Tuning-Math Archive Section 2: 1152

Message: 1152 - Contents - Hide Contents Date: Fri, 27 Jul 2001 23:59:42 Subject: some notes about my Dictionary (was: FAQ again (hear, hear!)) From: monz

> From: J Gill <JGill99@xxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, July 27, 2001 2:17 PM > Subject: [tuning-math] Fwd: Re: FAQ again (hear, hear!) > > ... > Monz does a commendable and excellent job of attempting to compile > the ongoing process of the definition and explanation of the many and > varied terms and phrases in the "vernacular" of tuning, as it > evolves. However, the more the merrier, and these things (like many > subjects) are enhanced by a variety of (hopefully not too divergent) > viewpoints from which the newcomer is able to consider these matters > from various veiwpoints in formulating a personal working > understanding of these esoteric, yet truly fascinating, subjects!

Thanks for all the compliments, Jay. Please note that my Dictionary is a cooperative effort, and many other list subscribers have contributed. Chief among them is John Chalmers, who handed over to me the entire glossary of his book _Divisions of the Tetrachord_ for use in the online Dictionary. I have yet to finish entering all of those terms. And now, of course, there's a whole new group of terms that needs to be defined: Unidala, Harmonidala, etc. :) Daniel Wolf commented a while back that he liked the way I present each individual's perspective as such, as opposed to the usual way a collaborative project goes, where a "collective" opinion is presented as fact. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1153 - Contents - Hide Contents Date: Sat, 28 Jul 2001 17:29:38 Subject: Re: Paul's Message #17 From: Paul Erlich --- In tuning-math@y..., "J Gill" <JGill99@i...> wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >

>> Hmm . . . have you read the _Gentle Introduction to Periodicity >> Blocks_, including the "excursion" (the third webpage in the > series)? >

> Yes, but my "absorption factor" remains "less than unity". > I will be re-reading it, with particular emphasis on your > periodicity block which includes those very interval ratios > (in the 12-tone JI system of which you ask is Ramos, and > Monz points out he believes it was from De Caus, with Manuel > Op de Coul pointing out it also Ellis' "Harmonic Duodene", > and your "excursion" in Part 3, as you suggested. >

>> It seems to me you may be confused about something. >

>>> Tempering does not decrease the number of unison vectors, > >

> one *might* add, "but does alter the individual values of those > unison vectors, whose number is determined by the prime limit of the > system"??? Exactly. > > I get the feeling that I may be simply be missing a fundamental rule > in the process of deriving unison vectors. I will run you through > what I have done, for your inspection and comment, below: > > "octave-reduced" interval ratios ordered in ascending numerical value > are: 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8. > > "tonal generators" (with redundancies omitted, and in ascending > value) are: 25/24, 135/128, 16/15, 27/25.

I'm not familiar with this concept of "tonal generators". Perhaps a term that would make more sense to me would be "step sizes"?

> > "unison vectors" (as I imagine them...with reduncies RETAINED) are: > 81/80, 2048/2025, and 81/80.

How are you obtaining that?

> > Could it be that I am simply unaware that redundant results > for "unison vectors" (as in the case of the "tonal generators") are > to be OMITTED?

I'm pretty sure, if I'm understanding you correctly, that the answer is "yes". However, for the 12-tone scale you mention, I'm surprised you didn't find 128/125 as one of the unison vectors.

> Thus, I would report 2, rather than 3, resultant > vectors emerging from the above calculation...???

There should be 2 . . . although 81/80, 125/128, and 2048/2025 are three unison vectors for this scale, each of these is just a linear combination of the other two, as can be seen very easily from the vector notation of these three intervals: 81/80 = (4 -1) 128/125 = (0 -3) 2048/2025 = (-4 -2) So clearly, 81/80 = 128/125 "-" 2048/2025, 128/125 = 81/80 "+" 2048/2025, 2048/2025 = 128/125 "-" 81/80, where "+" is vector addition but multiplication of the ratios, and "-" is vector subtraction but division of the ratios.

