lattice From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, January 31, 2002 1:12 PM Subject: [tuning-math] Re: new cylindrical meantone lattice> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Paul, *you* were the one who kept nagging me about "... but how >> are you going to lattice LucyTuning or Golden Meantone on this >> kind of lattice?". >> >> So if your point is that LucyTuning and 3/10-comma meantone >> are "functionally identical", then I can simply lattice >> LucyTuning *as* 3/10-comma meantone and call it a day. >> Not quite -- it would have to be *very slightly different*.OK, I think we're getting somewhere with this.>> Now, about that bit where I wrote: "... finding some way to >> represent pi in a universe where everything is factored by >> 3 and 5", I had an idea for a simpler beginning approach. >> >> Let's start with meantone-like EDOs instead. As an example, >> we want to lattice 1/3-comma meantone alongside 19-EDO. >> >> Can you devise some formula that would find the fractional >> powers of 3 and 5 that would be needed to plot 19-EDO on a >> trajectory that would follow closely alongside the 1/3-comma >> trajectory? Now, I think *that* would be a meaningful lattice! >> OK, I'm in favor of thinking in this direction, as it addresses at > least one of the three objections I raised in a post to you a few > minutes ago.And that objection would be this one, correct? :> (c) the fact that you can't yet plot > non-rational-fraction-of-a-comma meantones this way, > though the _angle_ part should be just as meaningful > for those.So, how about a formula that plots 19-EDO as, literally, a close cousin to 1/3-comma meantone spiral? How does take something that's roots of 2, and change it into "8ve"-equivalent fractional powers of 3 and 5? Now about your other two objections:> (a) the density of points along the line, which doesn't > appear to be meaningful;I'm hoping that the post I just sent before this one, about composer choosing particular flavors of meantone, addresses this one.> (b) the fact that you have to pin the spiral to a particular > "1/1" origin, which ruins the rotational symmetry of the > cylindrical meantone latticeI've already said elsewhere that the spiral doesn't have to be pinned to anything. It can float anywhere the user wants it. What's important is the angle of the spiral, as you've noted. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3715 - Contents - Hide Contents Date: Sat, 02 Feb 2002 03:34:04 Subject: Re: 7-limit MT reduced bases for ets From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 140: [2401/2400, 5120/5103, 15625/15552]I accidentally left off 171: [2401/2400, 4375/4374, 32805/32768] Wouldn't want to do that--look at those three high-powered commas!
Message: 3716 - Contents - Hide Contents Date: Sat, 02 Feb 2002 11:04:00 Subject: Re: 43-edo (was: 171-EDO...) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> you have to take the weighted mean of three 9/8 and two >> 10/9 whole tones. that means three 9/8s "plus" two 10/9s >> "divided" by five, that is, ( (9/8)^3 * (10/9)^2 )^(1/5). > >> so that's what manuel means by "geometric mean"? i would > have never understood it that way > > ok, i see it now ... but it took a little bit of work to > comprehend what's going on there ... perhaps you'd like to > reword that a bit, manuel?manuel worded it just fine. it's the geometric mean of three 9/8 whole tones and two 10/9 whole tones: ( (9/8) * (9/8) * (9/8) * (10/9) * (10/9) )^(1/5).
