Tuning-Math Archive Section 6: 5176

Message: 5176 - Contents - Hide Contents Date: Fri, 06 Sep 2002 02:58:38 Subject: [tuning] Re: Proposal: a high-order septimal schisma From: genewardsmith --- In tuning-math@y..., paul.hjelmstad@u... wrote:

> Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then

>> Why are there two "vectors?" What is p and what is q??

We need two of something to define a linear 7-limit temperament--two generators, two equal temperaments, two commas. In general two commas define something of codimension two, but in four dimensions this is the same. The above would be the two commas; so, for instance, we could define meantone using 81/80 and 126/125, and miracle using 225/224 and 1029/1024.

> p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4]

>> Got it. It's like some kind of dot product, with every combination of

> pairs of p and q?

So say it's the four-dimensional analog of the cross-product would be more correct. There's a web site which some people found useful for getting the gist of it: Grassmann Algebra Book * [with cont.] (Wayb.)

> Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation of 3, and > so forth; hence [12, 19, 28, 24] would be the usual 12-equal, and [31, 49, > 72, 87] the usual 31-et. >> Got it >

> The wedge now is > > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3]

>> OK, what is r and s (again?)

Two "vals", or duals to intervals; in the above example, the 12 and 31 equal temperaments.

> Whether we've computed in terms of commas or ets, the wedge product of the > linear temperament is exactly the same, up to sign.

>> So signs can be reversed?

Standardizing so the first non-zero entry is positive is what Graham and I have agreed on as a convention for the "wedgie", or wedge invariant of a temperament.

> If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by

>> Is this the same wedgie as above? (Based on r ^ s for example)?

It is the same numerically, and so for our purposes identical; in theory you get into Poincare duality, or you might do things as in the Grassman book I gave a url for. Did you find what you wanted re the Riemann Zeta function? Can I ask what your math background is?

Message: 5177 - Contents - Hide Contents Date: Fri, 06 Sep 2002 20:32:34 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4609]:

> At 02:13 PM 4/09/2002 -0700, George Secor wrote: >> [DK:] >>> For example:

>>> could be notated, relative to 12-ET, instead of relative to >> Pythagorean. >>

>> Okay, now I get your point. That would be a very useful capability for >> a notation, particularly if conventional instruments are used. But >> there's a problem, which I will address below. > > OK. > > >> ... I >> think it's important not to have any complicated symbols for these, so >> that would not be an obstacle that would preclude the notation from >> being considered by European microtonalists. >

> Fair enough. But lets wait until we look at my less ambitious proposal > below,before finalising the n*12-ETs. >

>>> ... I'm looking for a way to notate almost anything >> relative to

>>> 12-ET, but which still agrees as much as possible with the standard >> system.

>>> Do you think such a goal worthwhile? >> >> Of course! >>

>>> Maybe it can be done in a way that agrees with all that you propose >> here

>>> for the n*12-ETs. Care to put your mind to it? My spreadsheet might >> be made

>>> to generate all the notations you suggest, by tweaking ranges and >> precedences. >>

>> Okay, I'll have to give this some thought. (But I'm a bit skeptical >> about anything above 144.) > ...

> OK. I see now that that's way too ambitious. I'm happy to forget being able > to notate rational tunings precisely in this system and reduce the goal to > one of being able to notate any tuning to within about 2.5 cents. So, in > other words, we should have a way of interpreting a certain set of > single-shaft symbols (about 13 of them) as specific offsets from 12- ET > between about 2.5 and 60 cents (an alternative to writing plus or minus > cents next to the notes) while preserving their (preferably lowest product > complexity) comma meanings. > > Do you want to propose a set of symbols to do that?

Yes, and here it is! I have limited this to multiples of 12 through 96 and rational notation to the 13 limit. The commas are assigned the following values: 5 comma: 15 cents 7 comma: 31 cents 11 diesis: 50 cents 13 diesis: 38 cents |( comma: 14 cents The value of the 13 diesis is intended to approximate 8:13, 12:13, and 9:13. It is closest to making 12:13 exact and gives approximately equal (but opposite) error to 8:13 and 9:13. The value of 14 cents for the |( comma is a practical value midway between the sizes required for the two roles it plays in the 13 limit, as indicated in the following table: For 12 through 96-ET For rational notation --------------------- --------------------- |( not used 16 cents as 7/5 comma and 12 cents as 13/11 comma /| = 13 to 20 cents 15 cents (5 comma) (| not used 19 cents (11/7 comma) //| not used 30 cents (5+5 comma) |) = 25 to 33 cents 31 cents (7 comma) (|( not used 35 cents as 11/5 comma and 31 cents as 13/7 comma |\ not used 35 cents as 11-5 comma /|) = 38 to 43 cents 38 cents (13 diesis) /|\ = 50 cents 50 cents (11 diesis) (|\ = 57 to 63 cents 62 cents (13' diesis) /|| not used 65 cents as 11-5 comma complement ~||( not used 65 cents as 11/5 comma complement and 69 cents as 13/7 comma complement ||) = 67 to 65 cents 69 cents (7 comma complement) ~|| not used 70 cents (5+5 comma complement) )|~ not used 81 cents (11/7 comma complement) ||\ = 80 to 88 cents 85 cents (5 comma complement) /||) not used 84 cents as 7/5 comma complement and 88 cents as 13/11 comma complement /||\ = 100 cents 100 cents (apotome) This same information may be easier to digest if it is displayed graphically: 12: /||\ 100 24: /|\ /||\ 50 100 36: |) ||) /||\ 33 67 100 48: |) /|\ ||) /||\ 25 50 75 100 60: /| /|) (|\ ||\ /||\ 20 40 60 80 100 72: /| |) /|\ ||) ||\ /||\ 17 33 50 67 83 100 84: /| |) /|) (|\ ||) ||\ /||\ 14 29 43 57 71 86 100 96: /| |) /|) /|\ (|\ ||) ||\ /||\ 13 25 38 50 63 75 88 100 Ratl: /| |) /|) /|\ (|\ ||) ||\ /||\ 15 31 38 50 62 69 85 100 //| |\ /|| ~|| 30 35 65 70 |( (| (|( ~||( )||~ /||) 7/5 11/7 11/5 11/5 11/7 7/5 16 19 35 65 81 84 13/11 13/7 13/7 13/11 12 31 69 88 No attempt has been made to make the flag sizes add up so that /| + |) = /|) exactly. In order to do that, the 7 comma must become half of the 11 comma, or 25 cents (as long as the 11 comma and 11' comma are equal). Also, the 5 comma becomes 14 cents and the 13 diesis 39 cents. Then |( will be 11 cents as both the 7/5 and 13/11 comma, and (|( will be 36 cents as both the 11/5 (or 11'-5) and 13/7 (or 13'-7) comma. This is okay for the 5 comma and 13 diesis, but the value for the 7 comma is about 6 cents too small to make 7/4 just, and the error for 7/6 and 9/7 is even greater; 7/5 likewise suffers. So there is not much point in doing this. Does this look like it will work for what you had in mind? --George

Message: 5178 - Contents - Hide Contents Date: Fri, 06 Sep 2002 20:36:38 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "monz" <monz@a...> wrote:

