Yahoo Tuning Groups Ultimate Backup tuning A Rational Well Temperament

Hi everyone,

I thought about searching for a well temperament where the notes are
harmonics of a common fundamental. The criteria were that there would
be no harmonic waste (combined absolute 5-limit error is no greater
than in 12-equal) and the series of harmonics should be as low as
possible in the harmonic series.

I think that the following series fulfills these criteria:

116:123:130:138:146:155:164:174:184:195:207:219:232

Similar rational versions can be found for other equal temperaments
(and possibly for scales too). For example a corresponding rational
temperament for 22-equal with no 7-limit harmonic waste would be

149:154:159:164:169:174:180:186:192:198:204:
211:217:224:232:239:247:255:263:271:280:289

Kalle Aho

🔗Carl Lumma <carl@lumma.org>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > Hi everyone,
> >
> > I thought about searching for a well temperament where the notes
> > are harmonics of a common fundamental. The criteria were that
> > there would be no harmonic waste (combined absolute 5-limit error
> > is no greater than in 12-equal) and the series of harmonics
> > should be as low as possible in the harmonic series.
> >
> > I think that the following series fulfills these criteria:
> >
> > 116:123:130:138:146:155:164:174:184:195:207:219:232
>
> Cool! What happens with 12-ET and 7-limit harmonic waste?

Hi Carl and everybody,

Interestingly, there's no difference to the 5-limit case. Same with
9-limit.

22-equal and zero 9-limit harmonic waste gives a tuning which you can
get when you write the following in Scala:

equal 22
quantize/linear 1/381
key 21

Notice that by rotating the result (the "key 21" command) gives even
lower terms so

I'm not 100% sure this is as low in the harmonic series as possible. I
searched through all linear quantizations of 22-equal but I'm not sure
if there could still be a lower one which differs from
a linear quantization. A brute force search by a computer program that
takes this into account would take absurdly long time.

Kalle

🔗Carl Lumma <carl@lumma.org>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > > > I think that the following series fulfills these criteria:
> > > >
> > > > 116:123:130:138:146:155:164:174:184:195:207:219:232
> > >
> > > Cool! What happens with 12-ET and 7-limit harmonic waste?
> >
> > Hi Carl and everybody,
> >
> > Interestingly, there's no difference to the 5-limit case. Same
> > with 9-limit.
> //
> > I'm not 100% sure this is as low in the harmonic series as
> > possible.
>
> Hi Kalle,
>
> Are you just not sure about the 22-tone one, or also about
> the 12-tone one?

I did a computer search for 12-tone one where the linear quantizations
are offsetted by +1,0 and -1 so for every linear quantization the
program checks 3^11=177147 different possibilities. So I'm quite sure
about the 12-tone case. For 22-tone the program would have to go
through 3^21 possibilities for every linear quantization which is just
way too much.

The nth linear quantization (in case you're wondering what that is)
for x-equal you get with

round(n*2^(y/x)) where y goes from 0 to x

🔗Carl Lumma <carl@lumma.org>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kalle wrote...
>
> > I did a computer search for 12-tone one where the linear
> > quantizations are offsetted by +1,0 and -1 so for every
> > linear quantization the program checks 3^11=177147
> > different possibilities. So I'm quite sure about the
> > 12-tone case. For 22-tone the program would have to go
> > through 3^21 possibilities for every linear quantization
> > which is just way too much.
> >
> > The nth linear quantization (in case you're wondering
> > what that is) for x-equal you get with
> >
> > round(n*2^(y/x)) where y goes from 0 to x
>
> Cool. Smart work. That's a keeper.

Thanks, Carl!

By the way, 19-tone 5-limit tuning seems to be the following:

in Scala, type

equal 19
quantize/linear 1/1374
key 18

This tuning is much higher in the harmonic series than 12-tone and
22-tone ones.

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> > >
> > > > > I think that the following series fulfills these criteria:
> > > > >
> > > > > 116:123:130:138:146:155:164:174:184:195:207:219:232
> > > >
> > > > Cool! What happens with 12-ET and 7-limit harmonic waste?
> > >
> > > Hi Carl and everybody,
> > >
> > > Interestingly, there's no difference to the 5-limit case. Same
> > > with 9-limit.
> > //
> > > I'm not 100% sure this is as low in the harmonic series as
> > > possible.
> >
> > Hi Kalle,
> >
> > Are you just not sure about the 22-tone one, or also about
> > the 12-tone one?
>
> I did a computer search for 12-tone one where the linear quantizations
> are offsetted by +1,0 and -1 so for every linear quantization the
> program checks 3^11=177147 different possibilities. So I'm quite sure
> about the 12-tone case. For 22-tone the program would have to go
> through 3^21 possibilities for every linear quantization which is just
> way too much.
>
> The nth linear quantization (in case you're wondering what that is)
> for x-equal you get with
>
> round(n*2^(y/x)) where y goes from 0 to x

I just noticed that letting n be a real number gives more linear
quantizations. Searching this way gives the following approximate
values for n (two decimal places):

12-tone 5,7 and 9-limits 115.89
19-tone 5-limit 1324.71
22-tone 7-limit 124.13
22-tone 9-limit 369.03
13-tone Bohlen-Pierce 7-limit 104.65

Kalle Aho

🔗Carl Lumma <carl@lumma.org>

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

Hi kalle aho

Well temperaments where the notes are harmonics of a common fundamental
are best fiitted in my ADO and EDL tuning system :
http://240edo.googlepages.com/arithmeticrationaldivisionsofoctave

http://240edo.googlepages.com/equaldivisionsoflength(edl)

There you can find an Excel sheet to approximate any EDO with ADO and
EDL :

- How to approximate EDandADO systems with eachother?Download thisfile
<http://240edo.googlepages.com/ADOandEDO.xls>

- How to approximate EDO and EDL systems with each other?Download this
file <http://240edo.googlepages.com/EDOandEDL.xls>

For your examples , choose 22-EDO and 149-ADO or 12-EDO with 116-ADO.

Also you can use ADO-system in scala as varied interval scale based on
frequencies or string length find in file....new.

Best wishes for you

Shaahin mohajeri , Tombak player and microtonalist

My web site <http://240edo.googlepages.com/>

My farsi page in harmonytalk <http://www.harmonytalk.com/mohajeri>

Irandrumz ensemble

🔗Tom Dent <stringph@gmail.com>

Werckmeister's solution with monochord lengths (equivalent to
subharmonics of some extremely high frequency) was

196 186 175 165 156 147 139 131 124 117 110 104 98

Many of these numbers are closely related to Kalle's, but not quite.
The ratio 39/31 for a major 3rd is also familiar. If we invert it and
take the lowest possible integer representation we have

93 98 104 110 117 124 131 139 147 156 165 175 186

... the catch is that there is a tiny bit of 'harmonic waste' here.

What Kalle is doing is also equivalent to my attempts to build
temperaments with integer frequencies, if you allow a free choice of
pitch. I guess A=390 would have been a good one!

It is also equivalent to factoring the Pythagorean comma into
superparticular ratios (a pastime of Sparschuh's) as

261/260 . 585/584 . 657/656 . 369/368 . 621/620 . 465/464

Now what is the smallest 12-integer series you can have (with or
without 'harmonic waste') where no fifth is tempered more than 1/4
comma? I.e. the largest superparticular factor is 321/320 or smaller?