Message: 1154 - Contents - Hide Contents Date: Sat, 28 Jul 2001 17:35:33 Subject: Re: TGs, Unison Vectors, and Octaves From: Paul Erlich --- In tuning-math@y..., J Gill <JGill99@i...> wrote:

> Dave, > > Thank you for your response to my post. It did help me determine that the > my question, as it was framed, did not accurately reflect what I have been > wondering about here. > > While multiplying or dividing all of the interval ratios of a scale by a > common factor does not alter that scale's resultant "unison vectors", > movement (or "modulation") of a (physical) "pattern" of scale steps (on a > keyboard) certainly appears to do so. > > For instance (as I am certain that you must allready know): > > The 3-note scale made up of 4/5, 1/1, 5/4 where the (lowest) pitch of 4/5 > is the reference note, is equivalent to the scale 1/1, 5/4, > 25/16 (beginning at its reference pitch of 4/5 Hz [cps]), having a single > tonal generator of 5/4, and no unison vector.

Assuming "tonal generator" means "step size", and assuming the interval of equivalence is the octave (2:1), it seems you are missing something. There is an additional step size of 32:25, in addition to two of 5:4 (we like to use "/" for pitches and ":" for intervals).

> > Yet raising the 4/5 Hz pitch by one octave to become 8/5, where the new set > of 3-notes then becomes 1/1, 5/4, and 8/5, *and* referencing the scale to > the 1/1 Hz pitch (amounting to a "modulation") yields a scale which has the > tonal generators 5/4, 32/25, and a single unison vector of 128/125.

The single unison vector of 128/125 was there just the same in the first scale.

> > So, in thinking along the lines of the lowest pitch in a set of notes being > thought of as the "reference" tonic,

By thoughts on tonics are in a much different realm. I see periodicity blocks, etc. as fundamentally "pre-tonal", as the appearance of diatonic and chromatic scales in Pythagorean and meantone tuning preceded the appearance ot tonality in Western music.

Message: 1155 - Contents - Hide Contents Date: Sat, 28 Jul 2001 17:40:09 Subject: Re: 152-tET From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> Paul wrote: >

>>> I did notice 72 came up a lot in the higher limits. Here's what >> you get

>>> by combining the two: >>

>> Combining the two what? How? >

> That's how the program works. It takes two equal temperaments, and finds > the linear temperament consistent with both of them. It comes from all > linear temperaments being describable in terms of large and small scale > steps, as Dan Stearns mentions every now and then.

Hmm . . . please elaborate. I'd tend to think of this approach in terms of each equal temperament being consistent with its own set of unison vectors; those UVs that are consistent with both become the commatic unison vectors of the resulting linear temperament. I'd have trouble immediately seeing why a linear temperament would necessarily be the result, but in 2D, Herman Miller's graph makes it very clear: each ET is a point, and combining two ETs means drawing a line between those two points, and the slope of the line is the sole unison vector of the resulting unison vector . . . it's cool to be able to glimpse into this thicket from so many different directions.

Message: 1157 - Contents - Hide Contents Date: Sat, 28 Jul 2001 18:40:04 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Dave Keenan wrote, > > <<"hyper-MOS" as something that can be "generated" (in this case > presumably by a pair of generators).>> > > Good, that's what I thought.

I have no idea what this means. Reaching in the dark, I'd say my best guess of what hyper-MOS might turn out to mean might be something like something Clampitt brought up on the tuning list: For any generic interval (at least the ones with cardinality relatively prime with the cardinality of the scale), look at the cycle of that interval, and note the pattern of sizes. For some sensible (or perhaps all) mappings from the full range of sizes to two size "classes", the cycle expressed in terms of these "classes" is an MOS pattern. . . . ?

> > I'd say the answer is no then... certainly not 2D, or "hyper-MOS", as > it relates to the possible 1D "miracle" generators anyway (though I > don't have any expedient way to check that, and it's mostly just a > guess that's based on looking at it and having worked with these types > of things a bit).

Still can't see what a 2D generator could mean.

> > If I'm correct, doesn't that sink Paul's hypothesis, or at least > permanently unhinge it from the possibility of a simple single > generator to double generator, MOS to hyper-MOS?

