Tuning-Math Digests messages 4200 - 4224

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 5

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4200 - 4225 -



top of page bottom of page down


Message: 4200

Date: Tue, 12 Mar 2002 08:34:44

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > > ok. gene, once again, this means that in the 'gens' 
calculation, 
> > the 
> > > > number of generators in the 3:1 should be multiplied by log
(3), 
> > the 
> > > > number of generators in the 5:3 should be multiplied by log
(5), 
> > the 
> > > > number of generators in the 5:1 should be multiplied by log
(5).
> > > 
> > > And to make the result meaningful (i.e. comparable to the 
> > unweighted 
> > > values) then after you take the RMS of these weighted values 
you 
> > > should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).
> > 
> > why do you want them to be comparable to the unweighted values?
> 
> Obviously it doesn't matter as far as choosing lists, but I like 
for a 
> human (e.g. me) to be able to look at the error values and have 
them 
> mean something. i.e. to actually be in cents. So when you see 5 you 
> know kinda what a 5c mistuning sounds like.

but the units here are not cents, they're gens. what does 5 gens 
sound like? faggeddabbouddit.


top of page bottom of page up down


Message: 4201

Date: Tue, 12 Mar 2002 10:56:52

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > sounds like 'bastoni' (italian bread). fits in well with 'injera' 
> > (ethiopian bread). actually this works well, since 225/224^49/48 
> > (bastoni) is in the same 'aisle' as 81/80^50/49 (injera).
> 
> Next time I'm stuck for a name, I'll think bread.

or spiny mammal :)


top of page bottom of page up down


Message: 4202

Date: Tue, 12 Mar 2002 20:43:28

Subject: diminished temperament -- rimsky-korsakov, ravel, scriabin

From: paulerlich

Octatonic Scales *

V-2001 *

The Proxomitron Reveals... *


top of page bottom of page up down


Message: 4203

Date: Tue, 12 Mar 2002 08:39:32

Subject: Re: heuristic...

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Paul,
> 
> Do you have the original expository message number handy,
> or any good words to search for, or do you know what list
> you posted it to, or around when?
> 
> -Carl

i've lost track of the original one, but last year it used to be 
a 'conjecture', not a 'heuristic'

here's the most up-to-date exposition:
Yahoo groups: /tuning-math/message/2491 *
remember to click on 'message index' and then 'expand message' . . .


top of page bottom of page up down


Message: 4204

Date: Tue, 12 Mar 2002 11:00:01

Subject: skhismic

From: paulerlich

it's spelled this way sometimes. pronounce it. there's no reason one 
should think it might be spelled "shismic". how do you 
pronounce "schism"? like "shism"? naah . . .


top of page bottom of page up down


Message: 4205

Date: Tue, 12 Mar 2002 20:44:48

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >chromatic unison vectors.
> 
> You're suggesting this as a name for a temperament???

no!


top of page bottom of page up down


Message: 4207

Date: Tue, 12 Mar 2002 08:40:47

Subject: Re: amt

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> How was the name amt arrived at. Is it an abbreviation for something? 

It's an acronym for "acute minor third", from its generator.

> It could be called "fifth of eleventh".

Sounds like a borg.


top of page bottom of page up down


Message: 4208

Date: Tue, 12 Mar 2002 11:24:53

Subject: Re: skhismic

From: genewardsmith

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

> it's spelled this way sometimes. pronounce it. there's no reason one 
> should think it might be spelled "shismic". 

I used to spell it that way, but quit because everyone seemed to spell it differently. Still use "sk" to mean 2^(1/612), though.

how do you 
> pronounce "schism"? like "shism"?
naah . . .

Usage Note: The word schism, which was originally spelled scisme in
English, is traditionally pronounced (szm). However, in the 16th
century the word was respelled with an initial sch in order to conform
to its Latin and Greek forms. From this spelling arose the
pronunciation (skzm). Long regarded as incorrect, it became so common
in both British and American English that it gained acceptability as a
standard variant. Evidence indicates, however, that it is now the
preferred pronunciation, at least in American English. In a recent
survey 61 percent of the Usage Panel indicated that they use (skzm),
while 31 percent said they use (szm). A smaller number, 8 percent,
preferred a third pronunciation, (shzm).