>> From: "David C Keenan" <d.keenan@u...> >> To: "George Secor" <gdsecor@y...> >> Cc: <tuning-math@y...> >> Sent: Wednesday, September 04, 2002 4:01 PM >> Subject: [tuning-math] Re: A common notation for JI and ETs > > >> At 02:13 PM 4/09/2002 -0700, George Secor wrote: >> >> ... >>

>>> Julian Carrillo went up to 96, and I haven't heard >>> of anyone else going past that, except for suggestions >>> on the tuning list to use 144 for the 13 limit to >>> remedy a deficiency of 72. >>

>> That's my understanding as well. But you might check >> Joe Monzos Equal Temperament web page. Sorry I don't >> have the URL handy. > > > Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) > >

> also note that Dan Stearns was a prominent advocate of 144 > for a period back around 1999, and i joined with him. both > of us liked Dan's 144 notation not specifically for its > remediation of the 13-limit deficiency of 72, but rather > because we both felt that 144 was a useful representation > of the entire virtual pitch continuum. for example, i used > it as an aid in notating the very complex JI tuning in my > piece _A Noiseless Patient Spider_. > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > see my 144-EDO page: > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > > > > -monz > "all roads lead to n^0"

Monz, I didn't think anybody else besides Dave & me would be reading all of this stuff -- or did you find some magical way to home in on this? Anyway, thanks. Your information is very helpful. --George

Message: 5180 - Contents - Hide Contents Date: Fri, 06 Sep 2002 22:09:23 Subject: Re: ET tuning and rhythm From: wallyesterpaulrus --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> Background is that the process of octave identification of pitches (in n-TET: > calculating mod n)

i don't think many people hear this way. for example, in western diatonic music, intervals and motifs are predominantly classed into SEVEN categories, not twelve, and a fixed interval can sound completely different depending on the context which dictates which of the seven categories it "sounds like" -- for example minor thirds vs. augmented seconds. but ignoring that . . .

> Anyway, there is a possibility to verify or falsify this question: if Mazzola's > supposition holds, there would be an obvious consequence: in music where > pitches are not in 12-TET but in 10-TET or 5-TET, the case should be exactly > the other way round: improvisation and memorization of motifs on 5/8 should > be easier than 3/4, 4/4 or 6/8! > > So I would like to ask the experts for music in other tunings here: did you > ever notice a thing like that - that, depending on tuning, certain time > signatures appeared to be "easier" than others? I would really like to know! > > Hans Straub

absolutely not. tunings very close to 5-equal and 7-equal are very common all over the world, and those cultures show no predilection for quintuple or septuple meters -- let along a predilection for correlation those meters with the respective tunings! i feel that something's deeply wrong with mazzola's supposition or your interpretation of it . . . let me think . . .

Message: 5181 - Contents - Hide Contents Date: Fri, 06 Sep 2002 22:14:31 Subject: Re: ET tuning and rhythm From: wallyesterpaulrus --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> Musical motifs in n-TET tuning and in m time meter can thus be represented > as sets in Zm x Zn. For the properties and classification of sets in this > space, the numbers n and m (their prime factorization in

particular) are of

> high importance.

is mazzola influenced by balzano in this respect? i have a feeling a number of "set-theorists" are falling into a major trap, seduced by certain coincidences involving the number 12, while in reality the number 12 is of peripheral importance, if any, to our music (play any pre-beethoven common-practice western, thus inherently HEPTATONIC, piece of music in 19-equal or 31-equal or golden meantone instead of 12, and no one complains. where's 12? nowhere).

> I > hesitate to assert this since another explanation could be that we just prefer > "simplicity" (not only concerning interval ratios but also time ratios!).

that's more like it! (btw, the two processes are similar, but not exactly analogous.)

Message: 5182 - Contents - Hide Contents Date: Sun, 08 Sep 2002 02:25:42 Subject: Re: A common notation for JI and ETs From: David C Keenan At 01:39 PM 6/09/2002 -0700, George Secor wrote:

>[dk:] >> So, in

>> other words, we should have a way of interpreting a certain set of >> single-shaft symbols (about 13 of them) as specific offsets from >12-ET

>> between about 2.5 and 60 cents (an alternative to writing plus or >minus

>> cents next to the notes) while preserving their (preferably lowest >product

>> complexity) comma meanings. >> >> Do you want to propose a set of symbols to do that? >

>Yes, and here it is! > >I have limited this to multiples of 12 through 96 and rational notation >to the 13 limit. The commas are assigned the following values: > > 5 comma: 15 cents > 7 comma: 31 cents >11 diesis: 50 cents >13 diesis: 38 cents >|( comma: 14 cents > >The value of the 13 diesis is intended to approximate 8:13, 12:13, and >9:13. It is closest to making 12:13 exact and gives approximately >equal (but opposite) error to 8:13 and 9:13. > >The value of 14 cents for the |( comma is a practical value midway >between the sizes required for the two roles it plays in the 13 limit, >as indicated in the following table: > >For 12 through 96-ET For rational notation >--------------------- --------------------- > |( not used 16 cents as 7/5 comma > and 12 cents as 13/11 comma >/| = 13 to 20 cents 15 cents (5 comma) >(| not used 19 cents (11/7 comma) >//| not used 30 cents (5+5 comma) > |) = 25 to 33 cents 31 cents (7 comma) >(|( not used 35 cents as 11/5 comma > and 31 cents as 13/7 comma > |\ not used 35 cents as 11-5 comma >/|) = 38 to 43 cents 38 cents (13 diesis) >/|\ = 50 cents 50 cents (11 diesis) >(|\ = 57 to 63 cents 62 cents (13' diesis) >/|| not used 65 cents as 11-5 comma complement >~||( not used 65 cents as 11/5 comma complement > and 69 cents as 13/7 comma complement > ||) = 67 to 65 cents 69 cents (7 comma complement) >~|| not used 70 cents (5+5 comma complement) >)|~ not used 81 cents (11/7 comma complement) > ||\ = 80 to 88 cents 85 cents (5 comma complement) >/||) not used 84 cents as 7/5 comma complement > and 88 cents as 13/11 comma complement >/||\ = 100 cents 100 cents (apotome) > >This same information may be easier to digest if it is displayed >graphically: > >12: /||\ > 100 >24: /|\ /||\ > 50 100 >36: |) ||) /||\ > 33 67 100 >48: |) /|\ ||) /||\ > 25 50 75 100 >60: /| /|) (|\ ||\ /||\ > 20 40 60 80 100 >72: /| |) /|\ ||) ||\ /||\ > 17 33 50 67 83 100 >84: /| |) /|) (|\ ||) ||\ /||\ > 14 29 43 57 71 86 100 >96: /| |) /|) /|\ (|\ ||) ||\ /||\ > 13 25 38 50 63 75 88 100 > >Ratl: /| |) /|) /|\ (|\ ||) ||\ /||\ > 15 31 38 50 62 69 85 100 > //| |\ /|| ~|| > 30 35 65 70 > |( (| (|( ~||( )||~ /||) > 7/5 11/7 11/5 11/5 11/7 7/5 > 16 19 35 65 81 84 > 13/11 13/7 13/7 13/11 > 12 31 69 88 > >No attempt has been made to make the flag sizes add up so that /| + |) >= /|) exactly. In order to do that, the 7 comma must become half of >the 11 comma, or 25 cents (as long as the 11 comma and 11' comma are >equal). Also, the 5 comma becomes 14 cents and the 13 diesis 39 cents. > Then |( will be 11 cents as both the 7/5 and 13/11 comma, and (|( will >be 36 cents as both the 11/5 (or 11'-5) and 13/7 (or 13'-7) comma. >This is okay for the 5 comma and 13 diesis, but the value for the 7 >comma is about 6 cents too small to make 7/4 just, and the error for >7/6 and 9/7 is even greater; 7/5 likewise suffers. So there is not >much point in doing this. > >Does this look like it will work for what you had in mind?

