A temperament / adaptibility / ambiguity question

Ok, I'm sure this has been discussed before, but not sure how to
search for this exact answer.
I could figure this out myself too, but I bet someone already knows
and can tell me.

I'm wondering about the ambiguity of various temperaments. For
instance, 72ET is pretty good for 11-limit harmonies I know. But
while 31 has not so just "fifths" it still is clear that 2:3 is the
closest just ratio to its those fifths, within any reasonably low
number limit.

I'm saying this in contrast to the failure of many temperaments to be
unambiguous within a certain limit. For instance, 31ET has separate
notes approximating 9:5 and 7:4, but if I wanted 16-limit ratios, 31
has no 9:16...

Ok, now that I think about it, maybe what I'm asking is: which equal
temperaments do NOT temper out various commas? And In this case, I'm
NOT curious about whether or not it tempers out things like the
Pythagorean comma. I'm interested in whether it is capable of
distinguishing lower-integer basic ratios.

This is very similar to the common discussion about how accurately
tuned is a tempered pitch to a just ratio. Except I'm saying: "is it
simply distinct enough to clearly be closer to one or another
low-number ratio, regardless of tuning accuracy?"

Put another way: how about a list of ET's that have distinct notes
approximating 8-limit ratios, where every possible ratio with integers
less than 8 is represented by distinct notes in the temperament
(regardless of tuning error). And for higher and higher numbers...
what ET's include all 9-limit ratios this way? 10? 11? 12? 13? ...
Potentially interested up to 19, or even to 32...

Am I being clear? Has anyone ever made such a chart/list?

I'm thinking that such information would help with conceiving of the
minimum amount of generalized keys needed to indicate a desired
potential list of JI ratios, which could then potentially be sounded
via an adaptive algorithm and therefore be harmonically pure,
regardless of the tuning error of the temperament.
So pythagorean commas could exist or not, that's not important to me,
it is more about being able to specify clearly 19:16 versus 6:5 versus
7:6, something not possible with 12ET, but this example works with
many higher ETs. I just wonder which successfully do this for a given
set of low-integer limited ratios.

Best,
Aaron Wolf

Aaron wrote:
> For instance, 31ET has separate notes approximating 9:5 and 7:4,
> but if I wanted 16-limit ratios, 31 has no 9:16...
>
> Ok, now that I think about it, maybe what I'm asking is: which
> equal temperaments do NOT temper out various commas?

I have scheme code (pasted below) that finds all ETs smaller than
a given max ET which temper out a given comma. By extension, all
other ETs in that range don't temper out the comma.

> And In this case, I'm
> NOT curious about whether or not it tempers out things like the
> Pythagorean comma. I'm interested in whether it is capable of
> distinguishing lower-integer basic ratios.

Well then, maybe a table will suffice (make sure to view with
a constant-width font).

m- 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
n
|
3 3 3 5 5 22 26 29 58 80 94 94 282 282 282 311 311 311
5 9 9 12 22 26 29 58 80 94 94 282 282 282 311 311 311
7 27 27 31 41 41 58 80 94 94 282 282 282 311 311 311
9 41 41 41 41 58 80 94 94 282 282 282 311 311 311
11 58 58 58 58 80 94 94 282 282 282 311 311 311
13 87 87 94 94 94 94 282 282 282 311 311 311
15 111 111 111 111 282 282 282 282 311 311 311
17 149 217 217 282 282 282 282 311 311 311
19 217 217 282 282 282 282 311 311 311
21 282 282 282 282 282 311 311 311
23 282 282 282 282 311 311 311
25 388 388 388 388 388 388
27 388 388 388 388 388

This table is due to Paul Erlich. Each entry shows the smallest
ET that is m-limit consistent and n-limit unique. To make sure
an ET can distinguish 9/5 from 16/9 and 7/4, you need 9-limit
uniqueness. And the smallest ET that does it is 41.