In the ethnic bread department, the closest I can come to "chromic"
seems to be "chimichurri", an Argentine bread. Other great names
abound--lavosh, focaccia, ekmek, hapanleipa, panatone and malasadas.


top of page bottom of page up down


Message: 4209

Date: Tue, 12 Mar 2002 21:20:07

Subject: the underappreciated augmented system

From: paulerlich

Hexatonic thirds from Schubert to Schoenberg: Voice-leading, Harmonic Congruence, Prolongation *

404 Not Found * Search for http://theory.esm.rochester.edu/smt-99.abstracts/schuster-craig.html in Wayback Machine

404 Not Found * Search for http://www.scalamusica.com/sca/jz02.htm in Wayback Machine


top of page bottom of page up down


Message: 4210

Date: Tue, 12 Mar 2002 00:42:04

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: Carl Lumma

>neutral thirds
>meantone
>pelogic
>augmented
>porcupine
>diminished
>diaschismic
>small diesic
>quarter fourths
>kleismic
>twin meantone
>half meantone-fourth
>half meantone-fifth
>minimal diesic
>schismic
>wuerschmidt
>tiny diesic
>orwell
>amt

What you're calling small diesic is magic.  I wonder why 5-limit
miracle didn't make Gene's original list... otherwise the short
list has everything I care about.

-Carl


top of page bottom of page up down


Message: 4211

Date: Tue, 12 Mar 2002 11:37:56

Subject: bread

From: paulerlich

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

> Other great names abound--ekmek,

sounds like 'ekmelic' -- richter-herf's name for 72-equal.


>panatone

wow!


top of page bottom of page up down


Message: 4213

Date: Tue, 12 Mar 2002 08:44:49

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> What you're calling small diesic is magic. 

Sorry, my mistake.


top of page bottom of page up down


Message: 4214

Date: Tue, 12 Mar 2002 14:37 +0

Subject: Re: 32 best 5-limit linear temperaments redux

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a6jdeq+58k4@xxxxxxx.xxx>
paulerlich wrote:

> ok. gene, once again, this means that in the 'gens' calculation, the 
> number of generators in the 3:1 should be multiplied by log(3), the 
> number of generators in the 5:3 should be multiplied by log(5), the 
> number of generators in the 5:1 should be multiplied by log(5). this 
> will cause temperaments generated by the fifth to look better than 
> they currently do, relative to those that aren't. this is important 
> since augmented (especially) and diminished (a little less so) are 
> far harder for the ear to understand than meantone, even when all are 
> *tuned* in 12-equal, and the badness values would no longer put 
> meantone as the 'best'.

When was it decided these temperaments were "far harder for the ear to 
understand"?  Even if so, augmented is more complex than meantone if you 
measure by the simplest MOS to contain a consonant chord (8 compared to 5 
notes).

Still, for whatever reason you want to privilege fifths.  The rule you 
give, generalized to higher limits, will favour simpler ratios in general. 
 That'll give the usual bias towards temperaments that work with the 
limit lower than the one you asked for.  And it'll miss temperaments where 
a simple interval is close enough to use in modulations, but isn't the 
defined approximation to 3:2.


                      Graham


top of page bottom of page up down


Message: 4215

Date: Tue, 12 Mar 2002 23:20:31

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Oh I think I realise why you ask, since you wouldn't want it by 
your 
> > own badness measure. You think I'm violating my own badness 
measure. 
> 
> Nope. My 5-limit badness is 774, not something to set the world on 
fire, but it did pass your measure, as you say.
>

That was before I went back to using Paul's weighted complexity. Now 
it comes out badder than pelogic and semisuper (and with my new 
parameters 5.5c and 0.43 power, badder than seventh of major third 
[-9,7] and parakleismic).