I once found /tuning/topicId_58680.html#58680
E 315
F 335
F# 354
G 376
G# 398
A 421
Bb 447
B 472
C 502
C# 531
D 563
Eb 597
but I'm sure one can do 'better' in terms of reducing the size of the
numbers.
~~~T~~~

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> Hi everyone,
>
> I thought about searching for a well temperament where the notes are
> harmonics of a common fundamental. The criteria were that there would
> be no harmonic waste (combined absolute 5-limit error is no greater
> than in 12-equal) and the series of harmonics should be as low as
> possible in the harmonic series.
>
> I think that the following series fulfills these criteria:
>
> 116:123:130:138:146:155:164:174:184:195:207:219:232
>
> Similar rational versions can be found for other equal temperaments
> (and possibly for scales too). For example a corresponding rational
> temperament for 22-equal with no 7-limit harmonic waste would be
>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi kalle aho
>
>
>
> Well temperaments where the notes are harmonics of a common fundamental
> are best fiitted in my ADO and EDL tuning system :
> http://240edo.googlepages.com/arithmeticrationaldivisionsofoctave
>
>
>
> http://240edo.googlepages.com/equaldivisionsoflength(edl)
>
>
>
>
>
>
>
> There you can find an Excel sheet to approximate any EDO with ADO and
> EDL :
>
>
>
> - How to approximate EDandADO systems with eachother?Download thisfile
> <http://240edo.googlepages.com/ADOandEDO.xls>
>
>
>
> - How to approximate EDO and EDL systems with each other?Download this
> file <http://240edo.googlepages.com/EDOandEDL.xls>
>
>
>
> For your examples , choose 22-EDO and 149-ADO or 12-EDO with 116-ADO.

Hi Shaahin,

notice that there exists even lower 22-tone harmonic mode with no
7-limit harmonic waste where the series begins with 124 but your excel
sheet doesn't give this one because it doesn't take into account those
in between quantizations I talked about in message 75833.

Kalle Aho

🔗Tom Dent <stringph@gmail.com>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > > Now what is the smallest 12-integer series you can have (with or
> > > without 'harmonic waste') where no fifth is tempered more than 1/4
> > > comma? I.e. the largest superparticular factor is 321/320 or
smaller?
> >
> > Hi Tom,
> >
> > I think it is
> >
> > 90:95:101:107:113:120:127:135:143:151:160:169:180
> >
> >
> > Kalle Aho
> >
>
> No... there is the wide fifth 143/95 which is tempered by 286/285,
> about 2/7 comma. Perhaps you put a limit on the smallest narrow fifth
> but not on the largest wide one.
>
> Noting that 285=3*95, I think the smallest base number that allows all
> fifths to be tempered less than 321/320 will turn out to be over 107.

Oh, I'm sorry! What about

125:132:140:148:157:167:176:187:198:210:222:235:250?

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> > >
> > > > Now what is the smallest 12-integer series you can have (with or
> > > > without 'harmonic waste') where no fifth is tempered more than 1/4
> > > > comma? I.e. the largest superparticular factor is 321/320 or
> smaller?
> > >
> > > Hi Tom,
> > >
> > > I think it is
> > >
> > > 90:95:101:107:113:120:127:135:143:151:160:169:180
> > >
> > >
> > > Kalle Aho
> > >
> >
> > No... there is the wide fifth 143/95 which is tempered by 286/285,
> > about 2/7 comma. Perhaps you put a limit on the smallest narrow fifth
> > but not on the largest wide one.
> >
> > Noting that 285=3*95, I think the smallest base number that allows all
> > fifths to be tempered less than 321/320 will turn out to be over 107.
>
> Oh, I'm sorry! What about
>
> 125:132:140:148:157:167:176:187:198:210:222:235:250?

Wait! There's even better one

101:107:113:120:127:135:143:151:160:169:180:191:202

No fifth differs more than 1/4 pythagorean comma from just.

And with this

131:139:147:156:165:175:185:196:208:220:234:247:262

No fifth differs more than 321/320 from just.

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> > > > I think it is
> > > > 90:95:101:107:113:120:127:135:143:151:160:169:180

That's made by construction in 5ths:
C = 90 45
G = 3C = 135
D: (3G=405) > 404 202 101
A: (3D=303)> 302 151
E: (3A=453) > 452 226 113
B: (3e=339) > 338 169
F# (3B=507) < 508 254 127 attend the broade layperson's dog-5th
C# (3F#=381) > 380 190 95
G# (3C#=285) < 286 143 another even worser wide dog-5th
Eb (3G#=429) > 428 214 107
Bb (3Eb=321) > 320 160 80 40
F = 3Bb = 120 60 30 15
C = 3F = 45 cycle closed

>... there is the wide fifth 143/95 which is tempered by 286/285,
> about 2/7 comma.
that sound none well, but is still only good,
if you really intend there an 'open' 5th alike in meantonics.
> > >
> > > Noting that 285=3*95, I think the smallest base number that
allows all
> > > fifths to be tempered less than 321/320 will turn out to be over
107.
> > What about ?
> >
> > 125:132:140:148:157:167:176:187:198:210:222:235:250
>
in 5ths cycle:
C 125
G (375) > 374 187
D (561) > 560 280 140 70
A 210 105
E (315) > 314 157
B (471) > 470 235
F# (1175) < 1184 592 148 74 37 even worser wide than the above ex.
C# 222 111
G# (333) < 334 167 another problematic wide 5th
D# (501) > 500 250 125

in order to fix such ugly broade-5th bugs, just consult my:
http://www.strukturbildung.de/Andreas.Sparschuh/Mainz_1999.jpg
without such dog-5ths defects,
as found in some 'esotheric' reinterpretations,
that do not appear in my 1999 original 'discovery' version.

> Wait! There's even better one
> 101:107:113:120:127:135:143:151:160:169:180:191:202
> No fifth differs more than 1/4 pythagorean comma from just.

That's expanded:
C: 101
G: (303) > 302 151
D: (453) > 452 226 113
A: (339) > 338 169
E: (507) < 508 254 127 at least barely only one broade 5th
B: (381) > 382 191
F# (573) > 572 286 143
C# (429) > 428 214 107
G# (312) > 320 160 80 40 20 10 5
Eb 15
Bb 45
F: 135
C: (405) > 404 202 101
>
> And with this
>
> 131:139:147:156:165:175:185:196:208:220:234:247:262
> No fifth differs more than 321/320 from just.
>
C 131
G (393) > 392 196 98 49
D 147
A (441) > 440 220 110 55
E 165
B (495) > 494 247
F# (741) > 740 370 185 {Proposal in order to get rid of the dog}
C# (555) < 556 278 139 { here you'd better let 555 unchanged }
G# (417) > 416 ... 13 {...832 1664 < (1665 = 3*555) }
Eb 39
Bb 117
F: (351) > 350 175 instead W's choice 176=11*2^4
C: (525) > 524 262 131

{considering that little change converts your's originally
'open'-tuning into an
"well"-temperament in the sense of W. & Bach, without the dog}

Attend:
Already Werckmeister used almost about the same ratios in:
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings

A.S.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

Hi, Kalle. I was intrigued by some of these & plugged the numbers into
one of my spreadsheets. Unfortunately, ...

> Wait! There's even better one
>
> 101:107:113:120:127:135:143:151:160:169:180:191:202
>
> No fifth differs more than 1/4 pythagorean comma from just.

There's some harmonic waste in this one, since the fifths built on 127
and 169 are both wide (~3.4c & 4.5c, respectively).

> And with this
>
> 131:139:147:156:165:175:185:196:208:220:234:247:262
>
> No fifth differs more than 321/320 from just.

Likewise, the one has a wide fifth (~3.1c) on 185.

Interesting idea, though. Keep trying!

--George

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> Hi, Kalle. I was intrigued by some of these & plugged the numbers into
> one of my spreadsheets. Unfortunately, ...
>
> > Wait! There's even better one
> >
> > 101:107:113:120:127:135:143:151:160:169:180:191:202
> >
> > No fifth differs more than 1/4 pythagorean comma from just.
>
> There's some harmonic waste in this one, since the fifths built on 127
> and 169 are both wide (~3.4c & 4.5c, respectively).
>
> > And with this
> >
> > 131:139:147:156:165:175:185:196:208:220:234:247:262
> >
> > No fifth differs more than 321/320 from just.
>
> Likewise, the one has a wide fifth (~3.1c) on 185.
>
> Interesting idea, though. Keep trying!

Hi George,

these were just answers to Tom's question. I know they have harmonic
waste.

My original suggestion

116:123:130:138:146:155:164:174:184:195:207:219:232

has no harmonic waste.

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

🔗Carl Lumma <carl@lumma.org>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> > >
> > > > I think that the following series fulfills these criteria:
> > > >
> > > > 116:123:130:138:146:155:164:174:184:195:207:219:232
> > >
> > > Much to my surprise, this has a nearly perfect cycle of major
> > > third qualities, just as a traditional well temperament would.
> > > To get it to line up with traditional key centers, I would
> > > suggest "key 5" in Scala, thus...
> >
> > Carl,
> >
> > this
> >
> > 128:136:144:153:162:171:182:192:204:216:229:243:256
> >
> > has has monotonic third-sizes over circle of fifths!
> >
>
> Kalle- could I ask that you provide Scala files when sharing
> these? It makes it much easier to communicate, at least for me.

Here:

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

🔗Carl Lumma <carl@lumma.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > Hi, Kalle. I was intrigued by some of these & plugged the
numbers into
> > one of my spreadsheets. Unfortunately, ...
> >
> > > Wait! There's even better one
> > >
> > > 101:107:113:120:127:135:143:151:160:169:180:191:202
> > >
> > > No fifth differs more than 1/4 pythagorean comma from just.
> >
> > There's some harmonic waste in this one, since the fifths built
on 127
> > and 169 are both wide (~3.4c & 4.5c, respectively).
> >
> > > And with this
> > >
> > > 131:139:147:156:165:175:185:196:208:220:234:247:262
> > >
> > > No fifth differs more than 321/320 from just.
> >
> > Likewise, the one has a wide fifth (~3.1c) on 185.
> >
> > Interesting idea, though. Keep trying!
>
> Hi George,
>
> these were just answers to Tom's question. I know they have harmonic
> waste.
>
> My original suggestion
>
> 116:123:130:138:146:155:164:174:184:195:207:219:232
>
> has no harmonic waste.

Okay, I see now.

Regarding your original suggestion, I have both positive and negative
comments. (I'll give the negative first.)

At first I was a little hesitant to call this a well-temperament,