Note that this is for dyads -- larger chords are somewhat
more forgiving. i.e. it is possible that an ET smaller than
41 can uniquely represent all 9-limit triads...

> I'm thinking that such information would help with conceiving of
> the minimum amount of generalized keys needed to indicate a
> desired potential list of JI ratios, which could then potentially
> be sounded via an adaptive algorithm and therefore be harmonically
> pure, regardless of the tuning error of the temperament.

Yup, that's one application of this.

-Carl

;; Returns the monzo of a ratio.

(define monzo
(lambda (f)
(map cadr (factor-complete f))))

;; Returns the patent val for an ET over the given basis.

(define patent-val
(lambda (et basis)
(map (lambda (x)
(inexact->exact (round (* et (log2 x)))))
basis)))

;; Tunes a ratio using val.

(define val-image
(lambda (ratio val)
(apply + (map * val (monzo ratio)))))

;; Finds all patent vals beneath a given bound that send the
;; given comma to zero.

(define vals-that-temper
(lambda (comma highest-ET)
(mappend (lambda (x)
(if (= (car x) 0)
(cdr x)
'()))
(map (lambda (x)
(let ((v (patent-val
x
(map car
(factor-complete comma)))))
(list
(val-image comma v)
v)))
(reverse (num-ls highest-ET))))))

Carl,

That's a great chart of Paul's, thanks. Has anyone made an extension
of this that continues past the smallest ET? I'm sure that while 41
may have uniqueness up to 9-limit, not all ET's larger than 41 do.
Knowing which (up to maybe this same high ET or 311 or so) would be
interesting.
Oh, and I think I sorta get it, but what does consistent versus unique
mean here?

-Aaron W.

>
> Well then, maybe a table will suffice (make sure to view with
> a constant-width font).
>
> m- 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
> n
> |
> 3 3 3 5 5 22 26 29 58 80 94 94 282 282 282 311 311 311
> 5 9 9 12 22 26 29 58 80 94 94 282 282 282 311 311 311
> 7 27 27 31 41 41 58 80 94 94 282 282 282 311 311 311
> 9 41 41 41 41 58 80 94 94 282 282 282 311 311 311
> 11 58 58 58 58 80 94 94 282 282 282 311 311 311
> 13 87 87 94 94 94 94 282 282 282 311 311 311
> 15 111 111 111 111 282 282 282 282 311 311 311
> 17 149 217 217 282 282 282 282 311 311 311
> 19 217 217 282 282 282 282 311 311 311
> 21 282 282 282 282 282 311 311 311
> 23 282 282 282 282 311 311 311
> 25 388 388 388 388 388 388
> 27 388 388 388 388 388
>
> This table is due to Paul Erlich. Each entry shows the smallest
> ET that is m-limit consistent and n-limit unique. To make sure
> an ET can distinguish 9/5 from 16/9 and 7/4, you need 9-limit
> uniqueness. And the smallest ET that does it is 41.
>
> Note that this is for dyads -- larger chords are somewhat
> more forgiving. i.e. it is possible that an ET smaller than
> 41 can uniquely represent all 9-limit triads...
>
> > I'm thinking that such information would help with conceiving of
> > the minimum amount of generalized keys needed to indicate a
> > desired potential list of JI ratios, which could then potentially
> > be sounded via an adaptive algorithm and therefore be harmonically
> > pure, regardless of the tuning error of the temperament.
>
> Yup, that's one application of this.
>
> -Carl
>
> ;; Returns the monzo of a ratio.
>
> (define monzo
> (lambda (f)
> (map cadr (factor-complete f))))
>
> ;; Returns the patent val for an ET over the given basis.
>
> (define patent-val
> (lambda (et basis)
> (map (lambda (x)
> (inexact->exact (round (* et (log2 x)))))
> basis)))
>
> ;; Tunes a ratio using val.
>
> (define val-image
> (lambda (ratio val)
> (apply + (map * val (monzo ratio)))))
>
> ;; Finds all patent vals beneath a given bound that send the
> ;; given comma to zero.
>
> (define vals-that-temper
> (lambda (comma highest-ET)
> (mappend (lambda (x)
> (if (= (car x) 0)
> (cdr x)
> '()))
> (map (lambda (x)
> (let ((v (patent-val
> x
> (map car
> (factor-complete comma)))))
> (list
> (val-image comma v)
> v)))
> (reverse (num-ls highest-ET))))))
>

--- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@...> wrote:
>
> Carl,
>
> That's a great chart of Paul's, thanks. Has anyone made an
> extension of this that continues past the smallest ET? I'm sure
> that while 41 may have uniqueness up to 9-limit, not all ET's
> larger than 41 do.
> Knowing which (up to maybe this same high ET or 311 or so) would
> be interesting.
> Oh, and I think I sorta get it, but what does consistent versus
> unique mean here?
>
> -Aaron W.

Consistency is defined here:

http://library.wustl.edu/~manynote/music.html

-Carl

Ok, that's what I thought consistency meant. I got it.

However, these charts and the other aren't fully what I was after.

I used terms like "8-limit" in my initial post to discuss the
uniqueness aspect of an absolute limit. By not stopping at just odds
or primes, I'm saying that I am interested in the full list of integer
ratios 8 and less (in that example), thus eliminating the entire
discussion of consistency.

The chart I'd like to see would be a simple set of lists. It would
list all ET's that give 6-limit (no concern for ratios with any
integer >6), 7-limit, 8-limit, etc. maybe up to 32. So discussion of
81/84 being unique would not be in this list.

My thinking is that harmony is much more affected by subtle tuning
than melody, so... I could allow in an adaptive tuning for all sorts
of melodic error as long as the harmonies were good. I don't believe
that harmonies beyond even 20-something are consonant enough to give
an audibly special effect as a diad in most timbres. Sure, adding up
a bunch of higher harmonies that all belong to a series works well,
but that is a special circumstance with no limit at all.
So, for instance, I could use a 5-limit adaptive algorithm with 12ET,
and I would be able to be anywhere in the system and specify the
5-limit harmony I choose. Hence, Hermode tuning works for 5-limit.
If we go to 7-limit, then 12ET fails to be unique - there is no way to
tell a 7-limit algorithm that I want 7/6 vs 6/5 using a 12ET note-set.

So one question: what is the list of ET's that is 7-limit unique? The
chart you posted first seems to tell me that 27 31 41 58 80 94
282 & 311 are all included... but aren't there others in between these
as well, or are they only these and their multiples?

-Aaron W

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@> wrote:
> >
> > Carl,
> >
> > That's a great chart of Paul's, thanks. Has anyone made an
> > extension of this that continues past the smallest ET? I'm sure
> > that while 41 may have uniqueness up to 9-limit, not all ET's
> > larger than 41 do.
> > Knowing which (up to maybe this same high ET or 311 or so) would
> > be interesting.
> > Oh, and I think I sorta get it, but what does consistent versus
> > unique mean here?
> >
> > -Aaron W.
>
> Consistency is defined here:
>
> http://library.wustl.edu/~manynote/music.html
>
> -Carl
>

Dear Aaron,

Just to join in, that is a question I would like to see an answer for too. Which equal temperament distinguishes which ratio approximations or more generally which limits? So, which equal temperament can I use to compose in a certain limit when I later want change the tuning and fine-tune the intonation (e.g., by an adaptive tuning) without loosing the "harmonic meaning" of the notes.