The hypothesis itself doesn't mention hyper-MOS, let along 2D generators. Here's why the hypothesis should work. Take an n-dimensional lattice, and pick n independent unison vectors. Use these to divide the lattice into parallelograms or parallelepipeds or hyperparallelepipeds, Fokker style. Each one contains an identical copy of a single scale (the PB) with N notes. Any vector in the lattice now corresponds to a single generic interval in this scale no matter where the vector is placed (if the PB is CS, which it normally should be). Now suppose all but one of the unison vectors are tempered out. The "wolves" now divide the lattice into parallel strips, or layers, or hyper-layers. The "width" of each of these, along the direction of the chromatic unison vector (the one that remains untempered), is equal to the length of exactly one of this chromatic unison vector. Now let's go back to "any vector in the lattice". This vector, added to itself over and over, will land one back at a pitch in the same equivalence class as the pitch one started with, after N iterations (and more often if the vector represents a generic interval whose cardinality is not relatively prime with N). In general, the vector will have a length that is some fraction M/N of the width of one strip/layer/hyperlayer, measured in the direction of this vector (NOT in the direction of the chromatic unison vector). M must be an integer, since after N iterations, you're guaranteed to be in a point in the same equivalence class as where you started, hence you must be an exact integer M strips/layers/hyperlayers away. As a special example, the generator has length 1/N of the width of one strip/layer/hyperlayer, measured in the direction of the generator. Anyhow, each occurence of the vector will cross either floor(M/N) or ceiling(M/N) boundaries between strips/layers/hyperlayers. Now, each time one crosses one of these boundaries in a given direction, one shifts by a chromatic unison vector. Hence each specific occurence of the generic interval in question will be shifted by either floor(M/N) or ceiling(M/N) chromatic unison vectors. Thus there will be only two specific sizes of the interval in question, and their difference will be exactly 1 of the chromatic unison vector. And since the vectors in the chain are equally spaced and the boundaries are equally spaced, the pattern of these two sizes will be an MOS pattern. QED -- right? I'm quite confused as to why various 11-limit PB interpretations of the blackjack scale, put forth by various people, turned out _not_ to be equivalent to the blackjack MOS, when I calculated the hyperparallelepiped corresponding to these suggestions. This should be looked into. Anyway, a hyper-MOS should have the property that turning all but one of its chromatic unison vectors into commatic unison vectors (i.e., tempering them out) results in an MOS, _no matter which UV is chosen to remain chromatic_. Now the Clampitt-derived intuition I had in the last message is making a lot more sense to me . . . hopefully to you all too . . . see, any "reasonable" classificiation of specific sizes of a given generic interval into two classes should be able to be formulated as simply tempering out certain differences between the specific sizes. And since (as we have seen above, but extending from the 1D MOS case), any difference between the specific sizes of a generic interval must be a combination of one each of some subset of the set of chromatic unison vectors, tempering out any of these unison vectors will reduce the number of specific sizes that occur . . . another thought is that trivalency is a very special case . . . in general a D-dimensional hyper-MOS will have up to 2^D specific sizes for each generic interval, since there are D chromatic unison vectors, each of which defines a set of parallel boundaries in the lattice, and the number of these boundaries crossed by a specific instance of a given vector in the lattice has 2 possible values . . . may certain PB geometries allow one to derive a tighter upper bound than 2^D???

Message: 1159 - Contents - Hide Contents Date: Sat, 28 Jul 2001 01:07:18 Subject: Re: Hey Carl From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >> Hi Dave, >>

>> <<Is it possible that although the scale has 3 step sizes and is >> symmetrical, it is not a hyper-MOS?>> >> >> Right, that's sort of what I just posted, but then again I'm not > sure

>> of exactly what definition of "hyper-MOS" we're going by! >

> Perhaps Dave is trying to proceed by analogy from, say, the situation > where a scale like 2 2 1 2 1 2 2 in 12-tET has 2 step sizes and is > symmetrical, but is not an MOS?

Yes. "hyper-MOS" as something that can be "generated" (in this case presumably by a pair of generators).

Message: 1160 - Contents - Hide Contents Date: Sat, 28 Jul 2001 23:01 +0 Subject: Re: Magic lattices From: graham@xxxxxxxxxx.xx.xx Paul wrote:

>> 380.4 >

> I'm guessing this is the generator?

Yes. It's at the top of <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)>.

Message: 1161 - Contents - Hide Contents Date: Sat, 28 Jul 2001 01:22:10 Subject: Re: TGs, Unison Vectors, and Octaves From: Dave Keenan --- In tuning-math@y..., J Gill <JGill99@i...> wrote:

> When calculating tonal generators from the interval ratios of a given scale > (organized in an order of ascending pitch), followed by determining the > unison vectors (or the geometric steps of the geometric steps, organized in > a non-redundant order of ascending pitch, themselves), is it INVALID to > consider a (beginning, or starting) set of interval ratios which span > multiple octaves (as opposed to "octave-reducing" all such interval ratios > prior to performing the above described analysis)? > > Or, do such calculations remain (in some way) meaningful where "non-octave > reduced" interval ratios are utilized as the original "arguments" or > "independent variables" of the above described algorithm?