Yes. That looks very good. As far as it goes. There are obviously some big gaps, e.g. between 0 and 15 cents. We could use: Sym Approximate offset and Comma interpretation ------------------------------------------------ ~|( 3 cents as large 9:17, 3:17, 1:17 commas ~|~ 10 cents as 15:19, 5:19 commas /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas * (| 18 cents as 7:11 comma or |( 18 cents as 5:7 and 7:15 commas big gap, nothing near 22 cents ~|) 26 cents as 17 comma + 7 comma )|) 29 cents as 7:19 comma |) 33 cents as 7:9, 3:7, 1:7 commas * |\ 35 cents as 11 comma - 5 comma or (|( 36 cents as 5:11, 11:15 comma /|) 39 cents as 9:13, 3:13, 1:13 dieses * big gap, nothing near 44 cents. Seems a bad idea to use //| for 47 cents as 5:13, 13:15 commas /|\ 49 cents as 9:11, 3:11, 1:11 dieses * (|~ 54 cents as 11:19 comma (|\ 61 cents as large 9:13, 3:13, 1:13 dieses * For the cent values, I've taken the mean of the commas listed (19 limit) and rounded to the nearest cent. The asterisks are the ones that agree with the first line of your "rational" notation. I don't think we can call it rational. It's really only for approximation, of any tuning as cent offsets from 12-ET. For example, low numbered ETs that are not multiples of 12 could be notated approximately, using these symbols to represent offsets from 12-ET. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5184 - Contents - Hide Contents Date: Mon, 09 Sep 2002 19:18:52 Subject: Re: ET tuning and rhythm From: wallyesterpaulrus --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

>> --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> The formula, of course, applies only for music in 12-TET, hence indeed not > for pre-Beethoven.

viols and lutes and guitars were tuned in 12-equal centuries before beethoven. meanwhile, though 12-equal was fully worked out mathematically in 1585 by Simon Stevin and in 1636 by Marin Mersenne, it would have to wait until 1802-1817 (1850s in england and spain) to become a de facto standard. post-beethoven, essentially.

Message: 5187 - Contents - Hide Contents Date: Tue, 10 Sep 2002 20:38:54 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4619]

> At 01:39 PM 6/09/2002 -0700, George Secor wrote: >> [dk:] >>> So, in

>>> other words, we should have a way of interpreting a certain set of >>> single-shaft symbols (about 13 of them) as specific offsets from 12-ET >>> between about 2.5 and 60 cents (an alternative to writing plus or minus >>> cents next to the notes) while preserving their (preferably lowest product >>> complexity) comma meanings. >>> >>> Do you want to propose a set of symbols to do that? >>

>> Yes, and here it is! ... >> Does this look like it will work for what you had in mind? >

> Yes. That looks very good. As far as it goes. > > There are obviously some big gaps, e.g. between 0 and 15 cents. We could use: > > Sym Approximate offset and Comma interpretation > ------------------------------------------------ > ~|( 3 cents as large 9:17, 3:17, 1:17 commas > ~|~ 10 cents as 15:19, 5:19 commas > /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas * > (| 18 cents as 7:11 comma > or > |( 18 cents as 5:7 and 7:15 commas > big gap, nothing near 22 cents > ~|) 26 cents as 17 comma + 7 comma > )|) 29 cents as 7:19 comma > |) 33 cents as 7:9, 3:7, 1:7 commas * > |\ 35 cents as 11 comma - 5 comma > or > (|( 36 cents as 5:11, 11:15 comma > /|) 39 cents as 9:13, 3:13, 1:13 dieses * > big gap, nothing near 44 cents. Seems a bad idea to use > //| for 47 cents as 5:13, 13:15 commas > /|\ 49 cents as 9:11, 3:11, 1:11 dieses * > (|~ 54 cents as 11:19 comma > (|\ 61 cents as large 9:13, 3:13, 1:13 dieses * > > For the cent values, I've taken the mean of the commas listed (19 limit) > and rounded to the nearest cent.

This looks pretty good, except for the gaps that you noted, and it is in reasonable agreement with the latest 120 and 144 notation proposals. There are two 132-ET proposals that both use (|~ for 5deg (45 cents), which is in conflict with the above, but we could use /|~ for 5deg132 instead. I would make /|\ 50 cents instead of 49 (rounding 49.363 up instead of down); that way it's easier both to remember and to execute, so that, for example, E/|\ would be the same as F\!/. I think we will have to use //| to fill the gap between /|) and /|\. In looking for candidates for this position, I rejected (|( as the 17/11 and 19/11 commas, and then I came across //| as the 19/13 comma, which seems to validate its use as the 13/5 and 15/13 commas. There are two problems with this: 1) It's not intuitive relative to the /| comma. So what? The actual size of the //| comma is around 43 cents. We're just not using it as the 5+5 comma. 2) It's not compatible with the 108-ET notation: 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ 11 22 33 44 56 67 78 89 100 We could eliminate the 108 incompatibility by doing 108 this way: 108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ 11 22 33 44 56 67 78 89 100 This essentially writes off the ratios of 11 in this division. In a previous message I said that I thought that it was not very good to do something like this, but I don't see any other choice for what is admittedly not a very good division (not even 9-limit consistent). However, after writing the above I found another way. The ~|\ symbol could be interpreted as the 19'/11 (or 11'-19' comma, 297:304,~40.330 cents) comma (the difference between 19/11 and 27/16), for which the required alteration in 12-ET is ~46 cents. This would then allow us to have //| available as the 5+5 comma and to keep the 108 notation as in version a, above, if we wish. To fill the gap around 22 cents, I found ~|~ as the 23/17 (or 17+23) comma, which is ~23.3 cents in 12-ET, but we have already used this as the 19/5 comma. I then found that /|~ can approximate the 23/22 comma (as 24057:23552, ~36.729 cents, the difference between the apotome and 23/22); this symbol represents almost exactly 23 cents in 12-ET.

> The asterisks are the ones that agree with the first line of your > "rational" notation. I don't think we can call it rational. It's really > only for approximation, of any tuning as cent offsets from 12-ET.

I agree that "rational" is not the right term. How about "relational" or "12-relational"? We could abbreviate this as "12-R" notation.

> For > example, low numbered ETs that are not multiples of 12 could be notated > approximately, using these symbols to represent offsets from 12-ET.