because most of the pure 5ths alternate with tempered 5ths as you go
around the circle of 5ths. In a good well-temperament (pardon the
redundancy) most (if not all) of the pure 5ths should be consecutive
in order to achieve monotonic interval sizes around the circle of
5ths.

However, as Carl pointed out, there is a reasonable progression of
size for the major 3rds. If 116 is C, then your best major triads
are on Bb, C, and F (in order of increasing error), which tends to
favor the flat keys, and the worst is either F# or C# (depending on
whether you consider major-3rd error or total-triad error to be more
important). If you make 155 C, then the best triads are F, G, and C
(in that order) and the worst either C# or G#, which is not ideal.

I think that a listening test will be required to determine which
number would be preferable as C, and whether the differences among
the best triads are all that significant.

On the plus side:
1) I think you've outdone Robert Wendell in the synchronous well-
temperament category, because you've succeeded in getting fairly
simple brats in *all 24* major and minor triads. (RW completely
disregarded the minor triads.)
2) The cents errors seem to indicate that no triad is going to sound
significantly worse than 12-equal, which puts this tuning in the
synchronous modern well-temperament category (SMWT, for short) -- I
noticed that Carl saw this.

The bottom line: this one definitely deserves a listen.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >...
> > My original suggestion
> >
> > 116:123:130:138:146:155:164:174:184:195:207:219:232
> ...
> At first I was a little hesitant to call this a well-temperament,
> because most of the pure 5ths alternate with tempered 5ths as you
go
> around the circle of 5ths. In a good well-temperament (pardon the
> redundancy) most (if not all) of the pure 5ths should be
consecutive
> in order to achieve monotonic interval sizes around the circle of
> 5ths.

After writing that, I wondered how low the numbers could be for a WT
with all of the pure fifths occurring consecutively (and no harmonic
waste). I found this one, with five pure fifths:

364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728

Taking 364 as C, the best major triads are on C, G, and F, D, Bb, and
A (in that order) and the two worst are F# and C# (both Pythagorean).

The brats aren't quite as simple as in your synchronous modern well-
temperament (SMWT), but they're not so much more complicated as you
might expect, because only two of the harmonics in the above sequence
are odd numbers, and seven of them are divisible by 4. I'm going to
call this one my "24c", since it's the third one in my collection of
well-temperaments that has reasonably simple brats in all 24 major
and minor triads.

Since Carl wants a Scala file, here it is:

! WTPB-24c.scl
!
George Secor's 24-triad proportional-beating well-temperament (24c)
12
!
96/91
102/91
108/91
114/91
243/182
182/91
545/364
144/91
305/182
162/91
683/364
2/1

I was curious to compare both your SMWT and 24c with the two well-
temperaments (24a and 24b) that I found a couple of years ago:
/tuning-math/message/13733

Also see these links for answers to questions about 24a and 24b and
WT brats in general:
/tuning-math/message/13737
/tuning/topicId_59689.html#66380

Here's how they stack up, ranked in order of simplest brats:
major triads only: 24a, 24b, SMWT, 24c
minor triads only: SMWT, 24c, 24a, 24b
all 24 M&m triads: SMWT, 24a, 24c, 24b

I prefer 24b to 24a, because 24a, in spite of having monotonic 3rds,
does not have all pure 5ths consecutive in the circle of 5ths. Also,
24a has IMO too much contrast (i.e., the C major triad has too low an
error compared to the rest).

--George

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:

> > My original suggestion
> >
> > 116:123:130:138:146:155:164:174:184:195:207:219:232
> >
> > has no harmonic waste.
>
> Okay, I see now.
>
> Regarding your original suggestion, I have both positive and negative
> comments. (I'll give the negative first.)
>
> At first I was a little hesitant to call this a well-temperament,

Me too. It's a well-temperament only in the less strict sense that
there is no harmonic waste.

> because most of the pure 5ths alternate with tempered 5ths as you go
> around the circle of 5ths. In a good well-temperament (pardon the
> redundancy) most (if not all) of the pure 5ths should be consecutive
> in order to achieve monotonic interval sizes around the circle of
> 5ths.

> However, as Carl pointed out, there is a reasonable progression of
> size for the major 3rds. If 116 is C, then your best major triads
> are on Bb, C, and F (in order of increasing error), which tends to
> favor the flat keys, and the worst is either F# or C# (depending on
> whether you consider major-3rd error or total-triad error to be more
> important). If you make 155 C, then the best triads are F, G, and C
> (in that order) and the worst either C# or G#, which is not ideal.
>
> I think that a listening test will be required to determine which
> number would be preferable as C, and whether the differences among
> the best triads are all that significant.

> On the plus side:
> 1) I think you've outdone Robert Wendell in the synchronous well-
> temperament category, because you've succeeded in getting fairly
> simple brats in *all 24* major and minor triads. (RW completely
> disregarded the minor triads.)

And there are fairly simple beating ratios for other chords as well.

> 2) The cents errors seem to indicate that no triad is going to sound
> significantly worse than 12-equal, which puts this tuning in the
> synchronous modern well-temperament category (SMWT, for short) -- I
> noticed that Carl saw this.
>
> The bottom line: this one definitely deserves a listen.

I suggest trying it out!

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > > My original suggestion
> > >
> > > 116:123:130:138:146:155:164:174:184:195:207:219:232
> > >
> > > has no harmonic waste.
> >
> > Okay, I see now.
> >
> > Regarding your original suggestion, I have both positive and
negative
> > comments. (I'll give the negative first.)
> >
> > At first I was a little hesitant to call this a well-temperament,
>
> Me too. It's a well-temperament only in the less strict sense that
> there is no harmonic waste.

Well, I think there's more to it than that. The four best major 3rds
all occur consecutively on one side of the circle of 5ths, so there
is a general key contrast such as one expects in a WT. But since the
key contrast of this (modern) WT is rather low, I think that the
synchronous beating becomes a more notable feature, relatively
speaking.

> > ...
> > I think that a listening test will be required to determine which
> > number would be preferable as C, and whether the differences
among
> > the best triads are all that significant.

After taking another look at the numbers, I have a hunch that 116
will work best as C.

> > On the plus side:
> > 1) I think you've outdone Robert Wendell in the synchronous well-
> > temperament category, because you've succeeded in getting fairly
> > simple brats in *all 24* major and minor triads. (RW completely
> > disregarded the minor triads.)
>
> And there are fairly simple beating ratios for other chords as well.

Hmmm, good point -- I hadn't thought of that.

> > 2) The cents errors seem to indicate that no triad is going to
sound
> > significantly worse than 12-equal, which puts this tuning in the
> > synchronous modern well-temperament category (SMWT, for short) --
I
> > noticed that Carl saw this.
> >
> > The bottom line: this one definitely deserves a listen.
>
> I suggest trying it out!

I'll be spending many hours this weekend putting a new hardwood floor
in my kitchen, so I may not be able to listen to it for a few days.
But I'm looking forward to it!

--George

🔗Carl Lumma <carl@lumma.org>

🔗Magnus Jonsson <magnus@smartelectronix.com>

I just tried this scale out, and I think it is an clear improvement over 12-edo. Switching between the two, the difference is obvious.

12-edo sounds busy and shallow. It never feels ... I can't find the word I want in English, but transliterated from Swedish it is self-clear. Meaning something like "of course, this is how it should be".

The well-temperament sounds serene and deep. After playing for a while I'm starting to hear the reinforced low frequency beating, and I find it quite pleasant and soothing. Meditative you could say.

I would say this scale is a success.

/ Magnus

On Fri, 11 Apr 2008, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>>
>> Oops, thanks for spotting that! Here, then, is the corrected
>> listing:
>>
>> ! WTPB-24c.scl
>> !
>> George Secor's 24-triad proportional-beating well-temperament (24c)
>> 12
>> !
>> 96/91
>> 102/91
>> 108/91
>> 114/91
>> 243/182
>> 182/91
>> 545/364
>> 144/91
>> 305/182
>> 162/91
>> 683/364
>> 2/1
>
> That still has the error! :)
> But I've got it straight anyway. I'm going to be auditioning
> these in a few days. If anything interesting happens I'll
> post. -Carl
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>
>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

Hi Magnus,

did you try my original suggestion too?

12
!
123/116
65/58
69/58
73/58
155/116
41/29
3/2
46/29
195/116
207/116
219/116
2/1

(It's funny that there is a corresponding word for 'självklart' in Finnish
'itsestäänselvä' but not in English.)

Kalle

--- In tuning@yahoogroups.com, Magnus Jonsson <magnus@...> wrote:
>
> I just tried this scale out, and I think it is an clear improvement
over
> 12-edo. Switching between the two, the difference is obvious.
>
> 12-edo sounds busy and shallow. It never feels ... I can't find the
word I
> want in English, but transliterated from Swedish it is self-clear.
Meaning
> something like "of course, this is how it should be".
>
> The well-temperament sounds serene and deep. After playing for a while
> I'm starting to hear the reinforced low frequency beating, and I
find it
> quite pleasant and soothing. Meditative you could say.
>
> I would say this scale is a success.
>
> / Magnus
>
> On Fri, 11 Apr 2008, Carl Lumma wrote:
>
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >>
> >> Oops, thanks for spotting that! Here, then, is the corrected