I'm not sure exactly what you are asking here, but I think the answer is that its's perfectly valid. I assume we are talking only about scales that do repeat at the octave. In that case, factors of two are irrelevant to most calculations. You can leave them in or take them out. The two common conventions are: 1. All ratios are octave-reduced to between 1/1 and 2/1, 2. All factors of two are eliminated so that ratios only consist of odd numbers. I hope this helps. -- Dave Keenan

Message: 1162 - Contents - Hide Contents Date: Sat, 28 Jul 2001 19:47:30 Subject: Re: TGs, Unison Vectors, and Octaves From: monz

> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, July 28, 2001 10:35 AM > Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves > > > > [My?] thoughts on tonics are in a much different realm. > I see periodicity blocks, etc. as fundamentally "pre-tonal", > as the appearance of diatonic and chromatic scales in Pythagorean > and meantone tuning preceded the appearance [of] tonality > in Western music. Hey Paul,

This is a cool idea, and I agree with it. It seems to me like you're alluding to my idea of "finity", in that the composers make use of unison-vectors and the listeners pick that up, without anyone really being very conscious of it all. Am on I the right track? love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1163 - Contents - Hide Contents Date: Sat, 28 Jul 2001 19:52:33 Subject: Re: Hey Carl From: monz ----- Original Message ----- From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Saturday, July 28, 2001 11:40 AM Subject: [tuning-math] Re: Hey Carl

> Here's why the hypothesis should work. > > Take an n-dimensional lattice, and pick n independent > unison vectors. Use these to divide the lattice into > parallelograms or parallelepipeds or hyperparallelepipeds, > Fokker style. Each one contains an identical copy of > a single scale (the PB) with N notes. Any vector in the > lattice now corresponds to a single generic interval > in this scale no matter where the vector is placed > (if the PB is CS, which it normally should be). Now > suppose all but one of the unison vectors are tempered > out. The "wolves" now divide the lattice into parallel > strips, or layers, or hyper-layers. The "width" of each > of these, along the direction of the chromatic unison > vector (the one that remains untempered), is equal to > the length of exactly one of this chromatic unison vector. > <etc. ... snip>

Paul, could you draw this process on some lattices. I'll put it in a webpage if you put the whole hypothesis together with nice graphics. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1165 - Contents - Hide Contents Date: Sat, 28 Jul 2001 04:51:38 Subject: Re: Paul's Message #17 From: Paul Erlich --- In tuning-math@y..., J Gill <JGill99@i...> wrote:

> Paul, > > In tuning-math message #17, in describing your analysis of Monz's "24-tone > periodicity block you" (Monz) "came up with to derive the 22-shruti system > of Indian music" you stated: > > < But here's the rub. If the schisma is a unison vector, and the diesis is > a unison vector, then the schisma+diesis (multiply the ratios) is a unison > vector. But you can verify that the ratio for the schisma times the ratio > for the diesis is the square of the ratio of the syntonic comma. In other > words, it represents _two_ syntonic commas. Now, if _two_ of

anything is a

> unison vector, then the thing itself must be either a unison or a > half-octave. But in your scale, the syntonic comma separates pairs of > adjacent pitches, so it's clearly not acting as a half-octave. So it must > be a unison. In a sense, it's logically contradictory to say that the > schisma and diesis are both unison vectors while maintaining syntonic comma > differences in the scale. The scale is "degenerate", or perhaps more > accurately, it's a "double exposure" -- it seems to have twice as many > pitch classes than it really has.> > > I have been calculating the tonal generators and unison vectors for the > 12-tone scale made up of the interval ratios 1/1, 16/15, 9/8, 6/5, 5/4, > 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, and 15/8 (which is a subset of Monz's > periodicity block related to the Indian shruti system (1996), and which > consists of the 12 interval ratios which are centered (around the mid-point > between the 1/1 and 3/2 interval ratios) in a 3x4 rectangle of intervals > within Monz's matrix of interval ratios found on page 131 of his > "JustMusic: A New Harmony", 1996. The commatic unison vectors I get are > syntonic (81/80), diaschismic (2048/2025), and a second syntonic (81/80), > representing an (apparently) "ill-conditioned" situation for linear > algebraic analysis (as a 5-limit system containing 3, as opposed to 2, > commatic unison vectors,

Hmm . . . have you read the _Gentle Introduction to Periodicity Blocks_, including the "excursion" (the third webpage in the series)? It seems to me you may be confused about something.