I can see the value of this for notating approximations of rational intervals relative to 12-ET, but I am a bit skeptical about how well this would work for notating ETs other than multiples of 12. It's as if we're trying to shoehorn everything into a 12-ET framework to make it more convenient (at least initially) for the performer. In the process we end up with ET notations that are considerably more complicated than those that we have already worked out, and I would hate to see musicians become so dependent on relating microtonal intervals to 12 that they would be unable to think of them in any other way. And the worst case scenario would be refusal of performers to change to the simpler, more precise ET and rational notations because they had gotten accustomed to the 12-R notation and would not want to change. Why don't we try to do something like 19 or 22-ET in 12-R notation and see what it looks like before we go any farther with this? I am beginning to think that this is getting more complicated than I would like it to be. Perhaps we should just keep it simple by restricting the 12-R notation to the 13-limit symbols and let that serve as a gentle introduction to the sagittal symbols. Musicians could then learn how the same (now-familiar) symbols are used outside a 12-ET framework. If players of flexible-pitch 12-ET instruments need some sort of aid in remembering how many cents away from 12-ET the notes are, then numbers can be placed above the notes in their parts. (Hopefully this could eventually be done automatically with a computer.) For players of fixed-pitch (i.e., retuned) or specially built instruments, the symbols should suffice. --George

Message: 5188 - Contents - Hide Contents Date: Tue, 10 Sep 2002 05:50:02 Subject: [tuning] Re: Proposal: a high-order septimal schisma From: Gene Ward Smith --- In tuning-math@y..., paul.hjelmstad@u... wrote:

> > I've been studying the Grassman Algebra book-in-progress. So far, I > understand the calculation for r ^ s, and how it creates the wedgie at the > bottom, and the commas that are given at the very bottom make sense, but > alas, I am still not clear on p ^ q (wedge of commas?) Could you be so kind > as to plug in values for u1 to u4 and v1 to v4 so I could seem how the > resultant wedgie for p ^ q resembles the wedgie for r ^ s? I obtain (-1. > -4, -10, -12, -13, 4) for it.

Here are the two formulas for wedge product: Wedge of two intervals:

>> p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, >> u1*v3-v1*u3,u1*v4-v1*u4]

For example p = 126/125 and q=81/80, then p = 2^1 3^2 5^(-3) 7^1, so in vector form it is [1,2,-3,1]. Similarly, q=2^(-4) 3^4 5^(-1) 7^0, which in vector form is [-4,4,-1,0]. Wedging the two gives the wedgie for meantone, but 126/125 ^ 225/224, for example, will work also. Wedge of two vals (e.g., equal temperament mappings)

>> r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, >> u4*v2-u2*v4,u2*v3-v2*u3]

>>> OK, what is r and s (again?)

For instance h19=[19,30,44,53] and h12=[12,19,28,34] will work, the wedge product h19 ^ h12 being the same as the above, and the same as, for example, h50 ^ h31. We can also wedge together the mappings to primes of two generators of the temperament and get the wedgie (up to sign again.) For instance, the matrix [1 0] [1 1] [0 4] [-3 10] represents the mapping to primes of meantone in with generators of octave (first column) and fifth (second column.) Then [1,1,0,-3] ^ [0,1,4,10] = [1,4,10,12,-13,4] which is again the meantone wedgie.

Message: 5190 - Contents - Hide Contents Date: Wed, 11 Sep 2002 11:15:04 Subject: Re: A common notation for JI and ETs From: David C Keenan At 01:47 PM 10/09/2002 -0700, George Secor wrote:

>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4619]

>> There are obviously some big gaps, e.g. between 0 and 15 cents. We >could use: >>

>> Sym Approximate offset and Comma interpretation >> ------------------------------------------------ >> ~|( 3 cents as large 9:17, 3:17, 1:17 commas >> ~|~ 10 cents as 15:19, 5:19 commas >> /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas * >> (| 18 cents as 7:11 comma >> or >> |( 18 cents as 5:7 and 7:15 commas >> big gap, nothing near 22 cents >> ~|) 26 cents as 17 comma + 7 comma >> )|) 29 cents as 7:19 comma >> |) 33 cents as 7:9, 3:7, 1:7 commas * >> |\ 35 cents as 11 comma - 5 comma >> or >> (|( 36 cents as 5:11, 11:15 comma >> /|) 39 cents as 9:13, 3:13, 1:13 dieses * >> big gap, nothing near 44 cents. Seems a bad idea to use >> //| for 47 cents as 5:13, 13:15 commas >> /|\ 49 cents as 9:11, 3:11, 1:11 dieses * >> (|~ 54 cents as 11:19 comma >> (|\ 61 cents as large 9:13, 3:13, 1:13 dieses * >> >> For the cent values, I've taken the mean of the commas listed (19 >limit)

>> and rounded to the nearest cent. >

>This looks pretty good, except for the gaps that you noted, and it is >in reasonable agreement with the latest 120 and 144 notation proposals. > There are two 132-ET proposals that both use (|~ for 5deg (45 cents), >which is in conflict with the above, but we could use /|~ for 5deg132 >instead. > >I would make /|\ 50 cents instead of 49 (rounding 49.363 up instead of >down); that way it's easier both to remember and to execute, so that, >for example, E/|\ would be the same as F\!/.

Maybe we should round them all to the nearest 5 cents. Most are already within 1 cent. This has the side-effect of making the gap near 22 cents vanish. By the way, my apologies. Just when you start using the terminology such as "7/5 comma" that I suggested earlier, I decided that it was better to use the colon-based interval notation rather than slash-based pitch notation to refer to the commas.I figure we really are referring to intervals, not pitches. And I prefer to give them with no factors of 2 rather than in first octave form here, since then the two odd numbers involved can be read directly. What do you think?

>I think we will have to use //| to fill the gap between /|) and /|\. >In looking for candidates for this position, I rejected (|( as the >17/11 and 19/11 commas, and then I came across //| as the 19/13 comma, >which seems to validate its use as the 13/5 and 15/13 commas.

I don't understand "validate" here. In rational terms (i.e. relative to strict Pythagorean) the 13:19 comma (38:39) is 44.97 cents while the 1:25 comma (6400:6561) is 43.01 cents. That's a 1.96 cent schisma, far greater than any notational schisma we've accepted before. We can't use //| for the 13:19 comma anywhere.

> There >are two problems with this: > >1) It's not intuitive relative to the /| comma. > >So what? The actual size of the //| comma is around 43 cents. We're >just not using it as the 5+5 comma.

But we agreed we shouldn't use //| (at least for notating ETs) unless it _is_ the 5+5 comma. I don't see the lack of a 45 cent symbol in the 12-relative notation as a good enough reason to do something that is _so_ counterintuitive.

>2) It's not compatible with the 108-ET notation: > >108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ > 11 22 33 44 56 67 78 89 100 > >We could eliminate the 108 incompatibility by doing 108 this way: > >108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ > 11 22 33 44 56 67 78 89 100 > >This essentially writes off the ratios of 11 in this division. In a >previous message I said that I thought that it was not very good to do >something like this, but I don't see any other choice for what is >admittedly not a very good division (not even 9-limit consistent). > >However, after writing the above I found another way. The ~|\ symbol >could be interpreted as the 19'/11 (or 11'-19' comma, 297:304,~40.330 >cents) comma (the difference between 19/11 and 27/16), for which the >required alteration in 12-ET is ~46 cents. This would then allow us to >have //| available as the 5+5 comma and to keep the 108 notation as in >version a, above, if we wish.