Best
Torsten

On Dec 9, 2008, at 2:22 PM, Aaron Wolf wrote:

> Ok, that's what I thought consistency meant. I got it.
>
> However, these charts and the other aren't fully what I was after.
>
> I used terms like "8-limit" in my initial post to discuss the
> uniqueness aspect of an absolute limit. By not stopping at just odds
> or primes, I'm saying that I am interested in the full list of integer
> ratios 8 and less (in that example), thus eliminating the entire
> discussion of consistency.
>
> The chart I'd like to see would be a simple set of lists. It would
> list all ET's that give 6-limit (no concern for ratios with any
> integer >6), 7-limit, 8-limit, etc. maybe up to 32. So discussion of
> 81/84 being unique would not be in this list.
>
> My thinking is that harmony is much more affected by subtle tuning
> than melody, so... I could allow in an adaptive tuning for all sorts
> of melodic error as long as the harmonies were good. I don't believe
> that harmonies beyond even 20-something are consonant enough to give
> an audibly special effect as a diad in most timbres. Sure, adding up
> a bunch of higher harmonies that all belong to a series works well,
> but that is a special circumstance with no limit at all.
> So, for instance, I could use a 5-limit adaptive algorithm with 12ET,
> and I would be able to be anywhere in the system and specify the
> 5-limit harmony I choose. Hence, Hermode tuning works for 5-limit.
> If we go to 7-limit, then 12ET fails to be unique - there is no way to
> tell a 7-limit algorithm that I want 7/6 vs 6/5 using a 12ET note-set.
>
> So one question: what is the list of ET's that is 7-limit unique? The
> chart you posted first seems to tell me that 27 31 41 58 80 94
> 282 & 311 are all included... but aren't there others in between these
> as well, or are they only these and their multiples?
>
> -Aaron W
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@> wrote:
> > >
> > > Carl,
> > >
> > > That's a great chart of Paul's, thanks. Has anyone made an
> > > extension of this that continues past the smallest ET? I'm sure
> > > that while 41 may have uniqueness up to 9-limit, not all ET's
> > > larger than 41 do.
> > > Knowing which (up to maybe this same high ET or 311 or so) would
> > > be interesting.
> > > Oh, and I think I sorta get it, but what does consistent versus
> > > unique mean here?
> > >
> > > -Aaron W.
> >
> > Consistency is defined here:
> >
> > http://library.wustl.edu/~manynote/music.html
> >
> > -Carl
> >
>
>
>

--- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@...> wrote:
> I am interested in the full list of integer
> ratios 8 and less (in that example),

Then you want 7-odd-limit.

> thus eliminating the entire discussion of consistency.

??

> The chart I'd like to see would be a simple set of lists. It would
> list all ET's that give 6-limit (no concern for ratios with any
> integer >6), 7-limit, 8-limit, etc. maybe up to 32.

Give 6-limit... uniqueness? That's an infinite list. Anyway,
if you're interested in adaptive tuning, wouldn't you just want
the smallest number of notes that can uniquely specify the
intervals you're interested in? There'd be no reason to go
higher, since the adaptive tuning would be taking care of
the fine tuning anyway.

> So discussion of 81/84 being unique would not be in this list.

81/84 is a 7-prime-limit ratio and an 81-odd-limit ratio.

> I don't believe that harmonies beyond even 20-something are
> consonant enough to give an audibly special effect as a diad
> in most timbres. Sure, adding up a bunch of higher harmonies
> that all belong to a series works well, but that is a special
> circumstance

I agree.

> Hence, Hermode tuning works for 5-limit.

Yup. So does 9-ET. :)

> So one question: what is the list of ET's that is
> 7-limit unique?

It's an infinite list.

> The chart you posted first seems to tell me that
> 27 31 41 58 80 94 282 & 311 are all included... but
> aren't there others in between these as well, or are they
> only these and their multiples?

