Tuning-Math Archive Section 1: 625

Message: 625 - Contents - Hide Contents Date: Thu, 2 Aug 2001 09:47 +01 Subject: Re: Another BP linear temperament? From: graham@m... In-Reply-To: <9kafes+8qo4@e...> Paul wrote:

> Graham, or anyone else who can > "hear" 3:1 equivalence . . . does it > help to eliminate even partials > from the timbres?)

It's all in the timbre to me. Tritaves with odd-number partials have a stronger tritave equivalence than octaves with sawtooth timbres. The same is true of phi with a phi timbre. I'm guessing it would also be true of octaves with only even partials, but haven't tried yet. Graham

Message: 626 - Contents - Hide Contents Date: Thu, 2 Aug 2001 09:47 +01 Subject: Re: Another BP linear temperament? From: graham@m... In-Reply-To: <9kabno+v09n@e...> Dave Keenan wrote:

>> Here's the printout: >> >> 7/30, 279.0 cent generator >

> Oops! Your generator should be 442.1 cents. You've multiplied by cents > per octave when it should have been cents per tritave.

Yes, I mentioned this in a later post.

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" > in scare quotes, and rightly so. I'd call what you've given "7-limit > without 2s".

Oh, it does include the 2s, but not enough of them. It uses a 2-5-7 basis, but the consonance limit is the octave-equivalent 7-limit matrix. So it won't include 7:4 or 5:4.

> By analogy: > For an octave repeating scale, "7 limit" means ratios of all (whole > number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave > equivalents). > > So for a tritave repeating scale I think "7 limit" should mean ratios > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Sounds good, but it's a case of typing in the matrices. This was a quick experiment. Graham

Message: 628 - Contents - Hide Contents Date: Thu, 02 Aug 2001 19:39:09 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Paul wrote:

>> Your least squares optimization uses what mapping from generators to >> primes? And it considers the errors for intervals 3:5, 3:7, 3:11, >> 3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right? >

> Not exactly, it was an approximation to all pitches of the "11- limit" > scale given.

Now why would anyone want that? It's the consonant intervals, not the pitches, that we care about approximating well. (At least that's my philosophy.)

> A bit more work, but can do that too. Then the result > is a virtually equal scale

You mean virtually the 39th root of 3? That's more what I would have expected.

> with a LS generator of 780.2702 cents. > Differences are > 3:5 6.647 / 5.906 > 3:7 3.772 / 4.513 > 3:11 5.993 / 6.734 > 3:13 2.880 / 2.139 > 5:7 -2.876 / -2.135 > 5:11 -0.654 / 0.087 > 5:13 -3.768 / -4.509 > 7:11 2.222 / 1.481 > 7:13 -0.892 / -1.633 > 11:13 -3.114 / -3.855 > The first one is the optimised value, the second one is the one > occurring fewer times in the scale.

Can you explain what the latter means?

Message: 629 - Contents - Hide Contents Date: Thu, 2 Aug 2001 13:47:39 Subject: Ives stretched-8ve scales (was: Another BP linear temperament?) From: monz

> From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, August 01, 2001 7:55 PM > Subject: [tuning-math] Re: Another BP linear temperament? > > > Since the "orthodox" BP consonances are ratios of {3,5,7}, > and the triple-BP consonances are ratios of {3,5,7,11,13}, and > their tritave-equivalents, and these have been referred to as > "7-limit" and "13-limit", respectively, I think this could > lead to confusion . . . let's abandon the term "limit" when > leaving the octave-equivalent world and adopt more explicit > mathematical terminology. Hi Paul,

This caught my attention and reminded me of the webpage I made examining Charles Ives's stretched-8ve scales. A stretched-'octave' scale of Charles Ives, * [with cont.] (Wayb.) On that webpage, I propose 5 different tunings for the three types described by Ives, with equivalence-intervals of 15.00, 16.00, 15.50, 14.50, and ~14.94 Semitones. Can you guys (Paul, Graham, Dave) explain some more detailed mathematical analyses of these tunings? 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: 631 - Contents - Hide Contents Date: Fri, 03 Aug 2001 02:54:40 Subject: Re: Another BP linear temperament? From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., "Dave > Keenan" <D.KEENAN@U...> wrote:

>> So for a tritave repeating scale I think "7 limit" should mean ratios >> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}. >

> Since the "orthodox" BP > consonances are ratios of {3,5,7},

So isn't 7:9 included, or do you really mean ratios of {1,5,7} and their tritave equivalents?