> I want a name for it because my 7-limit badness score is 173, so we 
> have a decent 5-limit temperament which heats up in the 7-limit.

Being good at 7-limit is a good reason to have a name for it, but 
obviously should have no bearing on whether or not it's included in 
the 5-limit list.

If you modify your badness to use log-of-odd-limit weighted complexity 
you  will see that its requirement of 9 generators to the fifth 
combined with its so-so 3 cent rms error, gives it a badness of 1113 
that pushes it way down the list. The worst on my proposed list so far 
is only 625 by this badness (quadrafourths).

But I don't mind including it so long as it doesn't require you to 
include anything else. By your badness with weighted complexity, there 
may well be some others in between quadrafourths and "chrome" that I 
will find objectionable. There are certainly some that _you_ will find 
objectionable: the half and twin neutral thirds and kleismics (that 
I'm happy to omit).


top of page bottom of page up down


Message: 4216

Date: Tue, 12 Mar 2002 08:48:30

Subject: Re: Dave's 23 best 5-limit temperaments

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > 1990656/1953125 Extends to the 1029/1024^126/125 = 
> > [9,5,-3,-21,30,-13] system, and needs a name.)
> > 
> > map   [[0, 9, 5], [1, 1, 2]]
> > 
> > generators   77.96498962   1200
> > 
> > keenan   12.03289099   rms   2.983295872   g   6.377042156
> 
> How about "quarter major thirds"?

Not accurate; I think "chromic" would be a good name, since the generator is 21/20~25/24, the chromatic semitone or chroma.

> > 16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14] 
> system, and needs a name.)
> > 
> > map   [[0, -4, 3], [1, 2, 2]]
> > 
> > generators   126.2382718   1200
> > 
> > keenan   12.16857021   rms   5.942562596   g   4.966554810
> 
> I've called it "quarter fourths" in the past, but it could also be 
> "third of major thirds".

I think quadrafourths would be fine.


top of page bottom of page up down


Message: 4217

Date: Tue, 12 Mar 2002 14:37 +0

Subject: Re: 32 best 5-limit linear temperaments redux

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a6iupi+ksb1@xxxxxxx.xxx>
Dave K:
> > One of the reasons given for using rms was that if there was 
> an 
> > outlier the LT still got credit for the intervals it did well. But if 
> > we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is 
> > very wide, I don't think it is a good {1,3,5,7} temperament. Let it 
> be 
> > found to be a good {2,5,7} temperament or whatever.

Paul:
> it might not -- again, what if it's really a good {2, 3, 7/5} 
> temperament?

There shouldn't be anything special about 1.3.5 and 1.3.7 compared to 
1.3.7:5 as defining chords.  So far my script only works with the former, 
but don't make any plans assuming that will always be the case.  It should 
even be possible to automatically find and evaluate the simplest subset if 
that's all you want.


                         Graham


top of page bottom of page up down


Message: 4218

Date: Tue, 12 Mar 2002 23:46:46

Subject: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > ok. gene, once again, this means that in the 'gens' calculation, 
> the 
> > > number of generators in the 3:1 should be multiplied by log(3), 
> the 
> > > number of generators in the 5:3 should be multiplied by log(5), 
> the 
> > > number of generators in the 5:1 should be multiplied by log(5). 
> this 
> > > will cause temperaments generated by the fifth to look better 
> than 
> > > they currently do, relative to those that aren't.

Paul, don't you mean "divided by". log(3) is smaller than log(5) so if 
you want to favour fifths ...

