Full thread (26 messages)
From: Carl Lumma (2008-05-10)
Subject: harmonic series rwt?
Kalle, if you've got your ears on, can you tell us the
lowest subset of harmonics without harmonic waste when
interpreted as a WT *that has a mode in which all the
denominators are base-2 numbers*?
I assume it may be possible to do this with 128-256, as
your unrestricted result was done with 116-232.
Failing that, 256-512 must have enough resources.
Thanks!
-Carl
From: Kraig Grady (2008-05-10)
Subject: Re: [tuning] harmonic series rwt?
Helmholtz started at 32. no sense referring all 12 to a single harmonic
series, it just destroys the fell of modulation.
Personally when i modulate i want to modulate!
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Carl Lumma wrote:
>
> Kalle, if you've got your ears on, can you tell us the
> lowest subset of harmonics without harmonic waste when
> interpreted as a WT *that has a mode in which all the
> denominators are base-2 numbers*?
>
> I assume it may be possible to do this with 128-256, as
> your unrestricted result was done with 116-232.
> Failing that, 256-512 must have enough resources.
>
> Thanks!
>
> -Carl
>
>
From: Kalle Aho (2008-05-11)
Subject: Re: harmonic series rwt?
--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> Kalle, if you've got your ears on, can you tell us the
> lowest subset of harmonics without harmonic waste when
> interpreted as a WT *that has a mode in which all the
> denominators are base-2 numbers*?
>
> I assume it may be possible to do this with 128-256, as
> your unrestricted result was done with 116-232.
> Failing that, 256-512 must have enough resources.
>
> Thanks!
Hi Carl,
actually there are four of them!
128:136:144:152:162:171:182:192:203:216:228:243:256
128:136:144:152:162:171:182:192:204:216:228:243:256
128:136:144:153:162:171:182:192:204:216:228:243:256
128:136:144:153:162:171:182:192:204:216:229:243:256
Here are Scala files for these:
! ForCarl1.scl
!
12
!
17/16
9/8
19/16
81/64
171/128
91/64
3/2
203/128
27/16
57/32
243/128
2/1
! ForCarl2.scl
!
12
!
17/16
9/8
19/16
81/64
171/128
91/64
3/2
51/32
27/16
57/32
243/128
2/1
! ForCarl3.scl
!
12
!
17/16
9/8
153/128
81/64
171/128
91/64
3/2
51/32
27/16
57/32
243/128
2/1
! ForCarl4.scl
!
12
!
17/16
9/8
153/128
81/64
171/128
91/64
3/2
51/32
27/16
229/128
243/128
2/1
From: Carl Lumma (2008-05-11)
Subject: Re: harmonic series rwt?
> > Kalle, if you've got your ears on, can you tell us the
> > lowest subset of harmonics without harmonic waste when
> > interpreted as a WT *that has a mode in which all the
> > denominators are base-2 numbers*?
> >
> > I assume it may be possible to do this with 128-256, as
> > your unrestricted result was done with 116-232.
> > Failing that, 256-512 must have enough resources.
> >
> > Thanks!
>
> Hi Carl,
>
> actually there are four of them!
>
> 128:136:144:152:162:171:182:192:203:216:228:243:256
> 128:136:144:152:162:171:182:192:204:216:228:243:256
> 128:136:144:153:162:171:182:192:204:216:228:243:256
> 128:136:144:153:162:171:182:192:204:216:229:243:256
Thanks!
> Here are Scala files for these:
D'oh! Howabout with the constraint that all 3rds
be < 405cents? (Your original scale happened to
meet this.)
-Carl
From: Kalle Aho (2008-05-11)
Subject: Re: harmonic series rwt?
--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> > > Kalle, if you've got your ears on, can you tell us the
> > > lowest subset of harmonics without harmonic waste when
> > > interpreted as a WT *that has a mode in which all the
> > > denominators are base-2 numbers*?
> > >
> > > I assume it may be possible to do this with 128-256, as
> > > your unrestricted result was done with 116-232.
> > > Failing that, 256-512 must have enough resources.
> > >
> > > Thanks!
> >
> > Hi Carl,
> >
> > actually there are four of them!
> >
> > 128:136:144:152:162:171:182:192:203:216:228:243:256
> > 128:136:144:152:162:171:182:192:204:216:228:243:256
> > 128:136:144:153:162:171:182:192:204:216:228:243:256
> > 128:136:144:153:162:171:182:192:204:216:229:243:256
>
> Thanks!
>
> > Here are Scala files for these:
>
> D'oh! Howabout with the constraint that all 3rds
> be < 405cents? (Your original scale happened to
> meet this.)
Well, there are 25 of those in 256-512, the best (largest major third
is 402.844 cents) is this:
! ForCarl5.scl
!
12
!