Again, interpreting ~|\ as an 11:19 comma would involve a schisma of more than 2 cents relative to (19'-19)+5 and 23+5. Unacceptable.

>To fill the gap around 22 cents, I found ~|~ as the 23/17 (or 17+23) >comma, which is ~23.3 cents in 12-ET, but we have already used this as >the 19/5 comma. I then found that /|~ can approximate the 23/22 comma >(as 24057:23552, ~36.729 cents, the difference between the apotome and >23/22); this symbol represents almost exactly 23 cents in 12-ET

Apart from the fact that I'd rather not to go beyond 19-limit unless I really have to; using /|~ as the 11:23 comma (24057:23552) of 36.73 cents is out of the question since as the (11-5)+17 comma (4352:4455) /|~ is 40.5 cents (all relative to strict Pythagorean), a schisma of more than 3 cents. Anyway, as I said above, the gap near 22 cents disappears if we round to nearest 5 cents.

>I agree that "rational" is not the right term. How about "relational" or >"12-relational"? We could abbreviate this as "12-R" notation.

I suggest "12-relative", which can of course also be abbreviated 12-R.

>> For >> example, low numbered ETs that are not multiples of 12 could be >notated

>> approximately, using these symbols to represent offsets from 12-ET. >

>I can see the value of this for notating approximations of rational >intervals relative to 12-ET, but I am a bit skeptical about how well >this would work for notating ETs other than multiples of 12. It's as >if we're trying to shoehorn everything into a 12-ET framework to make >it more convenient (at least initially) for the performer. In the >process we end up with ET notations that are considerably more >complicated than those that we have already worked out, and I would >hate to see musicians become so dependent on relating microtonal >intervals to 12 that they would be unable to think of them in any other >way. And the worst case scenario would be refusal of performers to >change to the simpler, more precise ET and rational notations because >they had gotten accustomed to the 12-R notation and would not want to >change. > >Why don't we try to do something like 19 or 22-ET in 12-R notation and >see what it looks like before we go any farther with this?

Probably no need to bother. I think I can see that it will be complete garbage.

>I am >beginning to think that this is getting more complicated than I would >like it to be. Perhaps we should just keep it simple by restricting >the 12-R notation to the 13-limit symbols and let that serve as a >gentle introduction to the sagittal symbols. Musicians could then >learn how the same (now-familiar) symbols are used outside a 12-ET >framework. If players of flexible-pitch 12-ET instruments need some >sort of aid in remembering how many cents away from 12-ET the notes >are, then numbers can be placed above the notes in their parts. >(Hopefully this could eventually be done automatically with a >computer.) For players of fixed-pitch (i.e., retuned) or specially >built instruments, the symbols should suffice.

Excellent idea. Although I still think we should give the full list of 5-cent-resolution 19-limit symbols somewhere, accompanied by some commentary to the effect that "We don't really want you to use the sagittal symbols in this way, but we suspected some of you would try anyway, because you haven't yet escaped your 12-equal dependence, so we at least wanted to make sure that it is standardised and agrees as much as possible with the rest of the system. We must warn you that if you get stuck in this cul de sac, you will be missing out on the full generality and precision of the sagittal notation. We'd probably prefer you used cents written near the noteheads." On rereading, this sounds rather patronising. But perhaps we can instead make it humorous when worked into the mythology your daughter is working on. After going thru your proposals above, I decided it was more important to have a 45 cent symbol other than //|, than a 55 cent symbol, so I reassigned (|~ for 45 cents based on its 23-limit interpretation (which you proposed above, as agreeing with your proposed 132-ET notation). But then I threw in the 31' comma symbol, with two flags on one side, for 55 cents in case anyone needs it. Here's my current 12-R proposal, where 12-R notation is for approximate notation of pitches relative to 12-equal in a manner consistent with the general sagittal notation. ~|( 5 cents as large 9:17, 3:17, 1:17 commas ~|~ 10 cents as 15:19, 5:19 commas /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas |( 20 cents as 5:7 and 7:15 commas ~|) 25 cents as 17 comma + 7 comma )|) 30 cents as 7:19 comma |) 33 cents as 7:9, 3:7, 1:7 commas **** not rounded to nearest 5 **** (|( 35 cents as 5:11, 11:15 commas /|) 40 cents as 9:13, 3:13, 1:13 dieses (|~ 45 cents as 7:11' comma + 1:23 comma /|\ 50 cents as 9:11, 3:11, 1:11 dieses (/| 55 cents as 1:31' comma (|\ 60 cents as large 9:13, 3:13, 1:13 dieses Each symbol covers a +-2.5 cent range of pitches. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5192 - Contents - Hide Contents Date: Wed, 11 Sep 2002 04:07:30 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> Maybe we should round them all to the nearest 5 cents.

Doesn't that invalidate the whole idea?

Message: 5193 - Contents - Hide Contents Date: Wed, 11 Sep 2002 20:53:46 Subject: Fwd: Re: ET tuning and rhythm From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> The FRETS on guitars and such were built for > 12toneET, which is not so say that the strings > were more likely to be tuned to frets than to > harmonics. In most tonal music, a succussion > of guitar chords (tuned in harmonics) will contain > SOME arithmetic intervals both within each chord > and between successive chords. The guitar mediates > tuning systems.

still, any reasonable guitar tuning using harmonics would lead to an overall pitch set constrained to be far closer to 12-equal than those used on keyboard instruments in the 16th through 18th centuries. if you disagree, please propose a sample guitar tuning by harmonics.

Message: 5194 - Contents - Hide Contents Date: Wed, 11 Sep 2002 22:13:08 Subject: Re: Combinatorics and Tuning Systems? From: Gene Ward Smith --- In tuning-math@y..., <Josh@o...> wrote:

> Somehow, even the great serialists failed to much > exploit combinatoriality between sets of 5 and 7. > the 5-12/7-12 aggregate is particularly interesting > in that 7-12 does not actually include any forms of 5-12. > It's such an obvious candidate for serialist treatment... > ...ok, I'll drop that.

A t-(v,k,n) design is a set of subsets called "blocks" of a set of v elements, each of size k, and such that any t-tuple is contained in exactly n blocks; for example a projective plane of order q is a 2-(q^2+q+1,q+1,1) design. There are some very interesting designs related to the (sporadic, simple) Mathieu groups; in connection with the 12-et, the 5-(12,6,1) design related to the Mathieu group M12, a subgroup of the permutation group S12 of twelve elements, is especially interesting; it consists of 132 hexads, a set invariant under M12, such that if we know five elements of the hexad this determines the sixth. Some web stuff: Steiner System -- from MathWorld * [with cont.] Steiner Triple System -- from MathWorld * [with cont.] Dodecahedral Faces of the Mathieu group of deg... * [with cont.] (Wayb.) Hexad

Message: 5195 - Contents - Hide Contents Date: Wed, 11 Sep 2002 19:10:04 Subject: test From: wally paulrus hello from paul erlich --------------------------------- Do You Yahoo!? Yahoo! Finance - Get real-time stock quotes [This message contained attachments]

Message: 5196 - Contents - Hide Contents Date: Wed, 11 Sep 2002 23:54:10 Subject: Re: [tuning] Re: Proposal: a high-order septimal schisma From: monz hi Gene, From: "genewardsmith" <genewardsmith@xxxx.xxx> To: <tuning@xxxxxxxxxxx.xxx> Sent: Wednesday, September 04, 2002 11:27 AM Subject: [tuning] Re: Proposal: a high-order septimal schisma

> --- In tuning@y..., paul.hjelmstad@u... wrote: >>

>> Thanks. Hope this doesn't sound stupid, but could you tell me the >> significance of each number in the "wedge invariant"? (Being really literal >> please) Are they the powers of 2,3,5,7 or something? >

> It's hardly stupid, and in fact it's complicated enough I suggest further

discussion should take place on tuning-math, and not here. There *are* commas of various kinds hidden in it, for which the numbers are exponents, and as we just saw, if the first number is "1" then 5 and 7 can be expressed in terms of 2 and 3; however it really comes from multilinear algebra, and is not such a good thing to discuss here. There's quite a lot in the tuning-math archives about it.

> > Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4] > > Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation of 3,

and so forth; hence [12, 19, 28, 24] would be the usual 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is

> > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3] > > Whether we've computed in terms of commas or ets, the wedge product of the

linear temperament is exactly the same, up to sign.

> > If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by > > 2^u6 3^(-u2) 5^u1 > 2^u5 3^u3 7^(-u1) > 2^u4 5^(-u3) 7^u2 > 3^u4 5^u5 7^u6

at last, i finally understand how you're calculating wedgies! but that last bit has me a little confused. from the example meantone wedgie [1,4,10,12,-13,4] :

> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, September 09, 2002 10:50 PM > Subject: [tuning-math] [tuning] Re: Proposal: a high-order septimal schisma > > <snip> > > Wedge of two intervals:

>>> p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, >>> u1*v3-v1*u3,u1*v4-v1*u4] >

> For example p = 126/125 and q=81/80, then p = 2^1 3^2 5^(-3) 7^1, > so in vector form it is [1,2,-3,1]. Similarly, > q=2^(-4) 3^4 5^(-1) 7^0, which in vector form is [-4,4,-1,0]. > Wedging the two gives the wedgie for meantone, but 126/125 ^ 225/224, for

example, will work also. i calculated these commas [ 4 -4 1] = 80 / 81 [-13 10 -1] = 59049 / 57344 [ 12 -10 4] = 9834496 / 9765625 [ 12 -13 4] = 1275989841 / 1220703125 OK, so the syntonic comma (81/80) is there ... but what happened to

Message: 5197 - Contents - Hide Contents Date: Wed, 11 Sep 2002 23:59:19 Subject: commas from wedgies (was: Proposal: a high-order septimal schisma) From: monz (sorry ... the previous version of this post got away from me too soon; ignore it.) hi Gene,

> From: "genewardsmith" <genewardsmith@xxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Wednesday, September 04, 2002 11:27 AM > Subject: [tuning] Re: Proposal: a high-order septimal schisma > > > <snip> > > Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4] > > Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation > of 3, and so forth; hence [12, 19, 28, 24] would be the usual > 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is > > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3] > > Whether we've computed in terms of commas or ets, the wedge product > of the linear temperament is exactly the same, up to sign. > > If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by > > 2^u6 3^(-u2) 5^u1 > 2^u5 3^u3 7^(-u1) > 2^u4 5^(-u3) 7^u2 > 3^u4 5^u5 7^u6

at last, i finally understand how you're calculating wedgies! but that last bit has me a little confused. from the example meantone wedgie [1,4,10,12,-13,4] which you gave here:

>>> p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, >>> u1*v3-v1*u3,u1*v4-v1*u4] >

i calculated these commas [ 4 -4 1] = 80 / 81 [-13 10 -1] = 59049 / 57344 [ 12 -10 4] = 9834496 / 9765625 [ 12 -13 4] = 1275989841 / 1220703125 OK, so the syntonic comma (81/80) is there ... but what happened to 126/125 and 225/224? why are they not in this list, and why are the other ones there? -monz "all roads lead to n^0"

Message: 5198 - Contents - Hide Contents Date: Thu, 12 Sep 2002 17:21:54 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4633]:

> Hi Gene, > > Good to hear from you in this thread. I'm glad you're checking to make sure > we don't betray your original concept of notating both rational tunings and > ETs using accidentals representing one comma per prime. > > This is certainly still possible with the sagittal notation as it currently > stands, and I intend it to always be posssible (for primes up to

31). To do

> this one takes certain prime-comma interpretations of certain symbols, and > treats these symbols as atomic, taking no notice of the fact that they are > composed of various "flags" or half-arrow-heads, which don't quite add up > if considered as individual commas. In any case, the extent of their > not-adding-up is less than 0.5 cents. > > In many cases, this way of using the notation will require multiple > prime-comma accidentals against a single note, often pointing in opposite > directions (in addition to any sharps or flats, the 3-comma symbols). Here > they are: > > /| 5-comma 80;81 > |) 7-comma 63;64 > /|\ 11-diesis 32;33 > /|) 13-diesis 1024;1053 > ~| 17-comma 2176;2187 > )| 19-comma 512;513 > |~ 23-comma 729;736 > (| 29-comma 256;261 > )|\ 31-comma 243;248

Notice that up to 29, all of these commas are notated with a vertical shaft (pipe symbol in ascii) plus a single flag symbol placed to one side. The 11 and 13-defining symbols are larger than what one would ordinarily call a comma, and they are presented here as dieses, for which the symbols contain two flags (the sum of two commas), inasmuch as it is not economical to have a single flag representing an interval as large as a diesis. The |\ flag is therefore defined as the 11-5 comma, i.e., the 11 diesis minus the 5 comma.

> There are also some symbols for alternative commas for some primes, e.g. > ~|( for the 17'-comma 4096;4131.

This particular example makes it possible to write 17/16 (taking C as 1/1) either as C#~! -- C-sharp lowered by a 17-comma -- or Db~|( -- D- flat raised by a 17' comma. (In our ascii notation we have adopted Dave's proposal that the exclamation mark indicate downward alteration, while the pipe symbol is used for upward alteration and for no specific direction.)

> Of course the real symbols look much nicer than these ASCII representations > of them, and can be found in several .bmp files in George's or my folder, > in the files section of this yahoo group. > > However George and I have been concentrating on standardising the sagittal > notation for 15-limit JI and all ETs up to about 76 (and many

others up to

> about 300), in such a way that only _one_ sagittal accidental is ever > required. This involves redefining certain symbols, including some > high-prime comma symbols, as representing commas involving two (and > occasionally three) primes greater than 3. These redefinitions never > involve a change in value of more than 1 cent and are mostly less than 0.5 > cents.

This property holds for all consonances in the 17 limit, but not necessarily for anything above that. For example, the 29-comma symbol (| is redefined as the 7:11-comma

> 45056;45927, only 0.34 cents smaller. > As well as redefining a few high-prime symbols, several new symbols are > introduced (but with no new flags). For example, |( is the 5:7- comma > 5103;5120.