Here's the formula I'm using:

SQRT(((gens(1:3)/LN(3))^2+(gens(1:5)/LN(5))^2+(gens(3:5)/LN(5))^2)/(1/
LN(3)^2+1/LN(5)^2+1/LN(5)^2))*1200/period_in_cents


top of page bottom of page up down


Message: 4219

Date: Tue, 12 Mar 2002 00:37:03

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >Going by my map:
> >
> >          max int width  card(map)  me    g
> >5-limit   10             2          5     7.257
> >11-limit  11             4          2.75  ??
> >
> >I have no idea how Gene got 7.257, so I can't fill
> >g in for the 11-limit.
> 
> Dave's getting 7.3 too, with this:
> 
> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13
> 
> Anybody care to explain why this isn't total rubbish?  Putting
> both the individual gens per idenitity (E13 and F13) and the
> total width of the chain (E13-F13) together into the rms calc???

carl, if 'identity' is defined as 'consonant interval', then the 
*only* thing going in here is the individual gens per identity. 
that's all. E13-F13 is the major sixth or minor third.


top of page bottom of page up down


Message: 4220

Date: Tue, 12 Mar 2002 08:52:57

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> 
> > > 1990656/1953125 Extends to the 1029/1024^126/125 = 
> > > [9,5,-3,-21,30,-13] system, and needs a name.)
> > > 
> > > map   [[0, 9, 5], [1, 1, 2]]
> > > 
> > > generators   77.96498962   1200
> > > 
> > > keenan   12.03289099   rms   2.983295872   g   6.377042156
> > 
> > How about "quarter major thirds"?
> 
> Not accurate; I think "chromic" would be a good name, since the 
generator is 21/20~25/24, the chromatic semitone or chroma.

what about chromatic unison vectors?


top of page bottom of page up down


Message: 4221

Date: Tue, 12 Mar 2002 08:55:40

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I propose we go with either these 19 (in weighted-complexity order)

> neutral thirds
> meantone
> pelogic
> augmented
> porcupine
> diminished
> diaschismic
> small diesic = magic
> quarter fourths = quadrafourths
> kleismic
> twin meantone = garbage
> half meantone-fourth = ditto
> half meantone-fifth = ditto
> minimal diesic
> schismic
> wuerschmidt
> tiny diesic
> orwell
> amt

What happened to chromic?


top of page bottom of page up down


Message: 4222

Date: Tue, 12 Mar 2002 00:40:41

Subject: Re: Dave's 18 best 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Comma     Name                 Mapping           Gen   Period
> > -------------------------------------------------------------
> > 6561/6400 twin meantone	       [0,-1,-4] [2,4,8] 503.8 600
> 
> isn't this just meantone itself? isn't this an instance of torsion?

I have no idea what torsion is, but of course this is not simply 
meantone. It has a half-octave period. And thanks to Graham's list 
I've corrected its (lowest-terms) generator now to 96.2 cents.

> > 81/80     half meantone-fifth  [0,2,8]   [1,2,4] 348.1 1200
> > 81/80     half meantone-fourth [0,-2,-8] [1,2,4] 251.9 1200
> 
> these are the valid instances of contortion i was talking about. 
they 
> are not temperaments, but they are quite interesting. since 
> their 'badness' should simply be 4 or 8 times that of meantone, such 
> cases can be handled with an introductory passage and never 
mentioned 
> again -- it would be easy enough for the reader to supply them for 
> any given badness cutoff.

This assumes the reader is a mathematician. Are we planning to publish 
this in a math journal or a music journal? I don't see a problem with 
listing them in the appropriate place in the list, based on their 
complexity.

I don't see how you can disqualify them as 5-limit temperaments simply 
because all the 5-limit intervals are approximated by an even number 
of generators. What kind of a definition of temperament would disallow 
that?

Isn't a (octave-equivanlent) 5-limit temperament simply any scale or 
tuning system that approximates ratios of 1,3 and 5 and their octave 
equivalents?

There can be no argument that they are not _linear_. They have a 
single generator operating withing a whole-number fraction of an 
octave.

I think, Paul, that maybe you're being blinded by your "hypothesis", 
since here we have the same comma involved in different temperaments 
with different complexities. How about you modify your hypothesis to 
take care of that, rather than try to deny that these are 
temperaments.

And lest you are tempted to now claim as Gene has, that these are 
simply meantone, I claim that a temperament is fundamentally defined 
by its mapping of generators and periods to primes (with generator and 
period in lowest terms), not by the commas that vanish. different 
mapping = different temperament. Seems obvious enough to me.

