Tuning-Math Digests messages 3725 - 3749

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 4

Previous Next

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950

3700 - 3725 -



top of page bottom of page down


Message: 3725

Date: Sat, 2 Feb 2002 12:06:40

Subject: rational approximations to logarithms (was: simple math question)

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, February 02, 2002 11:44 AM
> Subject: Re: [tuning-math] simple math question
>
>
> there are some very good low-integer-ratio approximations to
> the log_10 of the lowest primes
> 
> 
> Examples:
> 
> log_10  ~fractional value 
>   of:     of logarithm
>       
>          more accurate  less accurate
> 
>      2      3/10
>      3     10/21  =   ~1/2
>      5      7/10
>      7     11/13  =   ~5/6
>     11     25/24
>     13     39/35  =  ~10/9
>     17     16/13  =  ~11/9
>     19     23/18  =  ~14/11  =  ~5/4


in fact, i observed before that 3/10 showed up in a sumerian
calculation which i think relates to tuning -- see the middle of
Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *

i had a hunch there that this had something to do with tempering,
and now i realize why i had that hunch

3/10 is so close to the log_10(2) that it would serve just fine
as an approximation in doing calculations by hand

hmmm ....



-monz



 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3726

Date: Sat, 2 Feb 2002 14:45 +00

Subject: Re: interval of equivalence, unison-vector, period

From: graham@xxxxxxxxxx.xx.xx

genewardsmith wrote:

> Because a "temperament" which sends
> 
> 1-9/8--5/4--4/3--3/2--5/3--15/8 to
> 
> 1--9--1/9--1/3--3--1/27--1/3
> 
> hardly seems worthy of the name. In any case, 2 isn't represented!

Why not?  Why is 2 special?


             Graham


top of page bottom of page up down


Message: 3727

Date: Sat, 2 Feb 2002 14:45 +00

Subject: Re: interval of equivalence, unison-vector, period

From: graham@xxxxxxxxxx.xx.xx

paulerlich wrote:

> You'd have to invoke "tritone-equivalence", which is clearly not a 
> recognized psychoacoustical phenomenon!

You certainly would.  Psychoacoustics will have to make its own mind up.


                        Graham


top of page bottom of page up down


Message: 3728

Date: Sat, 02 Feb 2002 20:28:23

Subject: Re: simple math question

From: genewardsmith

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

> particularly in tuning math, logs are often taken to base 2
> since 2 is the ratio of the "octave", and the author often
> assumes that the reader will know that and assume 2 as the base

Another base often found in these parts is 2^(1/1200); the log base the 1200 root of two defines interval size in cents.


top of page bottom of page up down


Message: 3729

Date: Sat, 2 Feb 2002 14:45 +00

Subject: Re: interval of equivalence, unison-vector, period

From: graham@xxxxxxxxxx.xx.xx

genewardsmith wrote:

> Is pajara the new official name? I'd like to get this settled. As for 
> this val, which defines only one of two required generator mappings 
> being a temperament, that's only if you layer on some interpretation 
> and perform the extra calculations to find a good choice for the second 
> generator; taken by itself, it isn't one. It's telling us to send the 
> octave to a unison, and 5 and 7 both to 1/9; it's only after you stick 
> in half-octaves and send 7 to some tuning of 64/9 and 5 to a 
> half-octave below that that pajara emerges. Read literally as a 
> temperament, it sends 2 to 1 and 5 and 7 to 1/9, and I don't think that 
> qualifies.

It defines one of the required mappings for an octave-specific linear 
temperament.  On it's own it is an equal temperament, but a somewhat 
strange one.  It could also be an octave-equivalent mapping for pajara.  
An equal temperament with no steps to the octave is an octave-equivalent 
linear temperament.  I don't see what other sense making an octave an 
identity vector could make.

Yes, it's telling us to map the octave to a unison which we could call 
"imposing octave equivalence" or making the octave an identity vector.  As 
it has torsion, that makes a tritone the real identity vector.