> >> listing:
> >>
> >> ! WTPB-24c.scl
> >> !
> >> George Secor's 24-triad proportional-beating well-temperament (24c)
> >> 12
> >> !
> >> 96/91
> >> 102/91
> >> 108/91
> >> 114/91
> >> 243/182
> >> 182/91
> >> 545/364
> >> 144/91
> >> 305/182
> >> 162/91
> >> 683/364
> >> 2/1
> >
> > That still has the error! :)
> > But I've got it straight anyway. I'm going to be auditioning
> > these in a few days. If anything interesting happens I'll
> > post. -Carl
> >
> >
> > ------------------------------------
> >
> > You can configure your subscription by sending an empty email to one
> > of these addresses (from the address at which you receive the list):
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - leave the group.
> > tuning-nomail@yahoogroups.com - turn off mail from the group.
> > tuning-digest@yahoogroups.com - set group to send daily digests.
> > tuning-normal@yahoogroups.com - set group to send individual emails.
> > tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> >
> >
> >
> >
>

🔗Carl Lumma <carl@lumma.org>

🔗Magnus Jonsson <magnus@smartelectronix.com>

🔗Petr Parízek <p.parizek@chello.cz>

🔗kraiggrady@anaphoria.com

I am curious about the difference between eastern and western finland music!

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Petr Parï¿½zek [mailto:p.parizek@chello.cz]
Sent: Saturday, April 12, 2008 02:07 AM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: A Rational Well Temperament

Kalle Aho wrote:

> (Funny that there is a corresponding world for 'sjï¿½lvklart' in Finnish
> 'itsestï¿½ï¿½nselvï¿½' but not in English.)

Petr

PS: Do you have any idea where I could find some folk music from eastern Finland?

🔗Petr Parízek <p.parizek@chello.cz>

🔗kraiggrady@anaphoria.com

well this makes sense as the so called british are really the result ofa bunch of vikings who kicked their ass and a few of the real britishare in Ireland. I have no objection to swedish or noewegian musicthough as they often use the neutral third. there are interestingintonational practices of the hardanger fiddle music whichin the upper octive will have an differnent intonation than thelower. it far exceeeds so much of the so called classical repetoire ofthe violin. I have been disapoined with the ocora and the La Chant deMonde etc. recordings i have gotten from finland. I would assume thatthe finnish being associated witrh russia would be a point ofcontention as they have been pretty sucessful in fighting them off!,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Petr Parízek [mailto:p.parizek@chello.cz]
Sent: Saturday, April 12, 2008 03:25 AM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: A Rational Well Temperament

Kraig wrote:

> I am curious about the difference between eastern and western finland music!

IIRC, western Finland was occupied by Swedes forquite some time and folk melodies from western Finland remind me of something between Swedish and Irish music. As far as I could find out on the web, eastern Finland, OTOH, was occupied by Russians for quite some time and eastern Finlands folk melodies sound like something between Russian and Bulgarian music, at least regarding the tiny excerpts that I found somewhere, having successfully forgotten the URLs.

Petr

🔗Petr Parízek <p.parizek@chello.cz>

Kraig wrote:

> there are interesting intonational practices of the hardanger fiddle music which in

> the upper octive will have an differnent intonation than the lower. it far exceeeds so much

> of the so called classical repetoire of the violin.

AFAIK, the hardanger fiddle has eight (or nine?) strings, only four of which are meant to be played. The rest are short resonating strings tuned to some higher pitches which are fifths or thirds from some of the lower tones, I don't know exactly how it works. The point is that this makes the player learn to play as much in tune as possible. A few years ago, there was a programme on the Czech tv where recordings of some hardanger violinists were played, I mean, not only in groups but also solo players. I was surprised how close their intonation was to 5-limit JI even with the tiny commatic differences -- one of them, when playing in E major, played B-G# almost like 5/3 and B-A almost like 16/9 (I didn't actually hear the 16/9 but I could clearly hear the upper voice rising from G# to A by a nice minor second which I heard almost like 16/15). When he then changed to G major, he played B-A not as 16/9 but very close to 9/5 (which was quite logical because he was playing B-G before and it would break the harmonic series if he raised the upper voice by 10/9 instead of 9/8).

What I know much more about is the overtone flute (meaning the one without finger holes) primarily because I like playing it myself. :-D It's surprising how many cultures také this instrument as their traditional folk instrument -- in Swedish "sälgflöjt", in Norwegian "seljeflöyte", in Slovak "koncovka" (sometimes we call it the same in Czech because some of the Moravian folk melodies are also played on this flute), also in Swahili it's "filimbi". Not knowing what it's called there, I've also read about it being used in the Solomon Islands.

I have several here at home -- one of them is played like a recorder, the rest are transverse flutes. When I first played an overtone flute in 2003, I very quickly got the opinion that the harmonic series is the nicest series of intervals that has ever existed in music. If you like to know something more about the flute, let me know.

> I have been disapoined with the ocora and the La Chant de Monde etc. recordings

> i have gotten from finland.

Why? What are they like?

> I would assume that the finnish being associated witrh russia would be a point of contention

> as they have been pretty sucessful in fighting them off!

Well, I really don't know almost anything about the history of the events.

Petr

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:

> Hey, Kalle, you are from Finland?

Hi Petr,

yes I am, originally from western Finland from the region of
Ostrobothnia, now living in Helsinki.

> PS: Do you have any idea where I could find some folk music from
eastern Finland?

I don't think this distinction is geographical as the instrument
kantele which belongs to the eastern influenced tradition of folk
music is also actively still played in western Finland too. While this
older tradition is eastern influenced the distinction is more temporal
than spatial so to speak.

from

http://www.fimic.fi/fimic/fimic.nsf/

"Finnish folk music can be roughly divided into two historical eras
differing considerably from each other both in their music and culture
and in their use in the community. The earlier stratum is often called
the ancient Finnish period or the Kalevalaic era. It covers such
genres as the singing of Kalevalaic poetry, the chain dance, the
lament, and a distinctive brand of music performed on such instruments
as the five-stringed kantele, the bowed harp (or crwth), and a host of
folk wind instruments.

The later stratum of Finnish folk music is the period of agrarian or
pelimanni music and it is clearly rooted in Western culture."

I suggest you look for records made with specific instruments like
kantele or the bowed lyre. Those fimic-pages contain lists of records.

My father, grandfather and other relatives were or are accordionists
and associated with pelimanni music, that later scandinavia-influenced
tradition of folk music. There is even some research done about my
grandfather who was actually born in America for finnish parents:

http://www.markkulepisto.com/sitahavois.htm

Kalle

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

🔗lenderbee <lenderbee@yahoo.com>

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> well this makes sense as the so called british are really the
result ofa bunch of vikings who kicked their ass and a few of the
real britishare in Ireland. I have no objection to swedish or
noewegian musicthough as they often use the neutral third. there are
interestingintonational practices of the hardanger fiddle music
whichin the upper octive will have an differnent intonation than
thelower. it far exceeeds so much of the so called classical
repetoire ofthe violin. I have been disapoined with the ocora and the
La Chant deMonde etc. recordings i have gotten from finland. I would
assume thatthe finnish being associated witrh russia would be a point
ofcontention as they have been pretty sucessful in fighting them
off!,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Your finely nuanced understanding of history does your nation credit!

Peter

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> > >...
> > > My original suggestion
> > >
> > > 116:123:130:138:146:155:164:174:184:195:207:219:232
> > ...
> > At first I was a little hesitant to call this a well-temperament,
> > because most of the pure 5ths alternate with tempered 5ths as you
> go
> > around the circle of 5ths. In a good well-temperament (pardon the
> > redundancy) most (if not all) of the pure 5ths should be
> consecutive
> > in order to achieve monotonic interval sizes around the circle of
> > 5ths.
>
> After writing that, I wondered how low the numbers could be for a WT
> with all of the pure fifths occurring consecutively (and no harmonic
> waste). I found this one, with five pure fifths:
>
> 364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728
>
> Taking 364 as C, the best major triads are on C, G, and F, D, Bb, and
> A (in that order) and the two worst are F# and C# (both Pythagorean).
>
> The brats aren't quite as simple as in your synchronous modern well-
> temperament (SMWT), but they're not so much more complicated as you
> might expect, because only two of the harmonics in the above sequence
> are odd numbers, and seven of them are divisible by 4. I'm going to
> call this one my "24c", since it's the third one in my collection of
> well-temperaments that has reasonably simple brats in all 24 major
> and minor triads.

You might as well see how you fare with my old suggestion:

E 315
F 335
F# 354
G 376
G# 398
A 421
Bb 447
B 472
C 502
C# 531
D 563
Eb 597

this has 3 pure fifths at B-F#-C# and G#-Eb.
~~~T~~~