> and the triple-BP consonances > are ratios of {3,5,7,11,13}, and > their tritave-equivalents,

You mean {1,5,7,11,13} and their tritave-equivalents?

> these have been referred to as > "7-limit" and "13-limit", > respectively, I think this could > lead to confusion . . . let's > abandon the term "limit" when > leaving the octave-equivalent > world and adopt more explicit > mathematical terminology. Yes please!

Message: 632 - Contents - Hide Contents Date: Fri, 03 Aug 2001 03:02:59 Subject: Re: Another BP linear temperament? From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9kabno+v09n@e...> > Dave Keenan wrote:

>> So for a tritave repeating scale I think "7 limit" should mean ratios >> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}. >

> Sounds good, but it's a case of typing in the matrices. This was a quick > experiment.

Duh! I included 6 above (which is of course tritave equivalent to 2 and therefore redundant). This stuff is tricky. It should have been: So for a tritave repeating scale I think "7 limit" should mean ratios of all non-multiples of 3 up to 7, i.e. {1,2,4,5,7} (and of course their tritave equivalents). But I agree with Paul, just spell it out, like after the "i.e." above. Regards, -- Dave Keenan

Message: 633 - Contents - Hide Contents Date: Fri, 03 Aug 2001 03:10:47 Subject: Re: Ives stretched-8ve scales (was: Another BP linear temperament?) From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> On that webpage, I propose 5 different tunings for the three > types described by Ives, with equivalence-intervals of 15.00, > 16.00, 15.50, 14.50, and ~14.94 Semitones. > > Can you guys (Paul, Graham, Dave) explain some more detailed > mathematical analyses of these tunings?

Sorry Monz, I've got other things I should be doing at the moment. BTW, do you suppose you are the only person on the planet that routinely specifies intervals in semitones with two decimal places, rather than cents? :-) -- Dave Keenan

Message: 634 - Contents - Hide Contents Date: Thu, 2 Aug 2001 22:47:07 Subject: Re: Ives stretched-8ve scales (was: Another BP linear temperament?) From: monz

> From: Dave Keenan <D.KEENAN@U...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, August 02, 2001 8:10 PM > Subject: [tuning-math] Re: Ives stretched-8ve scales (was: Another BP linear temperament?) > > > BTW, do you suppose you are the only person on the planet that > routinely specifies intervals in semitones with two decimal places, > rather than cents? :-)

Yep. It sure is taking a long time for everyone else to catch on! ;-P I mean, really... if +/- 5 cents is pretty much an accepted margin of error in practice, then do we really need anything more accurate than 240-EDO?, let alone 1200-EDO (which is exactly what my "Semitones" specify). And if we *do* need more accuracy, why not use another widely accepted standard with finer resolution, such as "cawapus", or even "midipus", rather than having to deal with more decimal points after the cents? I really like Semitones (with two decimal places) because isolating the semitone component makes it very easy for me to see at a glance how a pitch or interval relates to what I'm already familiar with as a long-time practicing 12-EDO musician. 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: 635 - Contents - Hide Contents Date: Fri, 03 Aug 2001 05:59:17 Subject: Re: Magic lattices From: Dave Keenan --- In tuning-math@y..., graham@m... 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:

[ lattice deleted ]

> > The template is > > 5 > 1-----3-----9 > \ / > \ / > 7 > > Like Dave's new septimal-kleismic lattice, but unlike a normal 7-limit > lattice, pitch increases left-right for a 4:5:6:7:9 chord.

Mine doesn't strictly-increase from left to right, but yours does.

> That'd make it > good as a mapping for a hexagonal keyboard.

Yes indeed. In fact it is _THE_ mapping for such a keyboard. Try putting the period, generator and number of notes into my keyboard mapper spreadsheet (I think you have to wind the aspect-ratio up to max to get it to look hexagonal). http://uq.net.au/~zzdkeena/Music/KeyboardMapper.xls - Type Ok * [with cont.] (Wayb.)