271/256
287/256
19/16
161/128
171/128
181/128
383/256
203/128
215/128
57/32
483/256
2/1
From: Kraig Grady (2008-05-11)
Subject: Re: [tuning] Re: harmonic series rwt?
there is also the subharmonic versions
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>, "Carl
> Lumma" <carl@...> wrote:
> >
> > > > Kalle, if you've got your ears on, can you tell us the
> > > > lowest subset of harmonics without harmonic waste when
> > > > interpreted as a WT *that has a mode in which all the
> > > > denominators are base-2 numbers*?
> > > >
> > > > I assume it may be possible to do this with 128-256, as
> > > > your unrestricted result was done with 116-232.
> > > > Failing that, 256-512 must have enough resources.
> > > >
> > > > Thanks!
> > >
> > > Hi Carl,
> > >
> > > actually there are four of them!
> > >
> > > 128:136:144:152:162:171:182:192:203:216:228:243:256
> > > 128:136:144:152:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:229:243:256
> >
> > Thanks!
> >
> > > Here are Scala files for these:
> >
> > D'oh! Howabout with the constraint that all 3rds
> > be < 405cents? (Your original scale happened to
> > meet this.)
>
> Well, there are 25 of those in 256-512, the best (largest major third
> is 402.844 cents) is this:
>
> ! ForCarl5.scl
> !
>
> 12
> !
> 271/256
> 287/256
> 19/16
> 161/128
> 171/128
> 181/128
> 383/256
> 203/128
> 215/128
> 57/32
> 483/256
> 2/1
>
>
From: Kraig Grady (2008-05-12)
Subject: Re: [tuning] Re: harmonic series rwt?
actually if you look at page 7 of
anaphoria.com/meantone-mavila.PDF
there is a recurrent sequence that although does not approach equal,
does produce a 12 tone meantone scale, in fact if you take it up to 19
places you haven't gone that much further
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>, "Carl
> Lumma" <carl@...> wrote:
> >
> > > > Kalle, if you've got your ears on, can you tell us the
> > > > lowest subset of harmonics without harmonic waste when
> > > > interpreted as a WT *that has a mode in which all the
> > > > denominators are base-2 numbers*?
> > > >
> > > > I assume it may be possible to do this with 128-256, as
> > > > your unrestricted result was done with 116-232.
> > > > Failing that, 256-512 must have enough resources.
> > > >
> > > > Thanks!
> > >
> > > Hi Carl,
> > >
> > > actually there are four of them!
> > >
> > > 128:136:144:152:162:171:182:192:203:216:228:243:256
> > > 128:136:144:152:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:229:243:256
> >
> > Thanks!
> >
> > > Here are Scala files for these:
> >
> > D'oh! Howabout with the constraint that all 3rds
> > be < 405cents? (Your original scale happened to
> > meet this.)
>
> Well, there are 25 of those in 256-512, the best (largest major third
> is 402.844 cents) is this:
>
> ! ForCarl5.scl
> !
>
> 12
> !
> 271/256
> 287/256
> 19/16
> 161/128
> 171/128
> 181/128
> 383/256
> 203/128
> 215/128
> 57/32
> 483/256
> 2/1
>
>
From: George D. Secor (2008-05-12)
Subject: Re: harmonic series rwt?
--- In [email protected], "Kalle Aho" <kalleaho@...> wrote:
>
> --- In [email protected], "Carl Lumma" <carl@> wrote:
> >
> > > > Kalle, if you've got your ears on, can you tell us the
> > > > lowest subset of harmonics without harmonic waste when
> > > > interpreted as a WT *that has a mode in which all the
> > > > denominators are base-2 numbers*?
> > > >
> > > > I assume it may be possible to do this with 128-256, as
> > > > your unrestricted result was done with 116-232.
> > > > Failing that, 256-512 must have enough resources.
> > > >
> > > > Thanks!
> > >
> > > Hi Carl,
> > >
> > > actually there are four of them!
> > >
> > > 128:136:144:152:162:171:182:192:203:216:228:243:256
> > > 128:136:144:152:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:229:243:256
> >
> > Thanks!
> >
> > > Here are Scala files for these:
> >
> > D'oh! Howabout with the constraint that all 3rds
> > be < 405cents? (Your original scale happened to
> > meet this.)
>
> Well, there are 25 of those in 256-512, the best (largest major
third
> is 402.844 cents) is this:
>
> ! ForCarl5.scl
> !
>
> 12
> !
> 271/256
> 287/256
> 19/16
> 161/128
> 171/128
> 181/128
> 383/256
> 203/128
> 215/128
> 57/32
> 483/256
> 2/1
Hi Carl & Kalle,
I can't figure out whether Carl's looking for a "real" WT (something
that not only has simple brats, but also has good harmonic balance,
therefore making it musically useful) or if you fellas are just doing
this for fun (otherwise, why the requirement that the denominator be
a power of 2?).