Others as well, if you don't care about consistency.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@> wrote:
> > I am interested in the full list of integer
> > ratios 8 and less (in that example),
>
> Then you want 7-odd-limit.
>
> > thus eliminating the entire discussion of consistency.
>
> ??
>
> > The chart I'd like to see would be a simple set of lists. It would
> > list all ET's that give 6-limit (no concern for ratios with any
> > integer >6), 7-limit, 8-limit, etc. maybe up to 32.
>
> Give 6-limit... uniqueness? That's an infinite list. Anyway,
> if you're interested in adaptive tuning, wouldn't you just want
> the smallest number of notes that can uniquely specify the
> intervals you're interested in? There'd be no reason to go
> higher, since the adaptive tuning would be taking care of
> the fine tuning anyway.
>

Well, the smallest ET with a given uniqueness is DEFINITELY useful
info, but if I were to consider key layout or notation, there might be
some options that offer more potential symmetry. Or, perhaps a larger
ET would allow for some intentional dissonance that goes against the
general limit. But to consider those options, I need to limit myself
to at least those ET's that do have the uniqueness.

For example, for a 5-limit adaptive piece, 12ET works, but maybe I'd
like 24ET for intentional dissonance without losing the ability to
clearly indicate 5-limit harmony. The fact that 12ET works doesn't
mean 13 and 14 and 15 all do, so in deciding on something for 5-limit
beyond 12ET (again, in order to deviate from the 5-limit system
without concern for just ratios in that instance), I'd want to know
what the other options are still 5-limit unique.

I wouldn't want to go into infinite listings... I think my ideal list
stops somewhere in the lower hundreds, like not consider higher than
300 maybe.

> > So discussion of 81/84 being unique would not be in this list.
>
> 81/84 is a 7-prime-limit ratio and an 81-odd-limit ratio.
>

That was a typo, I meant 81/64 syntonic comma, and I was essentially
saying I care about odd-limits and not prime limits.

> > I don't believe that harmonies beyond even 20-something are
> > consonant enough to give an audibly special effect as a diad
> > in most timbres. Sure, adding up a bunch of higher harmonies
> > that all belong to a series works well, but that is a special
> > circumstance
>
> I agree.
>
> > Hence, Hermode tuning works for 5-limit.
>
> Yup. So does 9-ET. :)
>
> > So one question: what is the list of ET's that is
> > 7-limit unique?
>
> It's an infinite list.
>

Ok, but what is the list of those ETs less than 100, for instance?

> > The chart you posted first seems to tell me that
> > 27 31 41 58 80 94 282 & 311 are all included... but
> > aren't there others in between these as well, or are they
> > only these and their multiples?
>
> Others as well, if you don't care about consistency.
>

I don't care about consistency.

Is there any obvious pattern to the others that are in there besides
the obvious multiples or would I have to calculate them all to
discover the list?

> -Carl
>

Thanks, Carl for your help!

Best,
Aaron

--- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@...> wrote:

> I wouldn't want to go into infinite listings... I think my ideal
> list stops somewhere in the lower hundreds, like not consider
> higher than 300 maybe.

Howabout 200?

5 3
6 3
7 3
8 3
9 5
10 3
11 5
12 5
13 5
14 5
15 5
16 5
17 5
18 5
19 5
20 5
21 7
22 5
23 7
24 5
25 7
26 5
27 7
28 5
29 5
30 5
31 7
32 5
33 7
34 5
35 9
36 9
37 7
38 7
39 9
40 9
41 9
42 9
43 7
44 7
45 9
46 9
47 9
48 9
49 9
50 7
51 9
52 9
53 9
54 9
55 9
56 11
57 11
58 11
59 9
60 9
61 9
62 9
63 11
64 11
65 11
66 11
67 11
68 9
69 9
70 13
71 11
72 11
73 11
74 11
75 11
76 9
77 13
78 11
79 13
80 11
81 11
82 11
83 11
84 11
85 11
86 11
87 13
88 13
89 11
90 11
91 11
92 13
93 13
94 13
95 11
96 13
97 13
98 13
99 11
100 15
101 15
102 15
103 13
104 13
105 13
106 13
107 13
108 11
109 15
110 15
111 15
112 15
113 13
114 13
115 13
116 13
117 15
118 15
119 15
120 15
121 15
122 15
123 15
124 13
125 13
126 13
127 17
128 15
129 15
130 15
131 15
132 15
133 15
134 15
135 13
136 15
137 17
138 15
139 17
140 15
141 15
142 15
143 15
144 15
145 13
146 15
147 17
148 15
149 17
150 15
151 17
152 15
153 15
154 15
155 15
156 15
157 17
158 15
159 17
160 15
161 17
162 15
163 17
164 15
165 15
166 15
167 19
168 15
169 17
170 15
171 17
172 15
173 17
174 15
175 17
176 15
177 15
178 15
179 19
180 15
181 19
182 15
183 17
184 15
185 17
186 15
187 17
188 15
189 17
190 15
191 19
192 15
193 19
194 15
195 19
196 15
197 17
198 15
199 17
200 17