Yes, it sends 2 to 1.  They only differ by identity vectors.  That's what 
octave equivalent temperaments are like.  Also 5, 7 and 1/9 differ only by 
identity vectors.  So it's a tritone equivalent linear temperament.


> > The octave is acting as a unison, but it's more complicated than 
> > that.  As it has torsion, it's actually half an octave that's acting 
> > as a commatic unison vector.
> 
> I would say it's acting as a generator, but if you make 2 a unison it 
> becomes a torsion element, since its square is an octave.

It only acts as a generator if it isn't tempered out.  In this case it is 
being tempered out, so it must be an identity vector, which means it's 
like a unison vector.  Torsion is certainly involved, but I don't 
understand what you're saying there.


> > (BTW, in an octave-equivalent system, half a unison is a half-octave 
> > as well as a unison.  This is obvious if you think of 
> > octave-equivalent frequency space as a Hilbert space, and remember 
> > that half the pitch is the same as the square root of the frequency.)
> 
> You get a real Hilbert space if you allow anything of the form 
> 3^e3 5^e5 ... which can have an infinite number of prime exponents so
> long as e3^2 + e5^2 + ... converges. Is this what you mean? The result 
> isn't even guaranteed to be a real number, and I don't know what it 
> would be good for.

3 and 5 don't enter into it.  I mean octave equivalent frequency ratios 
behave like the complex numbers with modulus 1.  Partly because it's a 
circular system, and also because nth roots have n values.


> > >>> i0 = i2-h2
> > >>> i0.basis
> > [0, 1, -2, -2]
> > 
> > Hey, that's the same as g0 above!
> 
> And which I think hardly counts as a temperament. As I said, it's not 
> one I want to listen to.

There are all kinds of temperaments I wouldn't want to listen to.  Why 
single this one out for opprobrium?


                  Graham


top of page bottom of page up down


Message: 3730

Date: Sat, 02 Feb 2002 20:42:38

Subject: Re: simple math question

From: jpehrson2

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

Yahoo groups: /tuning-math/message/3185 *

> 
> hi joe,
> 
> 
> > From: jpehrson2 <jpehrson@r...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 10:56 AM
> > Subject: [tuning-math] simple math question
> >
> >
> > Could it possibly be said that a logarithm is a way to find 
> > the "exponent" of a number??
> > 
> > I mean, in the most simple case...
> > 
> > ??
> 
> 
> 
> the logarithm is  t h e  way to find the exponent
> 
> by definition, that's exactly its purpose
> 

***Thanks, everybody, for the answers...  It looks like I was "kinda 
right" and "kinda wrong..."

It seems to hinge mostly on *definitions...*

Thanks!

JP


top of page bottom of page up down


Message: 3731

Date: Sat, 02 Feb 2002 20:58:21

Subject: 7-limit MT reduced bases for meantone ets

From: genewardsmith

For all you meantone fans out there (and you know who you are), here
are some meantone systems not already discussed. Rational implications
are staring us in the face here.

24: [49/48, 81/80, 128/125]
26: [50/49, 81/80, 525/512]
38: [50/49, 81/80, 3125/3072]
45: [81/80, 525/512, 2401/2400]
50: [81/80, 126/125, 16807/16384]
55: [81/80, 686/675, 6144/6125]
67: [81/80, 1029/1024, 9604/9375]
74: [81/80, 126/125, 4194304/4117715]
81: [81/80, 126/125, 17294403/16777216]


top of page bottom of page up down


Message: 3732

Date: Sat, 02 Feb 2002 07:12:12

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: genewardsmith

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

> > 171: [2401/2400, 4375/4374, 32805/32768]
> >
> > Wouldn't want to do that--look at those three high-powered commas!

> (note
that the title is an homage to Helmholtz)

Helmholtz liked the schismic temperament, and Vogel goes him one
better by combining schismic with ennealimmal, which the above reduced
basis shows isone way of thiking about 171-et. You could temper either
53 tones or 72 tones with it, among other things.