Just in case it's the former, then I think that you're making the
constraints too tight.
I actually had an ulterior motive for answering this. ;-) I happened
to find an alternative to my Victorian rational well-temperament,
having much lower numbers, so as to give reasonably simple brats (no
more complicated, overall, than those in ForCarl5.scl) for all 24
major & minor triads. I've labeled it "24e":
! SecorVRWT-24e.scl
!
George Secor's 24-triad proportional-beating Victorian well-
temperament (24e), based on Ellis #2
12
!
581/550
28/25
653/550
345/275
367/275
31/22
412/275
871/550
461/275
89/50
47/25
2/1
It turns out that this one is closer to Ellis #2 than my previous
VRWT. Like Ellis's #2, it has no just fifths, and all of the thirds
are <405c.
--George
From: Kalle Aho (2008-05-12)
Subject: Re: harmonic series rwt?
--- In [email protected], "George D. Secor" <gdsecor@...> wrote:
> Hi Carl & Kalle,
>
> I can't figure out whether Carl's looking for a "real" WT (something
> that not only has simple brats, but also has good harmonic balance,
> therefore making it musically useful) or if you fellas are just doing
> this for fun
Are you implying that as such my suggestions are not musically useful?
If so, I disagree. They have other properties than just simple brats.
For one, one could choose the absolute tuning so that the fundamental
of the series coincides with the desired tempo of the music. The
beating patterns of all possible chords would be in a synchronous
relationship to the tempo.
But I agree with you that it's a stretch to call these
well-temperaments. Some other name should be coined.
> (otherwise, why the requirement that the denominator be
> a power of 2?).
I don't know, Carl should elaborate on this.
Kalle
From: Kraig Grady (2008-05-12)
Subject: Re: [tuning] Re: harmonic series rwt?
I see, i wasn't seeing what was being gone after
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>,
> "George D. Secor" <gdsecor@...> wrote:
>
> The
> beating patterns of all possible chords would be in a synchronous
> relationship to the tempo.
>
>
>
From: Carl Lumma (2008-05-12)
Subject: Re: harmonic series rwt?
> But I agree with you that it's a stretch to call these
> well-temperaments. Some other name should be coined.
I don't agree with that.
> > (otherwise, why the requirement that the denominator be
> > a power of 2?).
>
> I don't know, Carl should elaborate on this.
Just a diversion. Robert Martin mentioned on tuning-math
that he sometimes works with a subset of 128-256. Kalle
just showed that there's no Victorian WT in that segment
of the harmonic series without harmonic waste.
-Carl
From: George D. Secor (2008-05-13)
Subject: Re: harmonic series rwt?
--- In [email protected], "Kalle Aho" <kalleaho@...> wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@> wrote:
>
> > Hi Carl & Kalle,
> >
> > I can't figure out whether Carl's looking for a "real" WT
(something
> > that not only has simple brats, but also has good harmonic
balance,
> > therefore making it musically useful) or if you fellas are just
doing
> > this for fun
>
> Are you implying that as such my suggestions are not musically
useful?
No, sorry if I gave that impression. I just couldn't figure out how
that might be useful.
> If so, I disagree. They have other properties than just simple
brats.
> For one, one could choose the absolute tuning so that the
fundamental
> of the series coincides with the desired tempo of the music. The
> beating patterns of all possible chords would be in a synchronous
> relationship to the tempo.
Okay, I see. That sounds similar to the idea of having a vibrato
rate that's in a simple numerical relationship to the tempo.
> But I agree with you that it's a stretch to call these
> well-temperaments. Some other name should be coined.
>
> > (otherwise, why the requirement that the denominator be
> > a power of 2?).
>
> I don't know, Carl should elaborate on this.
Yep, looks like Carl's just having fun. ;-)
--George
From: Jacques Dudon (2008-05-14)
Subject: [tuning] Re: harmonic series rwt?
le 12/05/08 23:34, Kalle Aho à kalleaho@... a écrit :
They have other properties than just simple brats.
For one, one could choose the absolute tuning so that the fundamental
of the series coincides with the desired tempo of the music. The
beating patterns of all possible chords would be in a synchronous
relationship to the tempo.
I must say that's what I can't escape with my photosonic disks. It does not
bother me at all, and I often play with it.
Depending on the instrument I would see other qualities, such as the
possibility to have differential tones coherent within the scale lower
octaves - what I call "differential coherence"
(with photosonic disks it is easy to make differential tones perfectly
audible).
In this example you have many coherent minor thirds such as 203 - 171 = 32,
216 - 182 = 34,
182 - 144 = 38 or 171 - 144 = 27 etc.
But I agree with you that it's a stretch to call these
well-temperaments. Some other name should be coined.