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>...
> You might as well see how you fare with my old suggestion:
>
> E 315
> F 335
> F# 354
> G 376
> G# 398
> A 421
> Bb 447
> B 472
> C 502
> C# 531
> D 563
> Eb 597
>
> this has 3 pure fifths at B-F#-C# and G#-Eb.
> ~~~T~~~

Not bad. However, in spite of the somewhat lower numbers, the brats
aren't as quite as simple as with my 24c (with five consecutive pure
fifths):
364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728
for which I still haven't gotten the Scala listing right. Here it is
(again):

! WTPB-24c.scl
!
George Secor's 24-triad proportional-beating well-temperament (24c)
12
!
96/91
102/91
108/91
114/91
243/182
128/91
545/364
144/91
305/182
162/91
683/364
2/1

Tom, a few days ago, while looking through some WT's in my notes, I
happened to come across a well-temperament you suggested last
year, "a mash-up of Youngy and Neidhardty features" at the very end
of this message:
/tuning/topicId_68966.html#69506

In my reply (about halfway through), I said that I didn't see
anything "grossly inappropriate":
/tuning/topicId_68966.html#69529

Instead, I observed that "most of your triads (both major & minor)
come very close to being proportional-beating" and that the brats are
very similar to my 2/11-comma rational WT. In fact, the two WT's are
so much alike that all I needed for a rational version of your Young-
Neidhardt hybrid was change two notes (B & F#):

! Dent-YN-RWT.scl
!
Tom Dent's Young-Neidhardt well-temperament (rationalized by G. Secor)
12
!
560/531
2643/2360
70/59
74/59
315/236
1329/944
883/590
280/177
890/531
105/59
887/472
2/1

The brats aren't nearly as simple as with the harmonic temperaments
we've been looking at lately, but OTOH the fifths are much more
consistent in size (all tempered <4 cents).

I'm still working on my kitchen floor, so I still haven't had a
chance to listen to any of the latest ones. :-(

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
Hi George, Tom & all others,
...just compare my actual refined 'squiggle' versus your similar...
>...old suggestion:

313 > E 315
334 > F 335
352 > F# 354
375 > G 376
396 > G# 398
418 > A 421 cps or Hz
445.5 > Bb 447
469.5 > B 472
501 > C 502
528 > C# 531
560 > D 563
594 > Eb 597

!squiggle418cps.scl
12
Sparschuh's 2008 Cammerton a4=418Hz 'squiggle'
176/167 ! C#
560/501 ! D
198/167 ! Eb
626/501 ! E = (5/4)*(2504/2505)
4/3 ! F
704/501 ! F#
250/167 ! G = (3/2)*(500/501)
264/167 ! G#
839/501 ! A
297/167 ! Bb
313/167 ! B
2/1

obtained from tempering some 5ths sharper than pure by the amounts:
C 500/501 G 374/375 D 224/225 A 209/201 E 626/627 B 2816/2817 F#...
...F# - C# - G# - Eb - Bb 2672/2673 F - C.
>

> George's proposal:
> 364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728:
>
> ! WTPB-24c.scl
> !
> George Secor's 24-triad proportional-beating well-temperament (24c)
> 12
> !
> 96/91
> 102/91
> 108/91
> 114/91
> 243/182
> 128/91
> 545/364
> 144/91
> 305/182
> 162/91
> 683/364
> 2/1
>
that sounds really fine,
works similar on 91=13*7 alike my HIP:
http://en.wikipedia.org/wiki/Historically_informed_performance
Contius reconstruction:

F: 3^6 = 729 ( > 728 364 182 91 = 13*7 )
C: 3*91 = 273
G: (3*273 = 819 1638 >) 1637 ( > 1632 816 408 204 102 51)
D: (51*3 = 153 306 612 >) 611 ( >608 304 152 76 38 19)
A: 19*3 = 57
E: (57*3 = 171 342 684 >) 683
B: (3*683 = 2049 >) 2048 ... 1=3^0
F# 3
C# 9 = 3^2
G# 27 = 3^3
Eb 81 = 3^4
Bb 243 = 3^5
F: 729 = 3^6

all 5ths inbetween the accidentials or nominals are just 3/2 pure.

!ContiusHIP.scl
12
HIP reconstruction of Contius organ-tuning in Halle by A. Sparschuh
192/91 ! C#
611/546 ! D
108/91 ! Eb
683/546 ! E = (5/4)*(1366/1365)
243/182 ! F
128/91 ! F#
1637/1092 ! G = (3/2)*(1637/1638)
144/91 ! G#
456/273 ! A Choir-tone 456cps or Hz
162/91 ! Bb
512/273 ! B
2/1
!

Which one of that ones would you prefer or even improve?

A.S.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> Hi George, Tom & all others,

Hi Andreas,

I won't be able to listen to any of these yet, but here are my
comments from looking at the numbers.

> ...just compare my actual refined 'squiggle' versus your similar...
> >...old suggestion:
>
> 313 > E 315
> 334 > F 335
> 352 > F# 354
> 375 > G 376
> 396 > G# 398
> 418 > A 421 cps or Hz
> 445.5 > Bb 447
> 469.5 > B 472
> 501 > C 502
> 528 > C# 531
> 560 > D 563
> 594 > Eb 597

Not bad. This is probably the one I would prefer, because no fifth
is tempered excessively.

I see that the worst major triads are on B and E. This is similar to
Brad's "Bach squiggle" temperament, but since you have no wide
fifths, there's a smoother size progression for the major 3rds.

Since you asked what I might "improve", here goes. If you substitute
C# 530, Eb 596, F# 707/2, and G# 795/2, then D, A, E, and B will
improve at the expense of F#, C#, Ab, and Eb, and the worst major
triads will be on F# and C#. (Although there are now only 2 pure
fifths (instead of 3), the brats are actually somewhat improved in
the process.)

As to whether favoring the sharp keys over the flats by this much is
an "improvement" is, of course, a matter of opinion. I would have
preferred to make it less sharp-biased, but it didn't work out that
way with 502 as a common denominator.

> !squiggle418cps.scl
> 12
> Sparschuh's 2008 Cammerton a4=418Hz 'squiggle'
> 176/167 ! C#
> 560/501 ! D
> 198/167 ! Eb
> 626/501 ! E = (5/4)*(2504/2505)
> 4/3 ! F
> 704/501 ! F#
> 250/167 ! G = (3/2)*(500/501)
> 264/167 ! G#
> 839/501 ! A
> 297/167 ! Bb
> 313/167 ! B
> 2/1
>
> obtained from tempering some 5ths sharper than pure by the amounts:
> C 500/501 G 374/375 D 224/225 A 209/201 E 626/627 B 2816/2817 F#...
> ...F# - C# - G# - Eb - Bb 2672/2673 F - C.

This is a very high-contrast well-temperament, such that: 5 major
triads (E thru G#) have thirds with >20c error, and the C major triad
is so much better that all of the rest that I expect that it will
stick out like the proverbial sore thumb. Also, the fifth A-E is
very narrow (almost 9 cents).

If you substitute D 560/501 and E 627/501, then:
1) The A-E error is cut to 6.2c;
2) The brats are much improved;
3) The contrast is lowered (by making C-E wider, so it beats at the
same rate as C-G); and
4) There are now only 4 major triads (B thru G#) having thirds with
>20c error.

> > George's proposal:
> > 364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728:
> >
> > ! WTPB-24c.scl
> > !
> > George Secor's 24-triad proportional-beating well-temperament
(24c)
> > 12
> > !
> > 96/91
> > 102/91
> > 108/91
> > 114/91
> > 243/182
> > 128/91
> > 545/364
> > 144/91
> > 305/182
> > 162/91
> > 683/364
> > 2/1
> >
> that sounds really fine,
> works similar on 91=13*7 alike my HIP:
> http://en.wikipedia.org/wiki/Historically_informed_performance
> Contius reconstruction:
>
> F: 3^6 = 729 ( > 728 364 182 91 = 13*7 )
> C: 3*91 = 273
> G: (3*273 = 819 1638 >) 1637 ( > 1632 816 408 204 102 51)
> D: (51*3 = 153 306 612 >) 611 ( >608 304 152 76 38 19)
> A: 19*3 = 57
> E: (57*3 = 171 342 684 >) 683
> B: (3*683 = 2049 >) 2048 ... 1=3^0
> F# 3
> C# 9 = 3^2
> G# 27 = 3^3
> Eb 81 = 3^4
> Bb 243 = 3^5
> F: 729 = 3^6
>
> all 5ths inbetween the accidentials or nominals are just 3/2 pure.
>
> !ContiusHIP.scl
> 12
> HIP reconstruction of Contius organ-tuning in Halle by A. Sparschuh
> 192/91 ! C#
> 611/546 ! D
> 108/91 ! Eb
> 683/546 ! E = (5/4)*(1366/1365)
> 243/182 ! F
> 128/91 ! F#
> 1637/1092 ! G = (3/2)*(1637/1638)
> 144/91 ! G#
> 456/273 ! A Choir-tone 456cps or Hz
> 162/91 ! Bb
> 512/273 ! B
> 2/1
> !