Since I am now writing a piece in 46-et and just finished one in 
53-et, I'll also add these:

46: [126/125, 245/243, 1029/1024]
53: [225/224, 1728/1715, 3125/3087]

I'm finding the 43-et set of commas quite useful.


top of page bottom of page up down


Message: 3733

Date: Sat, 02 Feb 2002 07:21:55

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Well I confused the two things, which is completely my fault, but 
was 
> > not helped by Graham's opinion that the thing you declined to 
call a 
> > temperament was in fact pajara.
> 
> If you ever get around to trying 222223 with a period of 3/2 in the 
>22-et, tell us about it.

I think I will!


top of page bottom of page up down


Message: 3734

Date: Sat, 02 Feb 2002 07:24:05

Subject: Re: 7-limit MT reduced bases for ets

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > 9: [21/20, 27/25, 128/125]
> > 10: [25/24, 28/27, 49/48]
> > 12: [36/35, 50/49, 64/63]
> > 15: [28/27, 49/48, 126/125]
> > 19: [49/48, 81/80, 126/125]
> > 22: [50/49, 64/63, 245/243]
> > 27: [64/63, 126/125, 245/243]
> > 31: [81/80, 126/125, 1029/1024]
> > 41: [225/224, 245/243, 1029/1024]
> > 68: [245/243, 2048/2025, 2401/2400]
> > 72: [225/224, 1029/1024, 4375/4374]
> > 99: [2401/2400, 3136/3125, 4375/4374]
> > 130: [2401/2400, 3136/3125, 19683/19600]
> > 140: [2401/2400, 5120/5103, 15625/15552]
> 
> This is seriously cool.
> 
> >For any prime limit, we could consider the most characteristic
> >linear temperament of a particular et to be the one leaving off
> >the last member of the MT reduced basis.
> 
> Does it have to be prime (not odd) limit?
> 
> -Carl

Yeah, Carl, these are _tuning system building_ considerations and not 
_simultaneous consonance_ considerations.


top of page bottom of page up down


Message: 3735

Date: Sat, 02 Feb 2002 07:26:06

Subject: Re: 7-limit MT reduced bases for ets

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "clumma" <carl@l...> wrote:
> 
> > Does it have to be prime (not odd) limit?
> 
> Fraid so. It occurs to me another fun game to play with these is to 
>find the corresponding Fokker blocks.

Or better yet, the most compact blocks where ratio odd-limit measures 
distance from a central 1/1 (this is what I refer to as the van 
Prooijen metric). Remember, these are constrained to be periodicity 
blocks, so will _not_ be ellipsoids.


top of page bottom of page up down


Message: 3736

Date: Sat, 02 Feb 2002 07:27:47

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, February 01, 2002 3:56 PM
> > Subject: [tuning-math] Re: interval of equivalence, unison-
vector, period
> >
> >
> > > I would say it's acting as a generator, but if you make 2 a 
unison 
> > > it becomes a torsion element, since its square is an octave.
> > 
> > This, along with my message to Monzo this morning, seems to show 
the 
> > very real problems with considering 2 a unison!
> 
> 
> How is 2^2 an octave?  By definition, it's simply 2.
> Now you guys have really lost me.

Dude, what exactly are you referring to? I thought this was amazingly 
clear, but I guess I'm wrong!


top of page bottom of page up down


Message: 3737

Date: Sat, 02 Feb 2002 07:29:19

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > > 171: [2401/2400, 4375/4374, 32805/32768]
> > >
> > > Wouldn't want to do that--look at those three high-powered 
commas!
> 
> > (note that the title is an homage to Helmholtz)
> 
> Helmholtz liked the schismic temperament, and Vogel goes him one 
better by combining schismic with ennealimmal, which the above 
reduced basis shows is one way of thiking about 171-et. You could 
temper either 53 tones or 72 tones with it, among other things.
> 
> Since I am now writing a piece in 46-et and just finished one in 
> 53-et, I'll also add these:
> 
> 46: [126/125, 245/243, 1029/1024]
> 53: [225/224, 1728/1715, 3125/3087]
> 
> I'm finding the 43-et set of commas quite useful.