> (1) Could you explain further the meaning and implications of your above > statement, "Now, if _two_ of anything is a unison vector, then the thing > itself must be either a unison or a half-octave.";

Well, if two of something is a unison vector, then two of something is close to 0 cents, or close to 1200 cents, or close to 2400 cents, etc. (assuming the usual situation where the octave is the interval of equivalence). Then the something itself must be close to 0 cents, or close to 600 cents, or close to 1200 cents (which is equivalent to 0 cents), etc. I.e. it must be either a unison of a half-octave. and

> > (2) What can be done in tempering such a scale in order to reduce the > number of commatic unison vectors to 2, instead of 3 consisting of a "pair" > of syntonic commas?

Tempering does not decrease the number of unison vectors, though I still think you're confused about something that's leading you to count three unison vectors. 5-limit space with octave-equivalence is two-dimensional, so there can only be two different independent unison vectors for any periodicity blocks within that space (though additional, non-independent unison vectors, formed by "adding" or "subtracting" the two you start with, can be found, and in fact are relevant in many situations -- both the "excursion" and _The Forms Of Tonality_ give clear (I hope) examples of this.

> I recognize that there may be no one simple answer to > this question, but would appreciate your thoughts in general regarding what > you might endeavor to choose to do in such a case as this. >

Perhaps you could supply some crude diagrams to help clarify what you're seeing? It would help me a lot in trying to answer your questions.

Message: 1169 - Contents - Hide Contents Date: Sat, 28 Jul 2001 14:57 +0 Subject: Re: 152-tET From: graham@xxxxxxxxxx.xx.xx Paul wrote:

>> I did notice 72 came up a lot in the higher limits. Here's what > you get

>> by combining the two: >

> Combining the two what? How?

That's how the program works. It takes two equal temperaments, and finds the linear temperament consistent with both of them. It comes from all linear temperaments being describable in terms of large and small scale steps, as Dan Stearns mentions every now and then. Graham