In this example the ratio 7/5 (relative to C) may be notated as Gb!( - - a (pythagorean) G-flat lowered by a symbol |( that, under one interpretation, is the 17' comma ~|( minus the 17 comma ~|, which leaves 288:289, ~6.001 cents. But here it functions as the 7 comma |), 63:64, minus the 5 comma /|, 80:81, resulting in the 5:7 comma, 5103:5120, ~5.758 cents. You can see that all of the time we have spent discussing how many schismas can vanish on the point of a flag has not gone to waste.

> (|( is the 5:11-comma 44;45 and sometimes the 7:13-comma > 1664;1701 (which only differ by 0.84 cents). And we have defined > double-shaft symbols as the apotome-complements of these symbols.

The double-shaft symbols are for those (such as myself) who wish to have the ability to modify each note-head with only a single symbol. My contention is that this would make keyboard music easier to read, because the notation would be more compact and less cluttered. This comes at the price of abandoning the conventional sharp and flat symbols, as well as requiring a greater number of symbols in a music font.

> This one-accidental-per-note notation is the most difficult to decide upon. > It is then easy to decompose this into either dual-symbol, where at most > one saggital is used in conjunction with a sharp or flat, or multi- symbol > using one symbol per prime.

Since we disagree on which is better, we will offer both and let the end user decide. Each has its advantages, and perhaps both will be used. I would surmise that with computerized generation of scores and parts, that either option could be available for printing.

> Now to your question: > > Gene Ward Smith wrote:

>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>>> Maybe we should round them all to the nearest 5 cents. >>

>> Doesn't that invalidate the whole idea? >

> I suspect you haven't been following recent discussions closely, and I > don't blame you. > > This suggestion was in the context of a small digression from the main > effort (one that seemed wise to persue now, since we couldn't reach > agreement on 48-ET or 96-ET). This digression involved designing what we > call the 12-R notation, a kind of bastard child of the proper sagittal > notation. 12-R notation is only an approximate notation with a resolution > of 5 cents (max error of +-2.5 cents). But as such, it does allow one to > notate any tuning _relative_to_12_equal_ in a manner that agrees as much as > possible with the proper sagittal notation for most n*12-ETs.

I hadn't replied to this yet because I didn't want to make a hasty response without thinking through the ramifications of this proposal. I don't like having obscure symbols such as )|) and (|~ in this scheme, because 1) they don't represent any low-number ratios or even any simple primes, for that matter; 2) neophytes who take the time to memorize these might then become frustrated once they learn that these symbols aren't even important, but were just put there to fill in some gaps. Which brings me to the question, what is the purpose of having a 12-R notation with 5-cent resolution, anyway? Certainly we don't think that it would be very important to notate 240-ET (or any particular multiple of 12 over 100, for that matter). What we are left with, then, is multiples of 12 through 96. For these, the only symbols you need are: 12: /||\ 100 24: /|\ /||\ 50 100 36: |) ||) /||\ 33 67 100 48: |) /|\ ||) /||\ 25 50 75 100 60: /| /|) (|\ ||\ /||\ 20 40 60 80 100 72: /| |) /|\ ||) ||\ /||\ 17 33 50 67 83 100 84: /| |) /|) (|\ ||) ||\ /||\ 14 29 43 57 71 86 100 96: /| |) /|) /|\ (|\ ||) ||\ /||\ 13 25 38 50 63 75 88 100 12-R: /| |) /|) /|\ (|\ ||) ||\ /||\ 15 33 39 50 61 67 85 100 which requires nothing more than: Sym Approximate offset and Comma interpretation ------------------------------------------------ /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas |) 33 cents as 7:9, 3:7, 1:7 commas /|) 39 cents as 9:13, 3:13, 1:13 dieses /|\ 50 cents as 9:11, 3:11, 1:11 dieses (|\ 61 cents as large 9:13, 3:13, 1:13 dieses It would probably be desirable to include three more symbols that would complete the 11 limit notation: |( 18 cents as 5:7 and 7:15 commas (| 18 cents as 7:11 comma (|( 36 cents as 5:11, 11:15 comma You will observe that, except for the two 13 diesis, all of these come very close to 72-ET. And you might also want to include these, since they are simple enough to comprehend: //| 27 cents as 5+5 comma |\ 35 cents as 11-5 comma Anyway, I thought that //| would be a better option than: ~|) 26 cents as 17 comma + 7 comma I would say stop there and don't worry about whatever gaps remain. As long as they can notate the multiples of 12 through 96 and an 11- limit tonality diamond, I think a lot of people will be satisfied with this as a start. In order to complete the 13 limit, you need no new symbols, only additional uses for existing symbols that in 12-R are considerably different in size: |( 11 cents as 11:13 comma (2nd usage) (|( 28 cents as 7:13 comma (2nd usage) //| 47 cents as 5:13 and 13:15 commas (2nd usage) For these there is no question that you would need to write the number of cents near the notehead for performers using 12-ET instruments.

> We only want to do this because we figure people will try to use the > sagittal symbols in this way anyhow, and we wanted to standardise it. > Anyone who wants better than 5 cent resolution in a 12-relative notation, > should write the cents near the noteheads. Anyone who wants precise > notation of rational or ET tunings, should use the true sagittal notation > (in one of the three mutually compatible forms described above).

My recommendation is to have the cents written above the notes in any and every part for an instrument of flexible pitch. Those who don't need them can ignore them, and those who do will be able to memorize them as they become familiar with the notation.

> ... Hey George, > > Can you put together in one message, in numerical order notations for (1) > all the ETs we agree on, and (2) your proposals for all the ETs we have yet > to agree on, and I will respond? I'm going away for 4 months in 1.5 weeks time. > > Could you please list _all_ ETs in order and just write "as subset of > <whatever>" against those that are not notated with their native fifth, and > "prefer subset of <whatever>" when they have an optional native- fifth-based > notation.

Okay. It's been a challenge to keep track of all this, but I'll try get that together before the weekend. I still have the final installment of my reply to your message #4532 to send (I wrote it before our latest conversations but will send it as is), so look for that one in a little while. Plus there are a few other things in your latest messages that I haven't replied to, so I will try to cover those by the beginning of next week. Then after that I'll be able to start presenting what we have on the main tuning list. --George

Message: 5199 - Contents - Hide Contents Date: Thu, 12 Sep 2002 17:27:23 Subject: Re: A common notation for JI and ETs From: gdsecor (This is a continuation of my message #4604, which is in reply to Dave Keenan's message #4532.)