Message: 3717 - Contents - Hide Contents Date: Sat, 02 Feb 2002 18:56:19 Subject: simple math question From: jpehrson2 Could it possibly be said that a logarithm is a way to find the "exponent" of a number?? I mean, in the most simple case... ?? J. Pehrson
Message: 3719 - Contents - Hide Contents Date: Sat, 02 Feb 2002 04:07:37 Subject: Re: 7-limit MT reduced bases for ets From: clumma> 9: [21/20, 27/25, 128/125] > 10: [25/24, 28/27, 49/48] > 12: [36/35, 50/49, 64/63] > 15: [28/27, 49/48, 126/125] > 19: [49/48, 81/80, 126/125] > 22: [50/49, 64/63, 245/243] > 27: [64/63, 126/125, 245/243] > 31: [81/80, 126/125, 1029/1024] > 41: [225/224, 245/243, 1029/1024] > 68: [245/243, 2048/2025, 2401/2400] > 72: [225/224, 1029/1024, 4375/4374] > 99: [2401/2400, 3136/3125, 4375/4374] > 130: [2401/2400, 3136/3125, 19683/19600] > 140: [2401/2400, 5120/5103, 15625/15552]This is seriously cool.>For any prime limit, we could consider the most characteristic >linear temperament of a particular et to be the one leaving off >the last member of the MT reduced basis.Does it have to be prime (not odd) limit? -Carl
Message: 3720 - Contents - Hide Contents Date: Sat, 2 Feb 2002 14:45 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx monz wrote:> Oh, OK ... I think I get it. > > If 2 = a unison, then 2^2 = an octave. Yes?No, if 2 were a unison (which it isn't) 2^2 would also be a unison.> But I'm still confused, because if 2 is a unison, then > essentially for purposes of tuning math 2=1. So how does > squaring that get you to the octave?It doesn't. Except that if an octave is a unison, squaring gives you a unison which is also an octave because an octave is a unison. Graham
Message: 3722 - Contents - Hide Contents Date: Sat, 02 Feb 2002 04:41:14 Subject: Re: 7-limit MT reduced bases for ets From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Does it have to be prime (not odd) limit?Fraid so. It occurs to me another fun game to play with these is to find the corresponding Fokker blocks.
Message: 3723 - Contents - Hide Contents Date: Sat, 2 Feb 2002 11:44:01 Subject: Re: simple math question From: monz hi joe,> From: jpehrson2 <jpehrson@xxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 02, 2002 10:56 AM > Subject: [tuning-math] simple math question > > > Could it possibly be said that a logarithm is a way to find > the "exponent" of a number?? > > I mean, in the most simple case... > > ??the logarithm is t h e way to find the exponent by definition, that's exactly its purpose example: the log_10 (the underscore is the ASCII way of writing a subscript) -- which is read "log base 10" -- of 2, is ~0.30103 this simply means that 10^.30103 = ~2 of course, it's of utmost importance to know what the base of the logarithm is, and people don't always indicate that explicitly particularly in tuning math, logs are often taken to base 2 since 2 is the ratio of the "octave", and the author often assumes that the reader will know that and assume 2 as the base another thing to keep in mind is that logarithms can be taken out to an arbitrary number of decimal places because the math doesn't work out exactly as low-integer fractions ... hence my use of the tilde(~) above to indicate approximations however, i did notice something interesting early this morning just before going to bed ... there are some very good low-integer-ratio approximations to the log_10 of the lowest primes Examples: log_10 ~fractional value of: of logarithm more accurate less accurate 2 3/10 3 10/21 = ~1/2 5 7/10 7 11/13 = ~5/6 11 25/24 13 39/35 = ~10/9 17 16/13 = ~11/9 19 23/18 = ~14/11 = ~5/4 this is useful because these simple fractions provide a very easy way to work with approximate logarithms -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3724 - Contents - Hide Contents Date: Sat, 2 Feb 2002 14:45 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx Me:>> Well, can you think of a word for something that acts like a unison >> vector but isn't? To cover the meanings of "unison vector", >> "generator", "period" and "equivalence interval"? Gene:> What about kernel element? Of course, a period is a kernel element only > if you make it one, by having a corresponding mapping, but that is the > case here. The same would be true of an equivalence interval--if we > send the half-octave to 1, it is a kernel element, but if we send 2 to > 1 but not sqrt(2), then sqrt(2) is an element of order 2. One way we > get a cyclic group of order 11, the other way of order 22.Yes, that'll do. Although can the period and interval of equivalence both be kernel elements? From group theory, I think "identity" will do for things like unisons. Adding or subtracting unisons gives you what you started, which is like identities. So I'll go with "identity vector" for things like commatic unison vectors. I'm not sure we need a word for things like chromatic unison vectors. Will "generator" do instead? Graham
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