Message: 1170 - Contents - Hide Contents Date: Sat, 28 Jul 2001 15:03 +0 Subject: Magic lattices From: graham@xxxxxxxxxx.xx.xx I've discovered that "Multiple Approximations Generated Iteratively and Consistently" is an acronym for "MAGIC". What a coincidence! Here are some lattices for the 19 note MOS with meantone naming / / \ \ / / \ /\ /\ /\ / \ B--/------F# Eb-/---\--Bb-/------F---------C \ / \/ \/ / \/ \/ \/ / \ \ / / \ / /\ /\ / /\ /\ /\ / \ D#-/------A#--------E# \ / A---------E \ / \/ \/ / \ / / \ \ / \ \ / / \ / /\ /\ / \ G--/---\--D------ --\--Gb-/------Db--------Ab \ / \/ \/ \/ \ / \/ \/ / \ / / \ / /\ /\ /\ \ Cb /\ /\ / \ B--/------F#--------C#-------G# \ \ -----C \ / \/ \/ / \ / / \ \ / / \ / / / \ / /\ /\ / \ Eb-/---\--Bb-/------F- D#-/------A#--------E# \ / \/ \/ \/ \/ / \ \/ / \ / / \ \ / /\ /\ /\ /\ / /\ / \ G--/---\--D---------A---------E \ / Ab \ / \/ \/ \/ \ / / \ \ / / \ / \ \ / /\ /\ /\ \ Cb-/---\--Gb-/------Db----- --\--F#--------C#-------G# \ \/ \/ \/ / \ / \/ \ / / \ \ / / \ \ /\ /\ /\ / \ B- /\ \ Eb-/---\--Bb-/---\--F---------C \ / \/ \ / / \ \ / / \ /\ /\ /\ / \ B--/---\--F# Eb-/---\--Bb-/------F---------C \ / \/ \/ \/ \/ \/ \/ / \ \ / / \ / /\ /\ /\ /\ /\ /\ / \ D#-/------A#--------E#--------Cb A---------E \ / \/ \/ / \ / / \ \ / / \ \ / / \ / /\ /\ / \ G--/---\--D--/--- \ Gb-/------Db--------Ab \ / \/ \/ \/ \/ \/ \/ / \ / / \ \ / /\ /\ /\ /\ /\ /\ / \ B--/---\--F#--------C#-------G# \ -----C \ / \/ \/ \/ \ / / \ \ / / \ / / / \ / /\ /\ /\ \ Eb-/---\--Bb-/------F- D#-/------A#--------E#--------Cb \ \/ \/ \/ / \ \/ / \ / / \ \ / / \ /\ /\ /\ / /\ / \ G--/---\--D--/------A---------E \ / Ab \ / \/ \/ \/ \/ / \ \ / / \ / \ \ / /\ /\ /\ /\ / \ Gb-/------Db----- --\--F#--------C#-------G# \ / \/ \/ / \ / \/ \ / / \ \ / / \ \ / /\ /\ / \ B- /\ \ Eb-/---\--Bb-/---\--F---------C \ / \/ \ This is the 22 note MOS / / \ \ / / /\ /\ /\ /\ / \ B--/---\--F# Eb-/---\--Bb-/------F---------C \ / \/ \/ \/ \/ \/ \/ / / \ \ / / \ \ / /\ /\ /\ /\ /\ /\ G^-/---\- D#-/---\--A#--------E#--------Cb A---------E \ \/ \/ \/ \/ \ / / \ \ / / \ \ / / \ /\ /\ /\ /\ \ G--/---\--D--/--- --\--Gb-/------Db--------Ab--------Ev \/ \/ \/ \/ / \/ \/ / \ \ / / \ \ / / /\ /\ /\ /\ B# /\ /\ / \ B--/---\--F#-/------C#-------G# \/ \ -----C \ / \/ \/ \/ \/ / / \ \ / / /\ / / / \ \ / /\ /\ /\ /\ Eb-/---\--Bb-/------F- D#-/---\--A#--------E#--------Cb \/ \/ \/ \/ / / \ \/ \/ \ / / \ \ / / /\ /\ /\ /\ G^-/--- /\ /\ \ G--/---\--D--/------A---------E \ \/ Ab--------Ev \/ \/ \/ \/ / / \ \ / / \ /\ \ \ / / /\ /\ /\ /\ B#-/---\--Gb-/------Db----- --\--F#-/------C#-------G# \ \/ \/ \/ / \ \ / \/ \/ / / \ \ / / \ \ /\ /\ /\ / \ B- /\ /\ Eb-/---\--Bb-/---\--F---------C \ / \/ \ Now Dave Keenan's found an alternative simplified Miracle lattice, let's see if he can make anything of this. This is my 24 note keyboard mapping, as used in <http://www.microtonal.co.uk/magicpump.mp3 - Type Ok * [with cont.] (Wayb.)> with decimal names / / \ \ / / /\ /\ /\ /\ 5--/---\--1--/---\--7v 5v-/---\--0^-/------6^--------2^ \/ \/ \/ \/ / \/ \/ \/ \/ / / \ \ / / /\ /\ /\ /\ 9--/\-- /\ /\ /\ 8--/---\--4^-/------0---------6---------2v 0v--------5^ \/ \/ \/ \/ / / \ \ / / \ \ / / \ \ / / /\ /\ /\ /\ 2--/---\--8v-/---\--4--/--- --\--7--/------3^--------9^ \/ \/ \/ \/ \/ \/ / \/ \/ / / \ \ / / \ /\ /\ /\ /\ /\ /\ 1^ /\ /\ 5--/---\--1--/---\--7v--------3---------9v \/ \ -----2^ \/ \/ \/ \/ / \/ \ \ / / \ \ / / /\ / / / /\ /\ /\ /\ 9--/\--\--5v-/---\--0^-/------6^ 4^-/------0---------6---------2v \/ \/ \/ \/ / / \ \/ / / \ \ / / \ \ /\ / /\ /\ /\ /\ 8--/--- /\ 2--/---\--7^-/---\--4--/------0v--------5^ \/ \/ 9^ \/ \/ \/ \/ \/ \/ \ / / \ \ / / /\ /\ \ /\ /\ /\ /\ /\ /\ 1^-/---\--7--/------3^----- --\--7v--------3---------9v \/ \/ \/ \/ / / \ \ / / \/ \ \ / / \ \ / / \ /\ /\ /\ /\ 5--/---\--1- 9--/\--\--5v-/---\--0^-/---\--6^--------2^ \/ \/ \/ \ C C# D Eb E F F# G G# A Bb B C C# D Eb r r r r q p q r r r r r r q r 0 1 2 3 4 5 6 Eb E F F# G G# A Bb B C r r r r r q p q r 6 7 8 9 Cents for the minimax tuning are 0.0 58.8 117.7 176.5 235.3 262.7 294.1 321.6 380.4 439.2 498.0 556.9 615.7 674.5 702.0 760.8 819.6 878.4 937.3 105.4 1082.3 1113.7 1141.2 1200.0 As scale steps: (0 1 2 3 4 4 5 5 6 7 8 9 10 11 11 12 13 14 15 16 17 17 18 18 19) (0 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 21 22) Graham

