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Message: 8050 Date: Mon, 10 Nov 2003 06:35:59 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> What's the definition of "standard val"?

> >> > > >> >The vector consisting of round(n log2(p)) for primes p in > >> >ascending order up to the chosen prime limit, considered > >> >as defining a val.

> >> > >> What's n?

> > > >The number you are finding a standard val for.

> > Then what's p!?

A prime number <= n.

> Besides using two vals to find a lt, can you give an example > of what a single standard val would be good for?

It's easy to calculate and most of the time gives you the val you are most interested in. However, the gram val isn't hard to compute either, and is more likely to do that, so "standard" is really just a connvenience. Gram vals would be harder to explain--look at the trouble I am in now with the standard val.

Message: 8051 Date: Mon, 10 Nov 2003 22:07:55 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> However, I had forgotten until I > checked it again that the 11-limit doesn't map to anything;

because 3:1 and 9:3 map to different numbers of "steps" in a chord? i think that was similar to george secor's reasoning . . .

> While g12 isn't > what I've called the "standard" val h12, it in fact is preferable, > since it has a lower consistent badness score.

why don't we switch from standard vals to "lowest consistent badness" vals around here as a general rule? i think that would be a positive development . . .

Message: 8052 Date: Mon, 10 Nov 2003 06:39:42 Subject: Re: Enharmonic diesis? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> 125:128 = [0, -3]

Why not <0, -3> ?

Message: 8053 Date: Mon, 10 Nov 2003 22:17:37 Subject: Re: _Alternative Rock Chord_ and 43 (was: Eponyms) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> > traditional music-theory prohibits > > sounding the "4th" along with the "major-3rd", but > > that's exactly what a lot of alternative-rockers did > > in the 1990s.

> > ... and jazz pianists have been voicing 'sus' chords with both the > third and the fourth for quite a long time, where the third is > usually placed in a higher octave above the fourth. The inverse > voicing is also used, but not as often. > > AH

i don't know of instances of the third being voiced below the fourth in jazz, it seems to create a completely different chord which i've heard in rock (and various tunings of which are wonderfully explored in monz's piece) but not in jazz.

Message: 8054 Date: Mon, 10 Nov 2003 07:56:23 Subject: Re: Enharmonic diesis? From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Silly me. I didn't read far enough. I was just searching > on "enharmonic diesis" and saw the value of 62.565148 cents > and the terms "great" and "diminished second" associated > with it. But you are quite correct, in precise > 1/3-comma-meantone (but not 19-EDO) the tempered > size of 125;128 is precisely the untempered 625:648.

just for the benefit of those trying to keep track: the 1/3-comma meantone "enharmonic diesis" is the ratio 625:648 = ~62.565148 cents. one degree of 19edo is 2^(1/19) = ~63.15789474 cents. 19edo and 1/3-comma meantone are close enough in pitch that there is probably never an audible difference.

> So who was the turkey who called 125:128 the "great" diesis? > The sooner we lose that name the better. I notice Scala doesn't > use it.

i'm pretty sure that the first turkey who did that was Alexander Ellis, in one of his many voluminous appendices to his translation of Helmholtz's _On The Sensations Of Tone_.

> And is it correct to call it a diminished second? Isn't G# > to Ab a _doubly_ diminished second? But considered as a > doubly diminished second (i.e. a Pythagorean comma), its > tempered size in 1/3-comma meantone is _minus_ 62.565148 cents.

G#:A is a minor-2nd, so G#:Ab is simply a diminished-2nd. examples of a doubly-diminished-2nd would be G#:Abb or Gx:Ab . -monz

Message: 8055 Date: Mon, 10 Nov 2003 22:20:36 Subject: Re: Enharmonic diesis? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> So who was the turkey who called 125:128 the "great" diesis?

probably the same person (not bird) who called 3125:3072 the "small" diesis.