> I also found this: > > > G#--B+--D#--F#+-A# > / \ / \ > G+/ B \ D+/ F#\ A+ > / \ / \ > G---Bt--D---F+--A > \ / \ / > Gt\ Bb/ Dt\ Ft/ At > \ / \ / > Gb--Bbt-Db--Fbt-Ab > > With the template > > 5 > > 1-------3-------9---11 > \ / > \ / > \ / > 7 > > ... > And a unified neutral-second/neutral-third lattice > > > B--D+-F#-A+-C# > |\ |\ | > | \ | \ | > A C+ E G+ B > | \ | \ | > | \| \| > G--Bt-D--F+-A > |\ |\ | > | \ | \ | > F At C Et G > | \ | \ | > | \| \| > Eb-Gt-Bb-Dt-F > > > It's like Dave Keenan's new Miracle lattice, but with extra rows. > Template > > 7 > | > | > | > | > | > 5 | > | | > | | > | 11 | > | | > | | > 1-----3-----9--11 > > > I don't know if there's a simpler position for the 7 ...

Nor do I. All these tempered lattices coming out of the woodwork! After trying the full colour version of the 2D tempered lattice for Blackjack, I'm thinking that 3D may be easier to deal with in some ways. It's also useful to note that the keyboard mapping can always be treated as a 2D tempered lattice. I'm thinking now that one way to find the best 3D lattice for an 11-limit scale may be to put several periods of it into a spreadsheet as a 5D triangular lattice and look at a 3D projection of it in the same way I did with the dekany. Then rotate it until it is as thin as possible in the into-the-screen dimension, followed by minor adjustments so no note is obscured. But I don't have time. -- Dave Keenan

Message: 637 - Contents - Hide Contents Date: Fri, 03 Aug 2001 22:16:53 Subject: Re: Another BP linear temperament? From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

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

>> --- In tuning-math@y..., "Dave >> Keenan" <D.KEENAN@U...> wrote:

>>> So for a tritave repeating scale I think "7 limit" should mean > ratios

>>> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}. >>

>> Since the "orthodox" BP >> consonances are ratios of {3,5,7}, >

> So isn't 7:9 included, or do you really mean ratios of {1,5,7} and > their tritave equivalents? The latter. >

>> and the triple-BP consonances >> are ratios of {3,5,7,11,13}, and >> their tritave-equivalents, >

> You mean {1,5,7,11,13} and their tritave-equivalents?

Same thing -- right?

Message: 638 - Contents - Hide Contents Date: Fri, 03 Aug 2001 22:55:19 Subject: Re: Another BP linear temperament? From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>>> and the triple-BP consonances >>> are ratios of {3,5,7,11,13}, and >>> their tritave-equivalents, >>

>> You mean {1,5,7,11,13} and their tritave-equivalents? >

> Same thing -- right?

Yes, but not in lowest terms, so could be confusing. You wouldn't write "ratios of {2,3,5,7,9} and their octave equivalents", would you? And there will be times when folks will not want to always write "and their tritave equivalents" because they feel it is given by the context. In this case "3" or "9" in place of "1" would definitely suggest tritave-specific. Just being my usual pedantic self. Sorry. -- Dave Keenan

Message: 639 - Contents - Hide Contents Date: Fri, 03 Aug 2001 23:10:09 Subject: Re: Another BP linear temperament? From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> You wouldn't write "ratios of {2,3,5,7,9} and their octave > equivalents", would you?

Why not? I might even use 4 instead of 2.

Message: 640 - Contents - Hide Contents Date: Sat, 04 Aug 2001 01:47:50 Subject: Re: Another BP linear temperament? From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >

>> You wouldn't write "ratios of {2,3,5,7,9} and their octave >> equivalents", would you? >

> Why not? I might even use 4 instead of 2.

I thought I answered "Why not?" in the same message you quote from. Why would you?

Message: 642 - Contents - Hide Contents Date: Sat, 04 Aug 2001 08:05:31 Subject: Re: Another BP linear temperament From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave, > > The way I look at it, what I did was simply remap the unison vectors > so that the 2D comma is 118098/117649

Ok <goes and figures> that's (2^1 * 3^10 * 7^-6) = 6.7 cents

> and the primary chroma is 343/324.

That's (2^-2 * 3^-4 * 7^3) = 99 cents. Is this for the 9-note per tritave MOS?

> This means that the dimensions are 9/7 and 7/6 with a period > of 3/1 as opposed to the usual 3/2 and 5/4 with a period of 2/1.

Ok. But for this to be a "generalised meantone" in any sense other than the one which is much better described by the term "linear temperament", then a chain of _four_ 9/7's would have to be tritave-equivalent to a 7/6. This is not the case. It needs a chain of five according to your comma.