(Due to Graham Breed.)

> That was a typo, I meant 81/64 syntonic comma, and I was essentially
> saying I care about odd-limits and not prime limits.

Good, because that's the way uniqueness works. If it were prime
limit, no temperament could even be 3-limit unique, since prime
limits are infinite.

-Carl

I wrote:
> no temperament could even be 3-limit unique, since prime
> limits are infinite.

Equal temperament, anyhow. I suppose a P-limit linear temperament,
whose commas lack factors of p < P, would indeed be p-limit unique.
Z'that right, Graham?

-Carl

2008/12/12 Carl Lumma <carl@...>:
> I wrote:
>> no temperament could even be 3-limit unique, since prime
>> limits are infinite.
>
> Equal temperament, anyhow. I suppose a P-limit linear temperament,
> whose commas lack factors of p < P, would indeed be p-limit unique.
> Z'that right, Graham?

By some definitions of "temperament" it wouldn't qualify if it's
unique in that way. Pretty good definitions, in fact, like the group
theoretic one of being a homomorphism. You aren't really tempering
unless you make the system simpler as a result. A 5-limit temperament
can be unique in the 3-limit, though, as is the case with meantone and
schismatic.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Wolf" <aaron@> wrote:
>
> > I wouldn't want to go into infinite listings... I think my ideal
> > list stops somewhere in the lower hundreds, like not consider
> > higher than 300 maybe.
>
> Howabout 200?
>
> 5 3
> 6 3
> 7 3
> 8 3
> 9 5
> 10 3
> 11 5
> 12 5
> 13 5
> 14 5
> 15 5
> 16 5
> 17 5
> 18 5
> 19 5
> 20 5
> 21 7
> 22 5
> 23 7
> 24 5
> 25 7
> 26 5
> 27 7
> 28 5
> 29 5
> 30 5
> 31 7
> 32 5
> 33 7
> 34 5
> 35 9
> 36 9
> 37 7
> 38 7
> 39 9
> 40 9
> 41 9
> 42 9
> 43 7
> 44 7
> 45 9
> 46 9
> 47 9
> 48 9
> 49 9
> 50 7
> 51 9
> 52 9
> 53 9
> 54 9
> 55 9
> 56 11
> 57 11
> 58 11
> 59 9
> 60 9
> 61 9
> 62 9
> 63 11
> 64 11
> 65 11
> 66 11
> 67 11
> 68 9
> 69 9
> 70 13
> 71 11
> 72 11
> 73 11
> 74 11
> 75 11
> 76 9
> 77 13
> 78 11
> 79 13
> 80 11
> 81 11
> 82 11
> 83 11
> 84 11
> 85 11
> 86 11
> 87 13
> 88 13
> 89 11
> 90 11
> 91 11
> 92 13
> 93 13
> 94 13
> 95 11
> 96 13
> 97 13
> 98 13
> 99 11
> 100 15
> 101 15
> 102 15
> 103 13
> 104 13
> 105 13
> 106 13
> 107 13
> 108 11
> 109 15
> 110 15
> 111 15
> 112 15
> 113 13
> 114 13
> 115 13
> 116 13
> 117 15
> 118 15
> 119 15
> 120 15
> 121 15
> 122 15
> 123 15
> 124 13
> 125 13
> 126 13
> 127 17
> 128 15
> 129 15
> 130 15
> 131 15
> 132 15
> 133 15
> 134 15
> 135 13
> 136 15
> 137 17
> 138 15
> 139 17
> 140 15
> 141 15
> 142 15
> 143 15
> 144 15
> 145 13
> 146 15
> 147 17
> 148 15
> 149 17
> 150 15
> 151 17
> 152 15
> 153 15
> 154 15
> 155 15
> 156 15
> 157 17
> 158 15
> 159 17
> 160 15
> 161 17
> 162 15
> 163 17
> 164 15
> 165 15
> 166 15
> 167 19
> 168 15
> 169 17
> 170 15
> 171 17
> 172 15
> 173 17
> 174 15
> 175 17
> 176 15
> 177 15
> 178 15
> 179 19
> 180 15
> 181 19
> 182 15
> 183 17
> 184 15
> 185 17
> 186 15
> 187 17
> 188 15
> 189 17
> 190 15
> 191 19
> 192 15
> 193 19
> 194 15
> 195 19
> 196 15
> 197 17
> 198 15
> 199 17
> 200 17
>
> (Due to Graham Breed.)
>
> > That was a typo, I meant 81/64 syntonic comma, and I was essentially
> > saying I care about odd-limits and not prime limits.
>
> Good, because that's the way uniqueness works. If it were prime
> limit, no temperament could even be 3-limit unique, since prime
> limits are infinite.
>
> -Carl
>