If your algorithm fails to find certain kinds of temperament that 
other algorithms do find, you shouldn't try to deny their existence, 
you should fix your algorithm.


top of page bottom of page up down


Message: 4223

Date: Tue, 12 Mar 2002 09:52:22

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> but the units here are not cents, they're gens. what does 5 gens 
> sound like? faggeddabbouddit.

Oh dear I keep doing that don't I. Must be the Alzheimer's. :-)


top of page bottom of page up down


Message: 4224

Date: Tue, 12 Mar 2002 20:30:33

Subject: Re: Systematic naming of new temperaments (was: amt)

From: Carl Lumma

>Well yeah, but if possible, they should give you some clue as to what 
>the temperament is. This can be 

Well, crazy names won't tell you as much, but are more memorable,
and might make up for it.  See also my complaint about using diatonic
intervals for temperaments where the diatonic scale may not even
be supported.

>hemithirds
>quadrafourths/bestonic
>chrome

I still don't know the generators for these.

>augmented (indirectly)
>diminished (indirectly)
>pelogic (indirectly)

Where historical names exist, I think they should
be used.

>orwell (obscurely)

But you'll never forget it once you've heard it.

>and possibly semisuper if "semi" is meant to indicate the half-octave 
>period and "super" is short for super fourth.  However I don't think 
>most people distinguish between semi and hemi,

I'm not sure I do.  I'd guess semi is partial, while hemi is
exactly half.

Super is short for super fourth?

>(b)The comma that vanishes:
>schismic
>kleismic
>diaschismic
>wuerschmidt
>tiny diesic
>minimal diesic
>limmal
>meantone (indirectly) 

These are my favs -- I consider unison vectors far more informative
than generator size.  Except for stuff like "tiny" diesic vs. "minimal"
diesic, which I still don't know as I write this.

>or is "meantone" indicating the generator size indirectly?

Meantone is just historical.  It doesn't have to make sense.

>Unfortunately this method doesn't generalise well to higher limits 
>since more than one comma vanishes.

I was just going to mention that.

>We could stick to using the single comma that vanishes in its
>5-limit subset, but that won't work for temperaments that don't
>_have_ a 5-limit subset. 

Indeed.  That's when we resort to shamelessly locking in all of our
surnames!  :)

>starling

When was this ratified?

>There's nothing particularly magical or miraculuos about Magic and 
>Miracle at the 5-limit.

That's okay.  These names are historical now, too.

>>1. Three words.  Temperaments should have cool, single-word
>>names.
>
>I can't see any reason to stop calling neutral thirds by a two word 
>three syllable name.

I don't think any established (as in, more than a month or two)
names should be changed at all.

>I think number of syllable is more relevant than number of words.

I don't care if it's hard to say, I just want people to want to say
it.  Compare "Acute minor thirds" to "kleismic" here.  Characters
from novels, breads, etc., are also good.

>> 2. I find it perverse to name temperaments by their relation to
>> diatonic intervals.  I guess this counts against Amt too.
>
>I know what you mean, but it's not too perverse for 5-limit since the 
>JI diatonic is 5-limit. But hey, that's simply how we name intervals, 
>of any limit. I think we're stuck with it.

I'm willing to accept this, but I don't have to like it.  I try
to avoid it, at any rate.

>Tiny diesic becomes hemisixths.
>Minimal diesic becomes quadrafifths.
>4294967296/4271484375 becomes septathirds.
>
>Any valid generator could be used, not just the smallest one.
>So semisuper becomes twin tritenths or double tritenths.
>
>But it's no use calling pajara/twintone/paultone "twin fourths" or 
>"twin fifths", since that could apply to twin meantone as well.
>
>And Magic and Wuerschmidt would both be "major thirds" so they had 
>better stay as they are.

There's not enough variety in this naming scheme for my taste.  In
effect, I'm going to have to think about the name each time I hear
the temperament, whereas "orwell" lives in my mind as its own entity.

-Carl


top of page bottom of page up

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4200 - 4225 -

top of page