> And is it correct to call it a diminished second? Isn't G# to Ab a > _doubly_ diminished second?

no. G to Ab is a minor second, and G# to Ab is a diminished second.

> its tempered size in 1/3-comma > meantone is _minus_ 62.565148 cents.

no, G# is lower than Ab in 1/3-comma meantone.

Message: 8056 Date: Mon, 10 Nov 2003 00:42:50 Subject: Re: Linear temperament names? From: Carl Lumma

>> That was to Paul, but I gotta say you're approaching "intervention" >> territory with this stuff.

> >I don't understand what you mean by ""intervention" territory". Please >explain.

An "intervention" is the stupid politically-correct term for when all a person's friends corner him about some problem he's having -- drinking too much, eating too much, etc. -Carl

Message: 8057 Date: Mon, 10 Nov 2003 22:39:07 Subject: Re: Linear temperament names? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> I'm suspecting that blind faith in log-flat badness measures may

have

> something to do with this.

blind faith in log-flat badness and believing it's a useful measure for any purpose are two completely different issues.

> Don't bother with a "you just never know" type of answer, because if > you're going to clutter an already cluttered (but very clever)

diagram

> with lines corresponding to them, and give them meaningless or

cryptic

> names and fill up a huge table (43 temperaments!),

going to? i already did! or do you think i'm going to add further clutter (since you said already cluttered . . .)?

> then you'd better > do a lot better than that.

monz made tables for ETs as large as 4296 so i wanted to show how those ETs were related to one another. perhaps the best solution is to make different versions of the table to go with the different zoom levels, so that it's never presented as that huge.

> The same goes for 5-limit linear temperaments having any error

greater

> than the one that uses the same neutral third generator as both

major

> and minor third. Why would anyone care about such so-called 5-limit > temperaments?

for example, one can regard a given temperament as 'lame' in the sense of not fully functioning, to help understand the structures inherent in a distributionally even scale that is not equally tempered. if nothing else, the temperament will signify that a certain ratio is operating as a chromatic unison vector for distributionally even scales of interest with cardinalities that appear as numbers along the line representing the temperament.

> In my opinion, the diagram (and table) would be greatly improved if > these did not appear at all,

they pretty much don't appear in the diagrams you would be looking at! it's only at the zoom levels that don't interest you anyway where you would see these lines.

> But if you really think these 25 should stay, then I'll take you up

on

> your offer and ask that you refer to them with single letters,

except

> where they are 5-limit subsets

you don't really mean subsets, but rather 'incarnations', right?

> of higher-limit temperaments that _do_ > deserve a name, in which case their names should be followed by > "(5-limit)".

ok, if monz is on board we'll proceed to strip the personality out of this page :). i have no qualms about doing so.

Message: 8058 Date: Mon, 10 Nov 2003 08:50:58 Subject: Re: Enharmonic diesis? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Dave, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >

> > So who was the turkey who called 125:128 the "great" diesis? > > The sooner we lose that name the better. I notice Scala doesn't > > use it.

> > > i'm pretty sure that the first turkey who did that was > Alexander Ellis, in one of his many voluminous appendices > to his translation of Helmholtz's _On The Sensations Of Tone_.

Ellis's use of the qualifier "great" was to distinguish 125:128 from the small interval which we now call the "magic comma", 3,5-monzo [-1, 5] = ratio 3072:3125 = ~29.61356846 cents, which he dubbed "small diesis". Rameau, on the other hand, called 125:128 the "minor diesis", in contrast to his "major diesis", 3,5-monzo [-5, 3] = ratio 250:243 = ~49.16613727 cents. it's important to note these two historical usages, if only because (i assume) they've been repeated in so many other theoretical works. exactly *how* often they've been repeated, i couldn't say. i do agree that a newer, more systematic method of naming is called for, and am happy to be acknowledged as having set some sort of precedent in that endeavor. -monz

Message: 8059 Date: Mon, 10 Nov 2003 22:41:20 Subject: Re: Linear temperament names? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >Carl Lumma wrote:

> >> Nooooooooooooooooooooooo!