> The 1/6 comma meantone generalization is exactly analogous to standard > 1/5 comma meantone in terms of fractionalizing a comma and altering > the size of the generator in a 1D chain.

I suspect you mean standard 1/4 comma meantone. But this is just a linear temperament. How is it more specifically a "generalised meantone"? Is it merely because it seeks to approximate only two primes (other than the interval-of-equivalence), in this case 2 and 7? That certainly isn't enough for _me_ to consider it a "generalised meantone". What do others think? So this BP temperament still has an approximate 7:9 generator but it's typically a narrow one, where the previous BP temperament has a wide one. The mapping from primes to numbers of generators is: Prime No. generators ----- -------------- 2 -6 3 0 5 (don't care, but 2 is best) 7 -1 The MA optimum {1,2,7} generator is 434.0 cents giving a maximum error of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6 of the above comma. If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents. The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents (in between the previous two) but with errors of 14.1 cents. It has MOS cardinalities of 4 (5) 9 13 (22) 35 (48), (improper in parenthesis). -- Dave Keenan

Message: 644 - Contents - Hide Contents Date: Sat, 04 Aug 2001 18:50:13 Subject: HyperMOS From: carl@l... Finally, I'll take the time to say something on this fascinating thread... () Does "HyperMOS" refer to the generalized higher-D MOS's of Paul's hypothesis, or are they another type? () Paul, you said that 'as far as we can tell, CS and PB are the same thing'. But aren't there many irrational CS scales that don't have any sensible PB interp? () Paul's (hi, Paul!) hypothesis post may be the most feverishly dense and impressive post I've seen since his original harmonic entropy post in '97. I don't think a notion of HyperMOS is nearly as important as harmonic entropy, but who knows.... I'll quote the pertinent parts here, and maybe say something more later when (a) I understand more of it and (b) I don't have an apartment full of boxes to unpack. For now, please accept my once-over comments:

>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.

Paul, did you have to use the letter "n" twice? You seem to want to use case to distinguish them... I'll hold you to that.

>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). Check. >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. Check. >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). Okay. >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.

I don't see why M/N yet, but I can see what's happening.

>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.

Spot on, for the 1-D case I can vidi!!! -Carl

Message: 645 - Contents - Hide Contents Date: Sat, 04 Aug 2001 22:18:34 Subject: Re: Another BP linear temperament From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> <<But this is just a linear temperament. How is it more specifically a > "generalised meantone"?>> > > Again, it's a generalization of meantone because the "tone" and the > "mean" are generalized! The process of determining a comma and > fractionalizing and distributing it are also all exactly analogous if > one strips away the particulars.

Ok. Can you tell me what intervals are analogous to the major and minor tone in this temperament. I'm concerned that you're generalising "tone" to mean "an interval of any size", which I think would be a bad move.

> It's the process, not the specific tuning -- besides, if the octave > divided into 12 and 13 equidistant parts can rightfully be called > "equal temperament" in either instance, then I see no reason why what > I'm doing here can't be called "meantone"! > > That said, I never was happy with "generalized meantone" as a blanket > term for this sort of thing, and I did post several times a while back > trolling for better, less likely to cause confusion, types of terms. > Unfortunately this never really went anywhere. So, as I haven't > thought of anything better or less confusing myself, I've just stuck > with my initial intuitive feeling as that seems obvious and simple > enough for communication purposes...

So it's just a particular way of arriving at a linear temperament? Why not call the process "comma distribution"?

> Does any of this make any better sense now? If not, well, then I'm > convinced that Graham was actually right, and I am quite incapable of > communicating these things.

I don't think he said (or meant) that exactly. Yes, it makes better sense now and we are quite capable of communicating (it takes two). Regards, -- Dave Keenan

Message: 649 - Contents - Hide Contents Date: Mon, 06 Aug 2001 19:36:16 Subject: Re: Another BP linear temperament? From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

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

>> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>

>>> You wouldn't write "ratios of {2,3,5,7,9} and their octave >>> equivalents", would you? >>

>> Why not? I might even use 4 instead of 2. >

> I thought I answered "Why not?" in the same message you quote from. > > Why would you?

So that the resulting ratios would tend to be the familar voicings within one octave: 3:4, 4:5, 4:7 . . .