The fifths G-D and D-A are >8c narrow. Is that what you really want?

> Which one of that ones would you prefer or even improve?
>
> A.S.

BTW, I finally had a chance to listen to Kalle's synchronous WT and
to compare it with some other rational temperaments, and I have two
general observations:

1) Synchrony doesn't seem to be of much benefit when the thirds are
beating quickly;
2) The general effect is much better if the tempering of *all* of the
fifths is kept under some maximum, e.g., ~5 cents, or better yet,
~4.6 cents. (Achieving this in combination with simple brats is not
easy.)

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
Hi George, Tom & all others,
>
> ! Dent-YN-RWT.scl
> !
> Tom Dent's Young-Neidhardt well-temperament (rationalized by G. Secor)
> 12
> !
> 560/531 ! C#
> 2643/2360 ! D
> 70/59 ! Eb
> 74/59 ! E
> 315/236 ! F
> 1329/944 ! F#
> 883/590 ! G
> 280/177 ! G#
> 890/531 ! A
> 105/59 ! A#
> 887/472 ! B
> 2/1
>
Did you mean by that following corresponding absolute pitches,
or which other frequncies had you concrete in mind when
creating that particular one by the factor 59?

C: 531 := 59*9
C# 560 280 140 70 35 = 7*5
D: 594.675 := 264.3*9/4
Eb 630 315 = 7*5*9
E: 666 333 = 37*9
F: 708 354 177 = 59*3
F# 747.625 = 1329*9/16
G: 794.7 = 88.3*9
G# 840 = 240*3
A: 890 445Hz ? probably yours intend A4=445 cps ?
Bb 945 = 105*9
B: 997.875 = 887*9/8

Hence i do agree with your's observation:
> The brats aren't nearly as simple as with the harmonic temperaments
> we've been looking at lately, but OTOH the fifths are much more
> consistent in size (all tempered <4 cents).
>
Sincerely
A.S.

🔗George D. Secor <gdsecor@yahoo.com>

I didn't intend any particular absolute pitches when giving these
ratios. I expected that you would use whatever pitch reference you
like.

--George

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> BTW, I finally had a chance to listen to Kalle's synchronous WT and
> to compare it with some other rational temperaments, and I have two
> general observations:
>
> 1) Synchrony doesn't seem to be of much benefit when the thirds are
> beating quickly;
> 2) The general effect is much better if the tempering of *all* of the
> fifths is kept under some maximum, e.g., ~5 cents, or better yet,
> ~4.6 cents. (Achieving this in combination with simple brats is not
> easy.)

Hi George,

why 4.6 cents?

Anyway, the following tuning has the errors of all fifths under 4.6 cents:

12
!
224/211
237/211
252/211
266/211
282/211
299/211
316/211
336/211
355/211
377/211
399/211
2/1

Kalle Aho

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
Hi George, Tom & all others,
> > ! Dent-YN-RWT.scl
> > !
> > Tom Dent's Young-Neidhardt well-temperament (rationalized by G.
> Secor)
> > 12
> > !
> > 560/531 ! C#
> > 2643/2360 ! D
> > 70/59 ! Eb
> > 74/59 ! E
> > 315/236 ! F
> > 1329/944 ! F#
> > 883/590 ! G
> > 280/177 ! G#
> > 890/531 ! A
> > 105/59 ! A#
> > 887/472 ! B
> > 2/1
> > >
Actually i realized that on my on piano in the absoute-pitches:
> >
starting in chromatically order from:
http://en.wikipedia.org/wiki/Middle_C
Tenor-C:
> C: 531 := 59*9
> C# 560 280 140 70 35 = 7*5
> D: 594.675 := 264.3*9/4
> Eb 630 315 = 7*5*9
> E: 666 333 = 37*9
> F: 708 354 177 = 59*3
> F# 747.625 = 1329*9/16
> G: 794.7 = 88.3*9
> G# 840 = 240*3
> A: 890 445Hz ? probably yours intend A4=445 cps ?
> Bb 945 = 105*9
> B: 997.875 = 887*9/8
>
> I didn't intend any particular absolute pitches when giving these
> ratios.
I.m.h.o. any tuning for a conrete instrument
appears to me as yet incomplete,
if the start pitch is lacking,
when intervals are given barely in relative ratios alone,
without definition of any concrete reference-tone intension,
that can be deduced from an absoute fixed standard normal-pitch:
alike:
http://en.wikipedia.org/wiki/A440
(in german:)
http://de.wikipedia.org/wiki/Kammerton

> I expected that you would use whatever pitch reference you
> like.
Hence for your's tuning i do prefer the correspoding above:
http://upload.wikimedia.org/wikipedia/commons/d/d7/445.ogg

That beats againt the normal with
5Hz := 445Hz - 440Hz sharper
or with
300 Metronome beats = 5Hz *60 beats/min
synchrone to an
http://en.wikipedia.org/wiki/Metronome

445HZ was broadly used by my friend:
the late
http://en.wikipedia.org/wiki/Herbert_von_Karajan
critziesed by
http://en.wikipedia.org/wiki/Andr%C3%A1s_Schiff
in
http://www.schillerinstitute.org/fid_02-06/021-2schiff.html
"For example, his fight against the absurdly high "Karajan-tuning,"
which he broadened with a new battle on the sidelines of the last
Salzburg Festival. Because of his invitation, members of the Berlin
and Vienna Philaharmonics (both of which orchestras play at extremely
high pitch, even above A=445 Hz) ,as well as opera singers, and even
conductors,...."

On which conrete pitch for a'=44?Hz
do you would realize yours above temperament
on yours own personal piano?

Sincerely
A.S.

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote/asked:
>
> why 4.6 cents?
>
Theoretical relative-ratios of:
>
!Kalle_C211Hz.scl
>
> 12
> !
> 224/211
> 237/211
> 252/211
> 266/211
> 282/211
> 299/211
> 316/211
> 336/211
> 355/211
> 377/211
> 399/211
> 2/1
>
>
That yields realized in concrete absolute-pitches:

C: 211
C# 224
D: 237
Eb 252
E: 266
F: 282
F# 299
G: 316
G# 336
A: 355 ~ Arnolt Schlick's low pitch
Bb 377
B: 399
c' 422

http://en.wikipedia.org/wiki/Arnolt_Schlick

or when expanded as cycle of the constructing epimoric 5ths:

c: 211 cps
g: (633>) 632 G=316 158 79
d: 237
a: (711>) 710 A4=355 Hz
e: (1065>) 1064 532 E=266 133
b: 399
f# (1197>) 1196 598 F#=299
c# (897>) 896 448 224 112...7
g# 336...21
eb 252 126 63
bb (189 378 >) Bb=377 (>376 188 94 47)
f: 282 141
c: (423>) 422 C=211 cps

Some 5ths turn out deviate flatter than 3/2 JI by the amounts:

C 632/633 G-D 710/711 A 1064/1065 E-B 1196/1197 F# 896/897 C#-G#-Eb...
Eb 377/378 Bb... @ 1200*ln(377/378)/ln(2) = ~-4.586...Cents
Bb 376/377 F ... @ 1200*ln(376/377)/ln(2) = ~-4.598...Cents
F 422/423 C

Congratulations:
That sounds really fine in my ears,
in deed very convincing,
especially in the flattend accidential keys,
-also even still acceptable-
in the sharp accidential keys,
as specified in A.Schlick's instructions ~500years ago.

Sincerely
A.S.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > BTW, I finally had a chance to listen to Kalle's synchronous WT
and
> > to compare it with some other rational temperaments, and I have
two
> > general observations:
> >
> > 1) Synchrony doesn't seem to be of much benefit when the thirds
are
> > beating quickly;
> > 2) The general effect is much better if the tempering of *all* of
the
> > fifths is kept under some maximum, e.g., ~5 cents, or better yet,
> > ~4.6 cents. (Achieving this in combination with simple brats is
not
> > easy.)
>
> Hi George,
>
> why 4.6 cents?

I should have said 4.7 cents. That's the maximum amount the fifths
are tempered in my 5/23-comma temperament extraordinaire to give C &
G major triads with equal beat rates for the 5th & M3, which are
arguably as good as 1/4-comma meantone triads (if not better). See:
/tuning/topicId_73833.html#74188
Since it has three wide 5ths, it's not a well-temperament.
Nevertheless, it's my alternative to modified meantone temperaments
(and also my favorite circulating 12-tone temperament).

> Anyway, the following tuning has the errors of all fifths under 4.6
cents:
>
> !
>
> 12
> !
> 224/211
> 237/211
> 252/211
> 266/211
> 282/211
> 299/211
> 316/211
> 336/211
> 355/211
> 377/211
> 399/211
> 2/1

Your best major triads are on Eb and Ab, and the worst are on B and
E, so this would benefit from a transposition. A nice feature is
that your worst M3s are <18.13c wide.