Do you really mean 43, or one of the above?


top of page bottom of page up down


Message: 3738

Date: Sat, 02 Feb 2002 08:05:50

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, February 01, 2002 11:27 PM
> > Subject: [tuning-math] Re: interval of equivalence, unison-
vector, period
> >
> >
> > > > > I would say it's acting as a generator, but if you
> > > > > make 2 a unison it becomes a torsion element, since
> > > > > its square is an octave.
> > > > 
> > > > This, along with my message to Monzo this morning,
> > > > seems to show the very real problems with considering
> > > > 2 a unison!
> > > 
> > > 
> > > How is 2^2 an octave?  By definition, it's simply 2.
> > > Now you guys have really lost me.
> > 
> > Dude, what exactly are you referring to? I thought this
> > was amazingly clear, but I guess I'm wrong!
> 
> 
> Oh, OK ... I think I get it.
> 
> If 2 = a unison, then 2^2 = an octave.  Yes?

Umm . . . not exactly. 2 is an octave, and 2^2 is a double octave.


top of page bottom of page up down


Message: 3739

Date: Sat, 02 Feb 2002 08:13:38

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: genewardsmith

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

> > I'm finding the 43-et set of commas quite useful.
> 
> Do you really mean 43, or one of the above?

I meant 46--just my usual terrific job of proof-reading.

43: [81/80, 126/125, 12288/12005]


top of page bottom of page up down


Message: 3740

Date: Sat, 2 Feb 2002 01:26:27

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: monz

------------

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, February 02, 2002 12:13 AM
> Subject: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced bases
for ets)
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > I'm finding the 43-et set of commas quite useful.
> >
> > Do you really mean 43, or one of the above?
>
> I meant 46--just my usual terrific job of proof-reading.
>
> 43: [81/80, 126/125, 12288/12005]



thanks for sending  t h a t  now anyway, gene!


43-edo was advocated as a tuning by Sauveur
file:///C:/interval/dict/meride.htm#43-1/5diff


i've updated that page now thanks to the attention
you caused me to give it, and included a link to your
post above


i'd like to change that link to point to something
that really explains what "7-limit MT reduced bases"
are, but i sure don't know to explain it, so if you
do then i'll let the link point to that post


thanks



from manuel's page:
Stichting Huygens-Fokker: Logarithmic Interval Measures *


>> méride: 1/43 part of an octave
>>
>> This name was chosen by Joseph Sauveur (1653-1716)
>> in 1696. The méride and eptaméride were the first
>> logarithmic interval measures proposed. Sauveur
>> favoured 43-tone equal temperament because the small
>> intervals are well represented in it. He had set the
>> comma to one step, then found a range of 2, 3 or 4
>> steps for the chromatic semitone, corresponding to 31,
>> 43 and 55 tones per octave. He found 43 to be optimal
>> because 4 steps is almost exactly a 16/15 minor second
>> and 7 steps almost exactly the geometric mean of
>> three 9/8 and two 10/9 whole tones. The chromatic scale
>> contained in 43-tET is virtually identical to 1/5-comma
>> meantone tuning.



-monz


------------




-monz







_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3741

Date: Sat, 2 Feb 2002 01:29:59

Subject: ---------- request for everyone to add line separators

From: monz

i have a request to make:


i've begun compiling hard-copy volumes of tuning-math
posts so that i can do serious studying of them

the easiest way i've found to do this is to go to
the "expand messages" version of the yahoo interface
for the post on the 1st of each month, copy what's
there, and keep clicking "next" until the month is
finished

when i print it, there's no easy way to see where
one post ends and another begins

would everyone please add something like this

---------- 

as a separator line at the beginning and end of
each post?  i'd really appreciate that



-monz






_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3742

Date: Sat, 02 Feb 2002 00:03:44

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > So why did you say "this was not a temperament"? 
> 
> Because a "temperament" which sends
> 
> 1-9/8--5/4--4/3--3/2--5/3--15/8 to
> 
> 1--9--1/9--1/3--3--1/27--1/3
> 
> hardly seems worthy of the name.