> > > >Thanks Carl. Rest assured that I've taken this carefully reasoned > >defense into account in the above. ;-) > > > >-- Dave Keenan

> > That was to Paul, but I gotta say you're approaching "intervention" > territory with this stuff. > > -Carl

that's ok, i must be at least a thousand times as guilty of "intervention" with respect to monz's pages. i think the fact that dave, graham, gene and i were hoping to write up a paper together makes this seem far less like an "intervention" and much more like a valid expression of opinion.

Message: 8060 Date: Mon, 10 Nov 2003 00:51:53 Subject: Re: Eponyms From: Carl Lumma

>> >> >The vector consisting of round(n log2(p)) for primes p in >> >> >ascending order up to the chosen prime limit, considered >> >> >as defining a val.

>> >> >> >> What's n?

>> > >> >The number you are finding a standard val for.

>> >> Then what's p!?

> >A prime number <= n. >

>> Besides using two vals to find a lt, can you give an example >> of what a single standard val would be good for?

> >It's easy to calculate and most of the time gives you the val you are >most interested in. However, the gram val isn't hard to compute >either, and is more likely to do that, so "standard" is really just a >connvenience. Gram vals would be harder to explain--look at the >trouble I am in now with the standard val.

It's just val I still don't understand. If a val is a homomorphism from the rationals to the integers, I can't fathom how a "number I'm finding a val for" could come into play. If the standard 5-limit val for 12-equal is [12 19 28] or something, how does it come from round(n log2(p))? Oh n is 12, eh? So vals are uniquely identified by this n? So how does one find a standard val for an odd limit (the start of this thread, which perhaps George is still following)? Where do you get your n? -Carl

Message: 8062 Date: Mon, 10 Nov 2003 08:53:26 Subject: Re: Enharmonic diesis? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> Ellis's use of the qualifier "great" was to distinguish > 125:128 from the small interval which we now call the > "magic comma", 3,5-monzo [-1, 5] = ratio 3072:3125 = > ~29.61356846 cents, which he dubbed "small diesis". > > > Rameau, on the other hand, called 125:128 the "minor diesis", > in contrast to his "major diesis", 3,5-monzo [-5, 3] = > ratio 250:243 = ~49.16613727 cents.

i'm quite happy to simply refer to 125:128 as the "enharmonic diesis", since that is its notational function in a JI analysis of "common-practice" (i.e., meantone-based) music. -monz

Message: 8064 Date: Mon, 10 Nov 2003 09:02:18 Subject: Re: Enharmonic diesis? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > > i'm quite happy to simply refer to 125:128 as the > "enharmonic diesis", since that is its notational > function in a JI analysis of "common-practice" > (i.e., meantone-based) music.

unfortunately, under the criteria of notation, the qualifier "enharmonic" also refers to other small intervals a syntonic-comma (or its multiples) larger and smaller than 125:128, such as ratio 625:648 = <2 3,5>-monzo [3 4, -4] = ~62.565148 cents, and ratio 2025:2048 = <2 3, 5>-monzo [11 -4, -2] = ~19.55256881 cents. i'm also trying to establish here a new standard usage of brackets. since using commas to separate groups of exponents in a monzo eliminates the need for contrasting brackets to indicate the presence or absence of prime-factor 2, i propose that we use angle-brackets for the prime-factors themselves, and square-brackets for the actual monzo of the exponents. -monz

Message: 8065 Date: Mon, 10 Nov 2003 15:11:08 Subject: Re: Linear temperament names? From: Carl Lumma

>ok, if monz is on board we'll proceed to strip the personality out of >this page :). i have no qualms about doing so.

Good thing I have local copies of these. I can't imagine a cooler calling card for this list than those names on those charts. -Carl

Message: 8066 Date: Mon, 10 Nov 2003 10:41:38 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:

> > 125:128 = [0, -3]

> > Why not <0, -3> ?