Perhaps many on this list will faint, but I've been calling these for years
"harmonic temperaments"... (because defective from harmonic series), but
only if they can fonction as some sort of unequal temperaments.
I've done harmonic meantones, harmonic well-temperaments, harmonic
quartertones scales, etc.
I can also approach equal temperaments of any kind, but I do not call them
however "harmonic equal temperaments", but "almost equal"... ;-)
At a quick glance your examples could be well temperaments, why not ? but
then 128 might not be the best key to refer to.
Except for the tempo, that's true.
- - - - - - - - - - -
Jacques Dudon
From: Kalle Aho (2008-05-14)
Subject: Re: harmonic series rwt?
Hi Jacques and everyone else,
--- In [email protected], Jacques Dudon <fotosonix@...> wrote:
> Perhaps many on this list will faint, but I've been calling these
for years
> "harmonic temperaments"... (because defective from harmonic series)
Great, that makes sense!
> , but
> only if they can fonction as some sort of unequal temperaments.
> I've done harmonic meantones, harmonic well-temperaments, harmonic
> quartertones scales, etc.
> I can also approach equal temperaments of any kind, but I do not
call them
> however "harmonic equal temperaments", but "almost equal"... ;-)
How about "harmonic quasi-equal temperaments"?
> At a quick glance your examples could be well temperaments, why not ?
Well, they have no harmonic waste (like traditional well temperaments)
but their key structure is not very traditional. But maybe that's not
essential. At least Scala claims they are well-temperaments.
> but
> then 128 might not be the best key to refer to.
> Except for the tempo, that's true.
Here
http://launch.groups.yahoo.com/group/tuning/message/75816
I describe the lowest harmonic well-temperament which starts with 116.
BTW the 7-limit 22-tone tuning discussed there is not the lowest one.
The lowest 22-tone harmonic temperament with no 7-limit harmonic waste is
124:128:132:136:141:145:150:155:160:165:170:
176:181:187:193:199:206:212:219:226:233:241
!
22
!
32/31
33/31
34/31
141/124
145/124
75/62
5/4
40/31
165/124
85/62
44/31
181/124
187/124
193/124
199/124
103/62
53/31
219/124
113/62
233/124
241/124
2/1
Kalle Aho
From: Kraig Grady (2008-05-14)
Subject: Re: [tuning] Re: harmonic series rwt?
it does seem recurrent sequences in general do the best job if you wants
beats in tune wih tempo. the harmonic series will produce related ones
but these can be of complicated subdivisions.
One could use these series as the basis of clusters ala Ligeti, except
one would now have defined some harmonic area allowing one to modulate
to other 'keys' (by modulating the actually series) within the same
range even.
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Jacques Dudon wrote:
>
> le 12/05/08 23:34, Kalle Aho \ufffd kalleaho@... a \ufffdcrit :
>
> They have other properties than just simple brats.
> For one, one could choose the absolute tuning so that the fundamental
> of the series coincides with the desired tempo of the music. The
> beating patterns of all possible chords would be in a synchronous
> relationship to the tempo.
>
> I must say that's what I can't escape with my photosonic disks. It
> does not bother me at all, and I often play with it.
> Depending on the instrument I would see other qualities, such as the
> possibility to have differential tones coherent within the scale lower
> octaves - what I call "differential coherence"
> (with photosonic disks it is easy to make differential tones perfectly
> audible).
> In this example you have many coherent minor thirds such as 203 - 171
> = 32, 216 - 182 = 34,
> 182 - 144 = 38 or 171 - 144 = 27 etc.
>
>
>
> But I agree with you that it's a stretch to call these
> well-temperaments. Some other name should be coined.
>
> Perhaps many on this list will faint, but I've been calling these for
> years "harmonic temperaments"... (because defective from harmonic
> series), but only if they can fonction as some sort of unequal
> temperaments.
> I've done harmonic meantones, harmonic well-temperaments, harmonic
> quartertones scales, etc.
> I can also approach equal temperaments of any kind, but I do not call
> them however "harmonic equal temperaments", but "almost equal"... ;-)
> At a quick glance your examples could be well temperaments, why not ?
> but then 128 might not be the best key to refer to.
> Except for the tempo, that's true.
>
> - - - - - - - - - - -
> Jacques Dudon
>
>
From: Kalle Aho (2008-05-15)
Subject: Re: harmonic series rwt?
--- In [email protected], Kraig Grady <kraiggrady@...> wrote:
>
> it does seem recurrent sequences in general do the best job if you
wants
> beats in tune wih tempo. the harmonic series will produce related ones
> but these can be of complicated subdivisions.
>
> One could use these series as the basis of clusters ala Ligeti, except
> one would now have defined some harmonic area allowing one to
modulate
> to other 'keys' (by modulating the actually series) within the same
> range even.
Hi Kraig,
can you show some examples of scales produced by recurrent sequences?