Message: 1171 - Contents - Hide Contents Date: Sun, 29 Jul 2001 08:22:01 Subject: Re: Magic lattices From: monz

> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, July 29, 2001 3:56 AM > Subject: [tuning-math] Re: Magic lattices > > > I wrote: >

>> Now Dave Keenan's found an alternative simplified Miracle lattice, >> let's see if he can make anything of this. >

> I came up with something overnight: > <etc.> Hi Graham,

*Please* provide a legend for your notation. There are so many different ones being used now that I'm not sure what you mean by "Bt" etc. (Yes, I openly admit my guilt in being one of the advocates of a "non-standard" notation, and thus one of the reason why legends are required...) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1172 - Contents - Hide Contents Date: Sun, 29 Jul 2001 22:27 +0 Subject: Re: Magic lattices From: graham@xxxxxxxxxx.xx.xx monz wrote:

> *Please* provide a legend for your notation. > There are so many different ones being used now > that I'm not sure what you mean by "Bt" etc.

t is a half-flat, + is a half-sharp. I'm now looking at lattices where the 3-direction is reversed, and the primary 5-limit chord is 3:4:5 rather than 4:5:6. This gives a 7-limit template of 7 \ ---5 / \ / 3---1 using the septimal kleisma. Ct--Ft--A#--D#--G# / \ / \ / \ / \ / \ G#--C#--F#--B---E---A \ / \ / \ / \ / \ / A---D---G---C---F / \ / \ / \ / \ / F---Bb--Eb--Ab--Db \ / \ / \ / \ / Gb--B+--E+--A+ The clever thing is that one diagonal is the miracle generator, and the other is the magic generator. The horizontal is obviously the meantone or schismic generator. You could bring the 11-limit in using neutral seconds, but it breaks the melodic pattern. Neutral thirds wouldn't fit. There is another 7-limit mapping: 5 / 7 / 3-----1 C#----F# / \ Eb/ \ / Ct\ / Ft\ A-----D-----G \ B+/ \ E+/ \ / C$\ / Bb----Eb You can set Ct==B+ and E+==Ft by splitting the fourth into equal parts. That means 7:6 and 8:7 become equal. This is the famous interval class with no name. It works in 29= and others. I think 26. Looks like this would make a good ZTar mapping for such temperaments. Graham

Message: 1173 - Contents - Hide Contents Date: Sun, 29 Jul 2001 15:39:25 Subject: Re: Magic lattices From: David C Keenan --- In tuning-math@y..., graham@m... wrote:

> > I've discovered that "Multiple Approximations Generated Iteratively and > Consistently" is an acronym for "MAGIC". What a coincidence!

Tee hee! Yes I _had_ noticed that. By the way, you can delete the second ocurrence of it in your catalog. The 5-limit one. That was my fault. ...

> Now Dave Keenan's found an alternative simplified Miracle lattice, let's > see if he can make anything of this.

The lattice I gave for Miracle works just as well for this temperament (without the 11s of course), because the 224:225 is distributed in this temperament too. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 1174 - Contents - Hide Contents Date: Sun, 29 Jul 2001 19:20:01 Subject: Re: TGs, Unison Vectors, and Octaves From: monz

> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, July 29, 2001 5:46 PM > Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves > >> [monz:]

>> It seems to me like you're alluding to my idea of "finity", >> in that the composers make use of unison-vectors and the >> listeners pick that up, without anyone really being very >> conscious of it all. Am on I the right track? >

> Sure -- I've been implying something to that effect specifically on > the Tuning List, rather than here. And of course, your inquiries into > finity are what led me to study PBs in the first place.

Right... duh! on my part... of course I remember that -- I was *with* you when it all started! But as far as the *unconscious* aspect of it... you haven't really made that explicit, and I suppose that's what I was doing here. That's probably the thing I find most interesting about your avenue of exploration. It seems that there are mathematical ways of modeling our unconscious tonal habits after all. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)