See Yahoo groups: /tuning-math/message/7435 * [with cont.] Yahoo groups: /tuning-math/message/7436 * [with cont.] Yahoo groups: /tuning-math/message/7437 * [with cont.] Yahoo groups: /tuning-math/message/7444 * [with cont.]

Message: 8067 Date: Mon, 10 Nov 2003 15:13:29 Subject: Re: Linear temperament names? From: Carl Lumma

>that's ok, i must be at least a thousand times as guilty >of "intervention" with respect to monz's pages. i think the fact that >dave, graham, gene and i were hoping to write up a paper together >makes this seem far less like an "intervention" and much more like a >valid expression of opinion.

Dave didn't get the "intervention" reference either. I didn't mean he's intervening on you, I meant he's asking for an "intervention" for being such a negative nancy. Anyway, it was supposed to be funny. -Carl

Message: 8068 Date: Mon, 10 Nov 2003 10:47:34 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> G#:A is a minor-2nd, so G#:Ab is simply a diminished-2nd. > > examples of a doubly-diminished-2nd would be G#:Abb or Gx:Ab .

Yes, of course. Thanks for taking the time to correct me.

Message: 8069 Date: Mon, 10 Nov 2003 23:14:25 Subject: Re: _Alternative Rock Chord_ and 43 (was: Eponyms) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" > <perlich@a...> wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex"

> <cfaah@e...> wrote:

> > > > traditional music-theory prohibits > > > > sounding the "4th" along with the "major-3rd", but > > > > that's exactly what a lot of alternative-rockers did > > > > in the 1990s.

> > > > > > ... and jazz pianists have been voicing 'sus' chords with both

> the

> > > third and the fourth for quite a long time, where the third is > > > usually placed in a higher octave above the fourth. The

> inverse

> > > voicing is also used, but not as often. > > > > > > AH

> > > > i don't know of instances of the third being voiced below the

> fourth

> > in jazz, it seems to create a completely different chord which

> i've

> > heard in rock (and various tunings of which are wonderfully

> explored

> > in monz's piece) but not in jazz.

> > > The interpretation of the chord as 'sus' may be debated, but > without question, the voicing happens quite a bit in jazz. An > example would be, spelled up from the bottom, G2, G3, C4, F4, > A4, B4.

no, there the third is *above* the fourth, not below it.

> The question is: is this a sus chord or merely a dominant > 11? Mark Levine writes in The Jazz Piano Book, "A persistent > myth about sus chords is that 'the fourth takes the place of the > third.' Jazz pianists, however, often voice the third with a sus > chord ... the third is voiced above the fourth" p. 24 Levine also > made a similar statement in his Jazz Theory book. I like Levine's > description, because these chords are in fact often used in > context as sus chords.

you're avoiding my question, which was about the third being voiced *below* the fourth. re-read my comments above again.

> Many people take issue with Levine's > interpretation. For example, Robert Rawlins writes in his Music > theory Online review of Levine's theory book: > > "The problem that now arises--if the Mixolydian mode is > considered equivalent to the Vsus chord--is that the mode > contains an unwanted third. Levine circumvents this difficulty by > arguing that the third is not an undesirable note in sus chords: "A > persistent myth is that 'the 4th takes the place of the 3rd in a

sus

> chord.' This was true at one time, but in the 1960s, a growing > acceptance of dissonance led pianists and guitarists to play sus > voicings with both the 3rd and the 4th" (p. 46). Undeniably, jazz > musicians have explored this possibility. The question is how to > interpret the resulting chord. If a sus chord is to retain anything

of

> its presumed historical origin, then the absence of the leading > tone would seem to be requisite. If jazz theory, in practice, has > dispensed with the preparation and resolution of this > suspension, what must remain is at least the displacement of > the third of the chord. If the third is present, and we indeed have > a dominant triad with upper extensions, then it is not clear what > justifies pulling the 11th of the chord into the basic structure