Kalle
From: Kraig Grady (2008-05-16)
Subject: Re: [tuning] Re: harmonic series rwt?
I need to redo my introduction to these scales and will soon as it seems
it might be useful. forthcoming.
but my post was a bit confusing, i was referring to your scales being
able to be used for clusters in this way more than recurrent sequences.
Erv has some of his examples listed under the the section entitled
The Scales of Mt. Meru
under the Wilson Archives
http://anaphoria.com/wilson.html
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>, Kraig
> Grady <kraiggrady@...> wrote:
> >
> > it does seem recurrent sequences in general do the best job if you
> wants
> > beats in tune wih tempo. the harmonic series will produce related ones
> > but these can be of complicated subdivisions.
> >
> > One could use these series as the basis of clusters ala Ligeti, except
> > one would now have defined some harmonic area allowing one to
> modulate
> > to other 'keys' (by modulating the actually series) within the same
> > range even.
>
> Hi Kraig,
>
> can you show some examples of scales produced by recurrent sequences?
>
> Kalle
>
>
From: Jacques Dudon (2008-05-16)
Subject: [tuning] Re: harmonic series rwt?
le 14/05/08 23:10, Kalle Aho à kalleaho@... a écrit :
Hi Jacques and everyone else,
--- In [email protected] <mailto:tuning%40yahoogroups.com> , Jacques
Dudon <fotosonix@...> wrote:
> Perhaps many on this list will faint, but I've been calling these for years
> "harmonic temperaments"... (because defective from harmonic series)
Great, that makes sense!
> , but only if they can fonction as some sort of unequal temperaments.
> I've done harmonic meantones, harmonic well-temperaments, harmonic
> quartertones scales, etc.
> I can also approach equal temperaments of any kind, but I do not
> call them however "harmonic equal temperaments", but "almost equal"... ;-)
How about "harmonic quasi-equal temperaments"?
Very good, sounds better indeed. I buy it !
> At a quick glance your examples could be well temperaments, why not ?
Well, they have no harmonic waste (like traditional well temperaments)
but their key structure is not very traditional. But maybe that's not
essential. At least Scala claims they are well-temperaments.
I understand absence of harmonic waste in a 12 tones temperament as having
no fifths intervals wider than 3/2, and no major thirds intervals smaller
than 5/4 (for octavial temperaments). Is that right, or can you or someone
tell me how we should define it ?
For any temperament do you define it always for some harmonic limit, or
could it be special choices of harmonics ?
For example, here is a "harmonic "quasi-equal" 5 tones temperament" :
35:40:46:53:61
is it the lowest 5-tones harmonic temperament with no harmonic waste ?
> but
> then 128 might not be the best key to refer to.
> Except for the tempo, that's true.
Here
http://launch.groups.yahoo.com/group/tuning/message/75816
I describe the lowest harmonic well-temperament which starts with 116.
Well done !
With your permission I shall make a photosonic disk
it confirms my feelings towards 29 as being a very useful prime.
BTW the 7-limit 22-tone tuning discussed there is not the lowest one.
The lowest 22-tone harmonic temperament with no 7-limit harmonic waste is
124:128:132:136:141:145:150:155:160:165:170:
176:181:187:193:199:206:212:219:226:233:241
!
22
!
32/31
33/31
34/31
141/124
145/124
75/62
5/4
40/31
165/124
85/62
44/31
181/124
187/124
193/124
199/124
103/62
53/31
219/124
113/62
233/124
241/124
2/1
Kalle Aho
From: Kalle Aho (2008-05-16)
Subject: Re: harmonic series rwt?
--- In [email protected], Jacques Dudon <fotosonix@...> wrote:
> I understand absence of harmonic waste in a 12 tones temperament as
> having no fifths intervals wider than 3/2, and no major thirds
> intervals smaller than 5/4 (for octavial temperaments). Is that
> right, or can you or someone tell me how we should define it?
Yes, that's right. Maybe somebody else can explain it more rigorously
but as I understand it, having no harmonic waste amounts to not
tempering any intervals more than necessary. In 12-tone case having
fifths larger than 3:2 introduces errors in major and minor thirds
that are larger than necessary for syntonic comma to vanish.
Generalizing the concept of no harmonic waste to all n-tone tunings
means that all consonances are on the same side of the pure ratio
(positive or negative) as in the corresponding equal tuning. For
example, to have no 5-limit harmonic waste in 22-tone tuning all
"fifths" should be larger than or equal to 3:2 (=positive) and all
"major thirds" should be smaller than or equal to 5:4 (=negative).
Also "minor thirds" should be larger than or equal to 6:5 (=positive).
This is the opposite situation compared to 12-tone case.
> For any temperament do you define it always for some harmonic limit,
> or could it be special choices of harmonics ?
I think it could be defined for any collection of ratios.