I found a WT with simpler brats (with one 5th tempered 4.62c narrow):

! SecorSWT149
George Secor's 149-based synchronous WT
12
!
315/298
167/149
177/149
187/149
199/149
210/149
223/149
236/149
250/149
265/149
280/149
2/1

I originally found this one with 167 as a common denominator, with
the best major triad on Bb, so I transposed the ratios to put it on C.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> Hi George, Tom & all others,
> > > ! Dent-YN-RWT.scl
> > > !
> > > Tom Dent's Young-Neidhardt well-temperament (rationalized by G.
> > Secor)
> > > 12
> > > !
> > > 560/531 ! C#
> > > 2643/2360 ! D
> > > 70/59 ! Eb
> > > 74/59 ! E
> > > 315/236 ! F
> > > 1329/944 ! F#
> > > 883/590 ! G
> > > 280/177 ! G#
> > > 890/531 ! A
> > > 105/59 ! A#
> > > 887/472 ! B
> > > 2/1
> > > >
> Actually i realized that on my on piano in the absoute-pitches:
> > >
> starting in chromatically order from:
> http://en.wikipedia.org/wiki/Middle_C
> Tenor-C:
> > C: 531 := 59*9
> > C# 560 280 140 70 35 = 7*5
> > D: 594.675 := 264.3*9/4
> > Eb 630 315 = 7*5*9
> > E: 666 333 = 37*9
> > F: 708 354 177 = 59*3
> > F# 747.625 = 1329*9/16
> > G: 794.7 = 88.3*9
> > G# 840 = 240*3
> > A: 890 445Hz ? probably yours intend A4=445 cps ?
> > Bb 945 = 105*9
> > B: 997.875 = 887*9/8
> >
> > I didn't intend any particular absolute pitches when giving these
> > ratios.
> I.m.h.o. any tuning for a conrete instrument
> appears to me as yet incomplete,
> if the start pitch is lacking,
> when intervals are given barely in relative ratios alone,
> without definition of any concrete reference-tone intension,
> that can be deduced from an absoute fixed standard normal-pitch:
> alike:
> http://en.wikipedia.org/wiki/A440
> (in german:)
> http://de.wikipedia.org/wiki/Kammerton
>
> > I expected that you would use whatever pitch reference you
> > like.
> Hence for your's tuning i do prefer the correspoding above:
> http://upload.wikimedia.org/wikipedia/commons/d/d7/445.ogg
>
> That beats againt the normal with
> 5Hz := 445Hz - 440Hz sharper
> or with
> 300 Metronome beats = 5Hz *60 beats/min
> synchrone to an
> http://en.wikipedia.org/wiki/Metronome
>
> 445HZ was broadly used by my friend:
> the late
> http://en.wikipedia.org/wiki/Herbert_von_Karajan
> critziesed by
> http://en.wikipedia.org/wiki/Andr%C3%A1s_Schiff
> in
> http://www.schillerinstitute.org/fid_02-06/021-2schiff.html
> "For example, his fight against the absurdly high "Karajan-tuning,"
> which he broadened with a new battle on the sidelines of the last
> Salzburg Festival. Because of his invitation, members of the Berlin
> and Vienna Philaharmonics (both of which orchestras play at
extremely
> high pitch, even above A=445 Hz) ,as well as opera singers, and even
> conductors,...."
>
> On which conrete pitch for a'=44?Hz
> do you would realize yours above temperament
> on yours own personal piano?

A=440 is my preference. (I see no reason why the frequencies of all
of the pitches should be exact integers.) With most well-
temperaments C will be higher in pitch than in 12-equal with A=440;
for the rationalized Dent-Young-Neidhardt C will be ~262.5 Hz.

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> I found a WT with simpler brats (with one 5th tempered 4.62c narrow):
>
> ! SecorSWT149
> George Secor's 149-based synchronous WT
> 12
> !
> 315/298
> 167/149 ! D
> 177/149
> 187/149 ! E
> 199/149 ! F
> 210/149
> 223/149 ! G
> 236/149
> 250/149 ! A
> 265/149 ! Bb or proposal 531/298 in order to avoid here an dog-5th?
> 280/149 ! B
> 2/1
>
> I originally found this one with 167 as a common denominator, with
> the best major triad on Bb, so I transposed the ratios to put it on >C.
Dear George,

as cycle of 5ths:
C 149
g: (447>) 446 G=223
d: (669>) 668 334 D=167
a: (501>) 500 A=250 125
e: (375>) 374 E=187
b: (561>) 560 B=280 140 70 35
f# 210 105
c# 315
g# (945>) 944 472 G#236 118 59
Eb 177
???????????????????????????????????????????????????????????????????
bb (531>) 530 Bb=265 ??????????????????????????????????????????????
F: (795<) 796 398 F=199 did you really intend here an broade dog-5th?
???????????????????????????????????????????????????????????????????
c: (597>) 596 298 149

In order to get rid of the buggy wide dog-5th inbetween Bb and F,
there i do reccomend to arise Bb a little bit, so that Brad
Lehman's overtempering defect vanishes, simply in replacing
the formerly old Bb 265 by
the new more apt higher value: 265.5 := 531/2
....
Eb 177
bb 531 Bb=265.5
F: (1593>) 1592 796 398 F=199
c: (597>) 596 298 C=149

Do you agree with that improving suggestion?
/tuning/topicId_71865.html#71930

Yours Sincerely
A.S.

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> A=440 is my preference. (I see no reason why the frequencies of all
> of the pitches should be exact integers.) With most well-
> temperaments C will be higher in pitch than in 12-equal with A=440;
> for the rationalized Dent-Young-Neidhardt C will be ~262.5 Hz.
>
at the moment i do prefer
from Werckmeister's septenarian comma versus the SC

81/80 = (99/98)*(441/440)

divided into 3 epimoric subparts:

99/98 = (297/296)*(296/295)*(295/294)

tempering the 5ths G-D-A-E flattend by the corresponding amounts:

G 296/297 D 295/296 A 294/295 E

yielding on the violin empty stings the absolute pitches:

g3: 198 cps = 99*2 := 220*(9/10) a minor-tone below a3=440Hz/2
d4: 296
a4: 442.5 := 885/2
e5: 631.5 := 1323/2

as subset of the tuning procedere in 5ths on my piano:

C: 523Hz (>522 264 132 66 33) 'tenoor-C'
G: 99 (((> 98 49=7*7 overtaken from Werckmeister's "septenarius")))
D: (297>) 296 (>295 (>294 147=49*3)
A: 885 (>882 441=49*9)
E: 1323 = 49*27
B: (49*81 = 3969>) 3968 ... 496...31 through all 7 'B's on the keys
F# 93
C# 279 a semitone above 'middle-C'
G# 837
Eb 2511
Bb (7533>) 7532 3716 1883
F: (5649>) 5648 2824 1412 706 353
C: (1059>) 1058 529 = 23^2 cycle returend back to the above 'tenor-C'

!sparschuhPiano.scl
!
from Andreas Sparschuh's violin strings G 296/297 D 295/296 A 294/295
12
!
558/523 ! C#
598/523 ! D
628/523 ! Eb
1323/1058 ! E = 661.5/523
706/523 ! F
724/523 ! F#
792/523 ! G
837/523 ! G#
885/523 ! A absolute 442.5Hz
1883/1058 ! Bb = 941.5/523
992/523 ! B
2/1
!

have a lot of fun with that
A.S

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > I found a WT with simpler brats (with one 5th tempered 4.62c
narrow):
> >
> > ! SecorSWT149
> > George Secor's 149-based synchronous WT
> > 12
> > !
> > 315/298
> > 167/149 ! D
> > 177/149
> > 187/149 ! E
> > 199/149 ! F
> > 210/149
> > 223/149 ! G
> > 236/149
> > 250/149 ! A
> > 265/149 ! Bb or proposal 531/298 in order to avoid here an dog-
5th?
> > 280/149 ! B
> > 2/1
> >
> > I originally found this one with 167 as a common denominator,
with
> > the best major triad on Bb, so I transposed the ratios to put it
on >C.
> Dear George,
>
> as cycle of 5ths:
> C 149
> g: (447>) 446 G=223
> d: (669>) 668 334 D=167
> a: (501>) 500 A=250 125
> e: (375>) 374 E=187
> b: (561>) 560 B=280 140 70 35
> f# 210 105
> c# 315
> g# (945>) 944 472 G#236 118 59
> Eb 177
> ???????????????????????????????????????????????????????????????????
> bb (531>) 530 Bb=265 ??????????????????????????????????????????????
> F: (795<) 796 398 F=199 did you really intend here an broade dog-
5th?
> ???????????????????????????????????????????????????????????????????
> c: (597>) 596 298 149
>
> In order to get rid of the buggy wide dog-5th inbetween Bb and F,
> there i do reccomend to arise Bb a little bit, so that Brad
> Lehman's overtempering defect vanishes, simply in replacing
> the formerly old Bb 265 by
> the new more apt higher value: 265.5 := 531/2
> ....
> Eb 177
> bb 531 Bb=265.5
> F: (1593>) 1592 796 398 F=199
> c: (597>) 596 298 C=149
>
> Do you agree with that improving suggestion?
> /tuning/topicId_71865.html#71930

Yes, that's very good. Thank you for spotting that!