You'd have to invoke "tritone-equivalence", which is clearly not a 
recognized psychoacoustical phenomenon!


> > And isn't it true 
> > that, if you took it out to, say, 10 notes per approximate 
octave, 
> > and tuned the octaves pure, it would _not_ be an octave-repeating 
> > scale? This seems to be the point Graham is missing.
> 
> We seem to be talking about different things--what is "it"? If you 
> mean pajara, it's a temperament, not a scale.

What I thought we were talking about, and I thought you agreed, was 
the fact that you derived temperaments originally in terms of two 
generators, neither of which was guaranteed to be an octave, and then 
came up with a different basis for the temperament such that the 
octave was either a member of the basis or a power of a member of the 
basis. For example, didn't you originally state that a form of 
Blackjack which has a period of ~5/3 came out of your mechanism? I 
took that to mean that if there were no such phenomenon as octave 
equivalence, a Blackjack scale with a period of ~5/3 would better 
exploit the consonances than the standard Blackjack scale.


top of page bottom of page up down


Message: 3743

Date: Sat, 2 Feb 2002 01:50:10

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, February 02, 2002 1:26 AM
> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced
bases for ets)
>
>
> from manuel's page:
> Stichting Huygens-Fokker: Logarithmic Interval Measures *
>
> >> ... Sauveur ... found 43 to be optimal
> >> because 4 steps is almost exactly a 16/15 minor second
> >> and 7 steps almost exactly the geometric mean of
> >> three 9/8 and two 10/9 whole tones. The chromatic scale
> >> contained in 43-tET is virtually identical to 1/5-comma
> >> meantone tuning.




[-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
+
[ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
----------
[-7  2  2]  (= 225:128 "augmented 6th")


[7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents


but what significance does that have?  i don't get it

manuel?



-monz



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3744

Date: Sat, 02 Feb 2002 01:04:01

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

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

> > The thing he said wasn't a temperament has no notes to an octave,
> 
> No notes? He said it was generated by a fifth and a fifth-tritone -- 
> so it seems like it could have plenty of notes, up to an infinite 
> number,
in fact.

One is a basis for pajara/twintone. We have octaves in it, since
(15/14)^(-2) (3/2)^2 = 49/25 ~ 2. The other thing, which I declined to
call a temperament, doesn't even represent octaves, so it depends on
which thing you are talking about.


top of page bottom of page up down


Message: 3745

Date: Sat, 2 Feb 2002 02:01:31

Subject: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets))

From: monz

------------

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, February 02, 2002 1:50 AM
> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced
bases for ets)
>
> from manuel's page:
> Stichting Huygens-Fokker: Logarithmic Interval Measures *
>
> >> ... Sauveur ... found 43 to be optimal
> >> because 4 steps is almost exactly a 16/15 minor second
> >> and 7 steps almost exactly the geometric mean of
> >> three 9/8 and two 10/9 whole tones. The chromatic scale
> >> contained in 43-tET is virtually identical to 1/5-comma
> >> meantone tuning.





>
>
>
> [-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
> +
> [ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
> ----------
> [-7  2  2]  (= 225:128 "augmented 6th")
>
>
> [7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents
>
>
> but what significance does that have?  i don't get it
>
> manuel?



the only thing that i think i can see is some kind of
tritone-equivalence in action, because if you ignore
prime-factor 2 you get a mean for the 225:128 of 15:8,
which is 2^(1/2) higher than the above interval, and
which is the interval that is given exactly by 5 generators
of 1/5-comma meantone

but i really don't understand what's going on



-monz





-------------



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3746

Date: Sat, 02 Feb 2002 01:11:57

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > > The thing he said wasn't a temperament has no notes to an 
octave,
> > 
> > No notes? He said it was generated by a fifth and a fifth-
tritone -- 
> > so it seems like it could have plenty of notes, up to an infinite 
> > number, in fact.
> 
> One is a basis for pajara/twintone. We have octaves in it, since
> (15/14)^(-2) (3/2)^2 = 49/25 ~ 2.