Woah, thanks. That was just what I was thinking of!
-Aaron

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> >> no temperament could even be 3-limit unique, since prime
> >> limits are infinite.
> >
> > Equal temperament, anyhow. I suppose a P-limit linear
> > temperament, whose commas lack factors of p < P, would
> > indeed be p-limit unique. Z'that right, Graham?
>
> By some definitions of "temperament" it wouldn't qualify if it's
> unique in that way. Pretty good definitions, in fact, like the
> group theoretic one of being a homomorphism. You aren't really
> tempering unless you make the system simpler as a result. A
> 5-limit temperament can be unique in the 3-limit, though, as is
> the case with meantone and schismatic.
>
> Graham

That's what I was trying to say, except I got the bit about
the commas wrong. Meantone's kernel contains 81/80, and that
has factors of 3 and 5 (not just 5 as I was thinking).

-Carl

[ Attachment content not displayed ]

[ Attachment content not displayed ]

[ Attachment content not displayed ]

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, "Marcel de Velde" <m.develde@...> wrote:
>
> Won't work in any equal temperament that'll fit the midi
> standard 128 notes.

That's not a limitation for two reasons:

1. Channel split. One can use more than one channel at
a time.

2. MIDI retune messages. See the MIDI Tuning Standard,
midi.org.

-Carl

On Dec 13, 2008, at 1:14 AM, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Marcel de Velde" <m.develde@...> > wrote:
> >
> > Won't work in any equal temperament that'll fit the midi
> > standard 128 notes.
>
> That's not a limitation for two reasons:
>
> 1. Channel split. One can use more than one channel at
> a time.
>
> 2. MIDI retune messages. See the MIDI Tuning Standard,
> midi.org.
>

Also, some MIDI sound generators provide programming capabilities where users can define how the tuning changes depending on any input message.

Moreover, why limit yourself to MIDI. Sound synthesis languages or music notation is not limited to any finite number of pitches.

Best
Torsten

On Dec 13, 2008, at 1:30 AM, Torsten Anders wrote:
> music notation is not limited to any finite number of pitches.

Oops, I actually meant "acoustic instruments are not limited to any finite number of pitches". Its rather late here too..

Best
Torsten

--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:

> Moreover, why limit yourself to MIDI. Sound synthesis languages or
> music notation is not limited to any finite number of pitches.

Indeed. -Carl