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>> Now 55 is a real problem, because nothing is really very good for >> 1deg. The only single flags that will work are |( (17'-17) or (| (as >> the 29 comma), and the only primes that are 1,3,5,n-consistent are >> 17, 23, and 29. >> >> If I wanted to minimize the number of flags, I could do it by >> introducing only one new flag: >> >> 55: ~|\ /|\ ~|| /||\ >> >> so that 1deg55 is represented by the larger version of the 23' comma >> symbol. Or doing it another way would introduce only two new flags: >> >> 55: ~|~ /|\ ~||~ /||\ >> >> The latter has for 1deg the 17+23 symbol, and its actual size (~25.3 >> cents) is fairly close to 1deg55 (~21.8 cents). Besides, the symbols >> are very easy to remember. So this would be my choice. >

> I would not use a 23 comma to notate this when it can be done in 17-

limit. Luckily ~|\ works for 1 step as the 17+(11-5) comma (which also agrees with 2 steps of 110-ET). So I go for your first (min flags) suggestion:

> > 55: ~|\ /|\

I didn't do the complement properly for that one (what I gave was left over from when we were doing inverse complements); ~|\ didn't even have a rational complement defined. With the proposals that I made in the previous message, ~|\ not only has a rational complement ~)||, but ~|\ also is *both* the 17+(11-5) comma and the 23' comma. That would make the symbol sequence: 55a: ~|\ /|\ ~)|| /||\ (RC) The flags in ~)|| don't really add up to the proper amount, but we aren't using )| in any other symbol, so there is no inconsistency in symbol arithmetic created by "forcing" the complement. There is a possibility in which the symbols for both 1deg and 3deg are rational complements consistent in 55: 55b: /|( /|\ ~||~ /||\ (RC) but this uses more flags. Instead we could just use an alternate complement to achieve matching symbols: 55c: /|( /|\ /||( /||\ (AC & MS) But if we forget about rational or alternate complements, we can have matching sequences, consistent symbol arithmetic, and a meaningful symbol (23' comma) in the first apotome with a minimum of new flags: 55d: ~|\ /|\ ~||\ /||\ (MS) Take your pick, but I would go with version d; I think it's the simplest.

> ...

>>> 69,76: |) ?? (|\ /||\ [13-comma] >>

>> Again, I wouldn't use |) by itself defined as a 13-comma symbol, but >> would choose /|) instead: >> >> 69,76: /|) )|\ (|\ /||\ [13-commas] >> >> For 2deg of either 69 or 76, )|\ is about the right size. > > Agreed. >

> I note that 62, 69 and 76 are all 1,3,9-inconsistent and might also

be notated as subsets of 2x or 3x ETs. I was going to say that now that we have (| as the 11'-7 comma, we could do both 69 and 76 this way: 69, 76: /|) (|~ (|\ /||\ [(11'-7)+23 comma] I think that (|~ is a more suitable size than )|\ for a half-apotome symbol, especially when it occurs between the 13 and 13' diesis symbols, which would keep the symbols in order of size. However, now I see that (| is not the same comma in (|~ and (|\, so that is not an option. So it must be done the way we had it: 69, 76: /|) )|\ (|\ /||\ (RC)

> ...

>>> 67,74: ~|) /|) (|\ ~||( /||\ >>

>> I'm certainly in agreement with the 2deg and 3deg symbols, and if you >> must do both ET's alike, then what you have for 1deg would be the >> only choice (apart from (| as the 29 comma). We both previously >> chose )|) for 1deg74 (see message #4412), presumably because it's the >> smallest symbol that will work, and I chose |( for 1deg67 (in #4346), >> which would give this: >> >> 67: |( /|) (|\ /||) /||\ >> 74: )|) /|) (|\ (||( /||\ >> >> So what do you prefer? >

> I prefer yours, but I'm uncertain about the complement used for 4

steps of 74. Add to this your latest observation about 67: << We agreed on |( for 1deg67 which is wrong (or at least not 1,3,5,7-consistently right) if |( is the 7-5 comma. I also proposed it for 93-ET (3*31) but we didn't agree on a notation for that. >> I now propose these as most memorable (fewest flags): 67: /|( /|) (|\ /||) /||\ (MM) 74: )|) /|) (|\ /||) /||\ (MM)

>>> 81,88: )|) /|) (|\ (||( /||\ [13-commas] >>

>> This is exactly what I have for 74, above. Should we do 67 as I did >> it above and do 74, 81, and 88 alike? > > Yes. >

>> On the other hand, why wouldn't 88 be done as a subset of 176? >

> I have a reason to do both 81 and 88 as subsets, apart from the

fact that they are 1,3,9-inconsistent. When using their native fifths they need a single shaft symbol for 4 steps and none is available.

>> It is with some surprise that I find that |( is 1deg in both 67 and >> 81, so 81 could also be done the same way as I have for 67, above. >

> Better to do it the same as 74 and 88 (or as a subset).

Your observation that |( for 1deg is wrong if |( is the 7-5 comma also holds here. I think that the simplest notation (fewest flags) for both 81 and 88 is: 81, 88: )|) /|) (|\ )||\ /||\ [13 commas] (MM)

>>> 6 steps per apotome >>> 37,44,51: )| /| /|) ||\ (||\ /||\ [13-commas] >>> or >>> 37,44,51: |) )|) /|) (||( ||) /||\ [13-commas] >>

> So are you agreeing to one of these for 37 and 44? Presumably not

the second one because of |) not being the 7-comma. Yes I prefer the first one, but not with (||\ for 5deg (how did you get that?). With no new flags it could be: 37a, 44a: )| /| /|) ||\ )||\ /||\ [13-commas] (MM)

> And with rational complements?

With rational complements we would have this: 37b, 44b: )| /| /|) ||\ (||~ /||\ [13-commas] (RC) But I think I prefer version a -- fewer flags and easier to remember, whereas the rational complementation of version b doesn't really accomplish anything.

> ...

>>> 86,93,100: )|) |) )|\ (|\ (||( /||\ [13-commas] >>> or >>> 86,100: )|( |) )|\ (|\ (||) /||\ [13-commas] >>> 93: |( |) )|\ (|\ /||) /||\ [13-commas] >>

>> I would do 93-ET and 100-ET as subsets of 186-ET and 200-ET, >> respectively. >

> I can agree to that for 100-ET since there is no single-shaft

symbol for 5 steps, but it is of course 2*50, and 93 is 3*31, so the fifth sizes are quite acceptable.

>> For 86, I wouldn't use |) by itself as anything other than the 7 >> comma, as explained above, >

> I totally agree we should avoid this in all cases. >

>> but would use convex flags for symbols >> that are actual ratios of 13. So this is how I would do it: >> >> 86: ~|~ /|) (|~ (|\ ~||~ /||\ [13-commas and 23-comma] >> >> The two best primes are 13 and 23, so there is some basis for >> defining |~ as the 23 flag. In any event, I believe that (|~ can be >> a strong candidate for half an apotome if neither /|\ nor /|) nor (|\ >> can be used. >

> I have no argument about the even steps (they agree with 43 and 50-

ET). But again I don't see the need to use a 23-comma. We have already used )|\ for a half-apotome in the case of 69 and 76-ETs. It works here too. 86-ET is 1,3,7,13,19-consistent. So why not:

> 86,93,100: )|) /|) )|\ (|\ ?? /||\ [13-commas]

I see that (|~ will work here for 86, but not 93 or 100. But I agree that the )|\ symbol is better for minimizing the flags and especially for keeping commonality over the three divisions when there is no reason not to. For 5deg )||\ works for all three without adding any new flags: 86, 93, 100: )|) /|) )|\ (|\ )||\ /||\ (MM)

> We can now consider the 31-ET family. > > 31: /|\ /||\ > 62: /|) /|\ (|\ /||\ [13-commas] > 93: )|) /|) )|\ (|\ ?? /||\ [13-commas]

And we can fill in the blank with )||\ if you agree.

> and compare it to the 19-ET family > > 19: /||\ > 38: /|\ /||\ > 57: /|) (|\ /||\ [13-commas] > 76: /|) )|\ (|\ /||\ [13-commas] > > Whew!

And whew! to you, too. (End of reply to your message #4532.) --George