> For example, here is a "harmonic "quasi-equal" 5 tones temperament"
> 35:40:46:53:61
>
> is it the lowest 5-tones harmonic temperament with no harmonic
> waste?
Yes, it is! This has no 7-limit harmonic waste. It is also the lowest
when only looking at the ratios 3:2, 7:6 and 7:4.
> > but
> > then 128 might not be the best key to refer to.
> > Except for the tempo, that's true.
>
> Here
>
> http://launch.groups.yahoo.com/group/tuning/message/75816
>
> I describe the lowest harmonic well-temperament which starts with 116.
>
> Well done !
> With your permission I shall make a photosonic disk
> it confirms my feelings towards 29 as being a very useful prime.
You have my permission, this is an honour!
Kalle Aho
From: Kraig Grady (2008-05-16)
Subject: Re: [tuning] Re: harmonic series rwt?
the whole concept of harmonic waste i have a problem with. I am not
saying these are not interesting guidelines but to refer to it as
'waste' i think is overly bias.
What is waste to one is not to another.
there might be more to heaven and hell than tunings that approximate the
1-3-5- triad
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>,
> Jacques Dudon <fotosonix@...> wrote:
>
> > I understand absence of harmonic waste in a 12 tones temperament as
> > having no fifths intervals wider than 3/2, and no major thirds
> > intervals smaller than 5/4 (for octavial temperaments). Is that
> > right, or can you or someone tell me how we should define it?
>
> Yes, that's right. Maybe somebody else can explain it more rigorously
> but as I understand it, having no harmonic waste amounts to not
> tempering any intervals more than necessary. In 12-tone case having
> fifths larger than 3:2 introduces errors in major and minor thirds
> that are larger than necessary for syntonic comma to vanish.
> Generalizing the concept of no harmonic waste to all n-tone tunings
> means that all consonances are on the same side of the pure ratio
> (positive or negative) as in the corresponding equal tuning. For
> example, to have no 5-limit harmonic waste in 22-tone tuning all
> "fifths" should be larger than or equal to 3:2 (=positive) and all
> "major thirds" should be smaller than or equal to 5:4 (=negative).
> Also "minor thirds" should be larger than or equal to 6:5 (=positive).
> This is the opposite situation compared to 12-tone case.
>
> > For any temperament do you define it always for some harmonic limit,
> > or could it be special choices of harmonics ?
>
> I think it could be defined for any collection of ratios.
>
> > For example, here is a "harmonic "quasi-equal" 5 tones temperament"
> > 35:40:46:53:61
> >
> > is it the lowest 5-tones harmonic temperament with no harmonic
> > waste?
>
> Yes, it is! This has no 7-limit harmonic waste. It is also the lowest
> when only looking at the ratios 3:2, 7:6 and 7:4.
>
> > > but
> > > then 128 might not be the best key to refer to.
> > > Except for the tempo, that's true.
> >
> > Here
> >
> > http://launch.groups.yahoo.com/group/tuning/message/75816
> <http://launch.groups.yahoo.com/group/tuning/message/75816>
> >
> > I describe the lowest harmonic well-temperament which starts with 116.
> >
> > Well done !
> > With your permission I shall make a photosonic disk
> > it confirms my feelings towards 29 as being a very useful prime.
>
> You have my permission, this is an honour!
>
>
> Kalle Aho
>
>
From: Kalle Aho (2008-05-16)
Subject: Re: harmonic series rwt?
--- In [email protected], Kraig Grady <kraiggrady@...> wrote:
>
> the whole concept of harmonic waste i have a problem with. I am not
> saying these are not interesting guidelines but to refer to it as
> 'waste' i think is overly bias.
> What is waste to one is not to another.
> there might be more to heaven and hell than tunings that approximate
the
> 1-3-5- triad
Hi Kraig,
when I have to talk about *error* of an interval, I cringe because I
now think that all intervals have their own character. Maybe
'distance' would be a better word because it's value-neutral. I don't
think tempered intervals should be considered as simply faking more
simple ratios. Of course one can think like that if one wishes but
there are other more fruitful (I think) approaches. At the same time
their closeness to certain just intervals make them function in a
certain way. I'm convinced that for example intervals in the
neighborhood of 3:2 derive their harmonic function from this fact:
their closeness to 3:2. While sounding quite different from pure 3:2
the quarter comma meantone fifth functions as a rooting interval in
chords just like 3:2 does because it is sufficiently close to it. I'm
sure this has a psychoacoustic reason in the phenomenon of virtual
pitch.
The harmonic well temperaments integrate these two seemingly opposing
ideas, the short distance from ideal pure ratios and the interesting
characters of intervals and chords of the tuning heard not as fakes
but as sonorities with their own right to exist.
Of course one can go beyond this thinking and abandon all mention of
closeness of intervals to any other intervals. I believe your work
with recurrent sequences has this as a starting point, am I right?