and

> calling it a sus chord. If one were to argue that the voicings > generally employed in contexts where both the third and fourth > are present seem to suggest the sus chord, then it will have to > be attributed to intended ambiguity, much as a twentieth-century > composer might flirt with the ambiguity between major and > minor tonality. There is little to justify the conclusion that sus > chords implicitly contain the third, which is available anytime one > wishes to include it in the harmonic structure. Again, Levine is > avoiding the obvious: The Mixolydian mode is indeed roughly > equivalent with a Vsus chord, with the exception of the third, > which is completely foreign to theharmony." > > 404 Not Found * [with cont.] Search for http://www.societymusictheory.org/mto/issues/mto.00.6.1/mto.00 in Wayback Machine > .6.1.rawlins.html > > AH

still nothing relevant, i'm afraid.

Message: 8070 Date: Mon, 10 Nov 2003 11:06:08 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > > > > > i'm quite happy to simply refer to 125:128 as the > > "enharmonic diesis", since that is its notational > > function in a JI analysis of "common-practice" > > (i.e., meantone-based) music.

> > > > unfortunately, under the criteria of notation, the > qualifier "enharmonic" also refers to other small > intervals a syntonic-comma (or its multiples) larger > and smaller than 125:128, such as > > ratio 625:648 = <2 3,5>-monzo [3 4, -4] = ~62.565148 cents, and > ratio 2025:2048 = <2 3, 5>-monzo [11 -4, -2] = ~19.55256881 cents. > > > i'm also trying to establish here a new standard usage > of brackets. since using commas to separate groups of > exponents in a monzo eliminates the need for contrasting > brackets to indicate the presence or absence of prime-factor 2, > i propose that we use angle-brackets for the prime-factors > themselves, and square-brackets for the actual monzo of > the exponents.

Well I suppose you could, but I don't think this is necessary, and we could save the angle-brackets for something more important. There is obviously no need for selective use of commas between the primes themselves. They provide no additional information there. I would simply call these 2,3,5-monzos. Of course the important thing to know is whether they are 2,...-monzos (complete monzos, octave specific monzos) or 3,..-monzos (2-free monzos, octave equivalent monzos). Maybe a good use for angle-brackets would be for wedgies since, as I understand it, these are in a _very_ different domain from that of monzos, and the angle-brackets are suggestive of wedges themselves.

Message: 8071 Date: Mon, 10 Nov 2003 23:15:23 Subject: Re: _Alternative Rock Chord_ and 43 (was: Eponyms) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> sorry, that example should be: > G2, G3, B3, F4, C5

where did you find this? on page 24 of the jazz piano book, you say? i'll check when i get home . . .

Message: 8073 Date: Mon, 10 Nov 2003 23:33:37 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >

> > Tossing out powers such as 6561/6400 is what I'd recommend also,

> > i thought that went without saying. >

> > though Dave might not go for it.

> > why not? dave has expressed interest in tuning systems where a single > just lattice does not suffice as a derivation for all the pitches, > but tempering out 6561/6400 simply leads to torsion and not to such a > tuning system.

I call this "twin meantone" and consider it to have the same errors but twice the complexity of meantone, and so I rank it above some things like pelogic and Blackwood (quintuple thirds). It seems to me that it would have two unconnected just lattices. But don't take any notice of me, I can't even tell my torsions from my contorsions. :-)

Message: 8074 Date: Tue, 11 Nov 2003 03:33:43 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"

<gwsmith@s...>

> wrote: >

> > However, I had forgotten until I > > checked it again that the 11-limit doesn't map to anything;

> > because 3:1 and 9:3 map to different numbers of "steps" in a chord?

i

> think that was similar to george secor's reasoning . . .

My reasoning is simply that trying to define one leads to an inconsistent system of linear equaltions.

> why don't we switch from standard vals to "lowest consistent

badness"

> vals around here as a general rule? i think that would be a

positive

> development . . .

Do you have a name?

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