This is the thing I was trying to call Graham's attention to earlier.

>The other thing, which I declined >to call a temperament, doesn't 
>even represent octaves, so it depends >on which thing you are 
>talking about.

Well I confused the two things, which is completely my fault, but was 
not helped by Graham's opinion that the thing you declined to call a 
temperament was in fact pajara.


top of page bottom of page up down


Message: 3747

Date: Sat, 02 Feb 2002 10:08:01

Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 1:26 AM
> > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT 
reduced
> bases for ets)
> >
> >
> > from manuel's page:
> > Stichting Huygens-Fokker: Logarithmic Interval Measures *
> >
> > >> ... Sauveur ... found 43 to be optimal
> > >> because 4 steps is almost exactly a 16/15 minor second
> > >> and 7 steps almost exactly the geometric mean of
> > >> three 9/8 and two 10/9 whole tones. The chromatic scale
> > >> contained in 43-tET is virtually identical to 1/5-comma
> > >> meantone tuning.
> 
> 
> 
> 
> [-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
> +
> [ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
> ----------
> [-7  2  2]  (= 225:128 "augmented 6th")
> 
> 
> [7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents
> 
> 
> but what significance does that have?  i don't get it
> 
> manuel?

what are you trying to do here, monz?


top of page bottom of page up down


Message: 3748

Date: Sat, 02 Feb 2002 01:12:36

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

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

> What I thought we were talking about, and I thought you agreed, was 
> the fact that you derived temperaments originally in terms of two 
> generators, neither of which was guaranteed to be an octave, and then 
> came up with a different basis for the temperament such that the 
> octave was either a member of the basis or a power of a member of the 
> basis.


That's a trick you can do with linear temperaments, and since this
form is both useful and commonly used by Graham, it seems like a good
one. This thread in my mind is partly about the point that the octave
does not have a special status, in that you can do exactly the same
for other intervals, such as a fifth.

For example, didn't you originally state that a form of 
> Blackjack which has a period of ~5/3 came out of your mechanism? 

That didn't come out of my mechanism, it came out of my
misunderstanding of a comment you made.

I 
> took that to mean that if there were no such phenomenon as octave 
> equivalence, a Blackjack scale with a period of ~5/3 would better 
> exploit the consonances than the standard Blackjack scale.

It seems like an interesting plan, at any rate. As I say, I can take
any linear temperament such as miracle, and use anything I like (and
5/3 would seem to be a particularly good choice) as one of the
generators, or at least as a power of one of the generators.


top of page bottom of page up down


Message: 3749

Date: Sat, 02 Feb 2002 10:08:34

Subject: Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases forets))

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> ------------
> 
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 1:50 AM
> > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT 
reduced
> bases for ets)
> >
> > from manuel's page:
> > Stichting Huygens-Fokker: Logarithmic Interval Measures *
> >
> > >> ... Sauveur ... found 43 to be optimal
> > >> because 4 steps is almost exactly a 16/15 minor second
> > >> and 7 steps almost exactly the geometric mean of
> > >> three 9/8 and two 10/9 whole tones. The chromatic scale
> > >> contained in 43-tET is virtually identical to 1/5-comma
> > >> meantone tuning.
> 
> 
> 
> 
> 
> >
> >
> >
> > [-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
> > +
> > [ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
> > ----------
> > [-7  2  2]  (= 225:128 "augmented 6th")
> >
> >
> > [7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents
> >
> >
> > but what significance does that have?  i don't get it
> >
> > manuel?
> 
> 
> 
> the only thing that i think i can see is some kind of
> tritone-equivalence in action, because if you ignore
> prime-factor 2 you get a mean for the 225:128 of 15:8,
> which is 2^(1/2) higher than the above interval, and
> which is the interval that is given exactly by 5 generators
> of 1/5-comma meantone
> 
> but i really don't understand what's going on

i don't understand what you're trying to do


top of page bottom of page up

Previous Next

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950

3700 - 3725 -

top of page