Kalle Aho
From: Andreas Sparschuh (2008-05-16)
Subject: Synchronously wohl_temperiert.scl on modern pianos,was Re: harmonic series rwt?
--- In [email protected], Jacques Dudon <fotosonix@...> wrote:
that he
>...understands absence of harmonic waste in a 12 tones temperament as
1. > having no fifths intervals wider than 3/2,
2. > and no major thirds intervals smaller than 5/4
> Is that right, or can you or someone
> tell me how we should define it ?
Hi Jaques & all others,
Just those 2 conditions define Werckmeister's german term
"wohl-temperiert" = 'well-tempered'
http://dict.leo.org/ende?lp=ende&p=wlqAU.&search=wohltemperiert
http://dict.leo.org/ende?lp=ende&p=wlqAU.&search=wohltemperierte
as used by J.S.Bach in Werckmeister's sense:
http://en.wikipedia.org/wiki/Well-Tempered_Clavier
for example by the following cycle of a dozen 5ths:
110A_2 220A_3 440HzA_4
329 E_4 (< 330 = 3*A_2)
493.5B_4 987B_5 = 3*E_4
185F#_3 370 F#_4 740F#_5...2960F#_7(<2961=3*B_5)
277.5C#_4 555C#_5 = 3*F#_3
104G#_2 208G#_3 416 G#_4 832G#_5 1664G#_6(<1665=3*C#_5)
156Eb_3 312 Eb_4 = 3*G#_2
117Bb_2 234Bb_3 468 Bb_4 = 3*Eb_3
351 F_4 = 3*Bb_2
(65.7 131.4<)131.5C_3 263middleC_4 526C_5 1052C_6 (<1053 = 3*F_4)
65.7*3 = 197.1G_3 394.2 G_4
(441/3=147 294 <) 294.5 D_4 589D_5 (<591.3 = 3*G_3)
440 A_4 (<441 = 3*147)
!wohltemperiert.scl
!for J.S.Bach's WTC invented by Andreas Sparschuh in 2008
12
!
C-major beats C:E:G = 4: 5*(1316/1315): 6*(1314/1315) synchronously
!
555/526 ! C# 277.5/263
589/526 ! D 294.5/263
312/263 ! Eb
329/263 ! E (5/4)*(1316/1315) ~1.316...Cents sharper than JI 5:4
351/263 ! F
370/263 ! F#
1971/1315 ! G 394.2/263 (3/2)*(1314/1315) ~-1.317...C lower than 3:2
416/263 ! G#
440/263 ! A
468/263 ! Bb
987/526 ! B 493.5/263
2/1
!
that's imho my preferred rational temperament,
that i personally do reccomend for playing in modern HIP
http://en.wikipedia.org/wiki/Historically_informed_performance
of J.S.Bach's music in all 24keys
on modern keyboard instruments with
http://en.wikipedia.org/wiki/A440
as choice of absolute pitch.
It replaces all my older proposals.
Quests:
1. Who in that group dares to try out that one on his/hers own piano?
2. Has anybody suggestions for better ratios in order to improve it?
asks sincerely
A.S.
From: robert thomas martin (2008-05-17)
Subject: Synchronously wohl_temperiert.scl on modern pianos,was Re: harmonic series rwt?
--- In [email protected], "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In [email protected], Jacques Dudon <fotosonix@> wrote:
> that he
> >...understands absence of harmonic waste in a 12 tones temperament
as
>
> 1. > having no fifths intervals wider than 3/2,
> 2. > and no major thirds intervals smaller than 5/4
>
> > Is that right, or can you or someone
> > tell me how we should define it ?
>
> Hi Jaques & all others,
> Just those 2 conditions define Werckmeister's german term
> "wohl-temperiert" = 'well-tempered'
>
> http://dict.leo.org/ende?lp=ende&p=wlqAU.&search=wohltemperiert
> http://dict.leo.org/ende?lp=ende&p=wlqAU.&search=wohltemperierte
>
> as used by J.S.Bach in Werckmeister's sense:
> http://en.wikipedia.org/wiki/Well-Tempered_Clavier
>
> for example by the following cycle of a dozen 5ths:
>
>
> 110A_2 220A_3 440HzA_4
> 329 E_4 (< 330 = 3*A_2)
> 493.5B_4 987B_5 = 3*E_4
> 185F#_3 370 F#_4 740F#_5...2960F#_7(<2961=3*B_5)
> 277.5C#_4 555C#_5 = 3*F#_3
> 104G#_2 208G#_3 416 G#_4 832G#_5 1664G#_6(<1665=3*C#_5)
> 156Eb_3 312 Eb_4 = 3*G#_2
> 117Bb_2 234Bb_3 468 Bb_4 = 3*Eb_3
> 351 F_4 = 3*Bb_2
> (65.7 131.4<)131.5C_3 263middleC_4 526C_5 1052C_6 (<1053 = 3*F_4)
> 65.7*3 = 197.1G_3 394.2 G_4
> (441/3=147 294 <) 294.5 D_4 589D_5 (<591.3 = 3*G_3)
> 440 A_4 (<441 = 3*147)
>
> !wohltemperiert.scl
> !for J.S.Bach's WTC invented by Andreas Sparschuh in 2008
> 12
> !
> C-major beats C:E:G = 4: 5*(1316/1315): 6*(1314/1315) synchronously
> !
> 555/526 ! C# 277.5/263
> 589/526 ! D 294.5/263
> 312/263 ! Eb
> 329/263 ! E (5/4)*(1316/1315) ~1.316...Cents sharper than JI 5:4
> 351/263 ! F
> 370/263 ! F#
> 1971/1315 ! G 394.2/263 (3/2)*(1314/1315) ~-1.317...C lower than
3:2
> 416/263 ! G#
> 440/263 ! A
> 468/263 ! Bb
> 987/526 ! B 493.5/263
> 2/1
> !
>
> that's imho my preferred rational temperament,
> that i personally do reccomend for playing in modern HIP
> http://en.wikipedia.org/wiki/Historically_informed_performance
> of J.S.Bach's music in all 24keys
> on modern keyboard instruments with
> http://en.wikipedia.org/wiki/A440
> as choice of absolute pitch.
>
> It replaces all my older proposals.
>
> Quests:
> 1. Who in that group dares to try out that one on his/hers own
piano?
> 2. Has anybody suggestions for better ratios in order to improve it?
>
> asks sincerely
> A.S.
>
From Robert. Try tuning-math message 17251 for another angle.
From: Herman Miller (2008-05-17)
Subject: Re: [tuning] Re: harmonic series rwt?
Kalle Aho wrote:
> Hi Kraig,
>
> when I have to talk about *error* of an interval, I cringe because I
> now think that all intervals have their own character. Maybe
> 'distance' would be a better word because it's value-neutral.
I generally use "deviation" in place of "error" in that context. For
intervals close to ratios of small integers, the amount of this
deviation does seem to have an effect on the character of the interval.
"Distance" might also work, but "error" could be misinterpreted.
From: Kalle Aho (2008-05-17)
Subject: Re: harmonic series rwt?
--- In [email protected], Herman Miller <hmiller@...> wrote:
>
> Kalle Aho wrote:
>
> > Hi Kraig,
> >
> > when I have to talk about *error* of an interval, I cringe because I
> > now think that all intervals have their own character. Maybe
> > 'distance' would be a better word because it's value-neutral.
>
> I generally use "deviation" in place of "error" in that context. For
> intervals close to ratios of small integers, the amount of this
> deviation does seem to have an effect on the character of the interval.
Yes, it has an effect on the interval's character but I oppose the
simple view that larger deviation always means worse sound even if
this is often true.
> "Distance" might also work, but "error" could be misinterpreted.
Yes.
From: Kraig Grady (2008-05-22)
Subject: Re: [tuning] Re: harmonic series rwt?
It stuck me this is almost the opposite of the Spectralist
in the sense that they imitate the harmonic series with somewhat
conventional scales .
here we have the harmonic series imitating just such a scale :)
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>
_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
Kalle Aho wrote:
>
> --- In [email protected] <mailto:tuning%40yahoogroups.com>, "Carl
> Lumma" <carl@...> wrote:
> >
> > > > Kalle, if you've got your ears on, can you tell us the
> > > > lowest subset of harmonics without harmonic waste when
> > > > interpreted as a WT *that has a mode in which all the
> > > > denominators are base-2 numbers*?
> > > >
> > > > I assume it may be possible to do this with 128-256, as
> > > > your unrestricted result was done with 116-232.
> > > > Failing that, 256-512 must have enough resources.
> > > >
> > > > Thanks!
> > >
> > > Hi Carl,
> > >
> > > actually there are four of them!
> > >
> > > 128:136:144:152:162:171:182:192:203:216:228:243:256
> > > 128:136:144:152:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:228:243:256
> > > 128:136:144:153:162:171:182:192:204:216:229:243:256
> >
> > Thanks!
> >
> > > Here are Scala files for these:
> >
> > D'oh! Howabout with the constraint that all 3rds
> > be < 405cents? (Your original scale happened to
> > meet this.)
>
> Well, there are 25 of those in 256-512, the best (largest major third
> is 402.844 cents) is this:
>
> ! ForCarl5.scl
> !
>
> 12
> !
> 271/256
> 287/256
> 19/16
> 161/128
> 171/128
> 181/128
> 383/256
> 203/128
> 215/128
> 57/32
> 483/256
> 2/1
>
>