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > A=440 is my preference. (I see no reason why the frequencies of
all
> > of the pitches should be exact integers.) With most well-
> > temperaments C will be higher in pitch than in 12-equal with
A=440;
> > for the rationalized Dent-Young-Neidhardt C will be ~262.5 Hz.
> >
> at the moment i do prefer
> from Werckmeister's septenarian comma versus the SC
>
> 81/80 = (99/98)*(441/440)
>
> divided into 3 epimoric subparts:
>
> 99/98 = (297/296)*(296/295)*(295/294)
>
> tempering the 5ths G-D-A-E flattend by the corresponding amounts:
>
> G 296/297 D 295/296 A 294/295 E
>
> yielding on the violin empty stings the absolute pitches:
>
> g3: 198 cps = 99*2 := 220*(9/10) a minor-tone below a3=440Hz/2
> d4: 296
> a4: 442.5 := 885/2
> e5: 631.5 := 1323/2
>
> as subset of the tuning procedere in 5ths on my piano:
>
> C: 523Hz (>522 264 132 66 33) 'tenoor-C'
> G: 99 (((> 98 49=7*7 overtaken from Werckmeister's "septenarius")))
> D: (297>) 296 (>295 (>294 147=49*3)
> A: 885 (>882 441=49*9)
> E: 1323 = 49*27
> B: (49*81 = 3969>) 3968 ... 496...31 through all 7 'B's on the keys
> F# 93
> C# 279 a semitone above 'middle-C'
> G# 837
> Eb 2511
> Bb (7533>) 7532 3716 1883
> F: (5649>) 5648 2824 1412 706 353
> C: (1059>) 1058 529 = 23^2 cycle returend back to the above 'tenor-
C'
>
> !sparschuhPiano.scl
> !
> from Andreas Sparschuh's violin strings G 296/297 D 295/296 A
294/295
> 12
> !
> 558/523 ! C#
> 598/523 ! D
> 628/523 ! Eb
> 1323/1058 ! E = 661.5/523
> 706/523 ! F
> 724/523 ! F#
> 792/523 ! G
> 837/523 ! G#
> 885/523 ! A absolute 442.5Hz
> 1883/1058 ! Bb = 941.5/523
> 992/523 ! B
> 2/1
> !
>
> have a lot of fun with that
> A.S

Either I don't understand this, or something is very wrong with the
numbers. The fifths D-A and A-E are tempered >23 cents, and D-F# is
tempered by >55 cents.

Or is this a joke (since you said, "have a lot of fun")? :-)

--George

🔗George D. Secor <gdsecor@yahoo.com>

(This replaces message #76092, which I've deleted, because there was
an error in the .scl listing.)

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > > ... The general effect is much better if the tempering of *all*
of
> the
> > > fifths is kept under some maximum, e.g., ~5 cents, or better
yet,
> > > ~4.6 cents. (Achieving this in combination with simple brats
is
> not
> > > easy.)
> >
> > Hi George,
> >
> > why 4.6 cents?
>
> I should have said 4.7 cents. That's the maximum amount the fifths
> are tempered in my 5/23-comma temperament extraordinaire to give C
&
> G major triads with equal beat rates for the 5th & M3, which are
> arguably as good as 1/4-comma meantone triads (if not better). See:
> /tuning/topicId_73833.html#74188
> Since it has three wide 5ths, it's not a well-temperament.
> Nevertheless, it's my alternative to modified meantone temperaments
> (and also my favorite circulating 12-tone temperament).

I've found a way to make this one even more extraordinaire -- with
reasonably simple brats for *all 24* major & minor triads:

! Secor5_23STX.scl
!
George Secor's synchronous 5/23-comma temperament extraordinaire
12
!
62/59
66/59
559/472
591/472
315/236
331/236
353/236
745/472
395/236
210/118
221/118
2/1

The narrowest fifths are now tempered ~4.9 cents, but that's still
somewhat better than 1/4-comma meantone.

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> from Werckmeister's
11-limit septenarian comma versus the SC
> >
> > 81/80 = (99/98)*(441/440)
3^4/5/16 = (11*3^2/7^2/2)*(7^2*3^2/11/5/2^3)

i do call the ratio:
441/440
http://www.petersontuners.com/index.cfm?category=85&sub=89
as "Werckmeister's 11-limit septenarian schisma"
Scheibler later in the early 19.th century used that interval
for defineing our todays still actual 440cps standard:
http://www.1911encyclopedia.org/Musical_Pitch
> >
Werckmeister's 11-limit-septenarian-comma becomes when
> > divided into 3 epimoric subparts:
> >
> > 99/98 = (297/296)*(296/295)*(295/294)
> >
That 3-fold decompostion of W's-comma can be used for
> > tempering the 5ths G-D-A-E flattend by the corresponding amounts:
> >
> > G 296/297 D 295/296 A 294/295 E
> >
> > yielding on the violin empty stings the absolute pitches:
>
G2=99Hz lowest violin__G3=198__string
G3=148 __G4=296__(<297=3*G2)
A3=221.25 __A4=442.5__ (<444=3*G3)
highest violin__E5=661.5__string (<663.75=3*A3)

> > g3: 198 cps = 99*2 := 220*(9/10) a minor-tone below a3=440Hz/2
> > d4: 296
> > a4: 442.5 := 885/2
> > e5: 661.5 := 1323/2
> >
or
When generalized to a dozen 5ths-cirlce on my own old piano:
> >
> > C_5: 523Hz (>522 264 132 66 33) 'tenor-C'
> > G_2: 99 (((> 98 49=7*7 taken from Werckmeister's "septenarius")))
> > D_4: (297>) 296 (>295 (>294 147=49*3)
> > A_5: 885 (>882 441=49*9)
> > E_6: 1323 = 49*27
> > B_0: (49*81 = 3969>) 3968 ... 496...31 use all 7 'B's on the keys
> > F#_2: 93
> > C#_4: 279 a semitone above 'middle-C'
> > G#_5: 837
> > Eb_7: 2511
> > Bb_6: (7533>) 7532 3716 1883
> > F_4: (5649>) 5648 2824 1412 706 353
> > C_5: (1059>) 1058 529 = 23^2 cycle returned back to 'tenor-C'
> >

Sorry, the old previous meassge contains here some typo-errors:
please read always 529 instead there formerly faulty 523.
I had confused that due to a mistake in all to much hurry.
Simply forget about the wrong numbers:

and study instead of that
again the now corrected version:

!well_Violin2Piano.scl
!by A.Sparschuh
temper from violin empty strings G 296/297 D 295/296 A 294/295 E
12
! middle_C 264.5Hz = 529cps/2
!
558/529 ! C#
598/529 ! D
2511/2116 ! Eb = 627.75/529
1323/1058 ! E = 661.5/529
706/529 ! F
724/529 ! F#
792/529 ! G
837/529 ! G#
885/523 ! A = 442.5Hz*2 absolute a4
1883/1058 ! Bb = 941.5/529
992/529 ! B
2/1
!
!
the relative deviation of the
5ths corresponds to the following epimoric decomposition

F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
B F# C# G# Eb 7532:7533 Bb 5648:5649 F

into the 8 superparticular subfactorization of the PC=3^12/2^19.

> Either I don't understand this,
or if you prefer the same distribution of the PC=~23.46cents
in logarithmically values as Cents approximation,
about the amounts:

F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F

correspodning to the above 8 epimoric ratios.

> or something is very wrong with the
> numbers.
In deed -i have to agree-
my first data were somewhat out of control.

Many thanks for making me aware of my blunder,
that had urgently demanded some bug-fixing.

> The fifths D-A and A-E are tempered >23 cents,
not anymore , but now
all that both 5ths are less tempered than PC^1/4 =~ 5.865 Cents
compareable to G-A in Werckmeister's#3.

> and D-F# is
> tempered by >55 cents.
that 3rd: D-F# is barely 186/185 ~9.33Cents wide
but attend the 3rd C-E with barely 2646/2645 ~0.654Cents wider
than 5/4, hence almost nearly to pure JI.
>
> Or is this a joke (since you said, "have a lot of fun")? :-)
That was never intened as hoax, even in its faulty version.
so,
now after that proof reading/checking/correction
that patched revision is really meant seriously adjusted
for properly usage.

Yours Sincerely
A.S.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>...
> Sorry, the old previous meassge contains here some typo-errors:
> please read always 529 instead there formerly faulty 523.
> I had confused that due to a mistake in all to much hurry.
> Simply forget about the wrong numbers:
>
> and study instead of that
> again the now corrected version:
>
> !well_Violin2Piano.scl
> !by A.Sparschuh
> temper from violin empty strings G 296/297 D 295/296 A 294/295 E
> 12
> ! middle_C 264.5Hz = 529cps/2
> !
> 558/529 ! C#
> 598/529 ! D
> 2511/2116 ! Eb = 627.75/529
> 1323/1058 ! E = 661.5/529
> 706/529 ! F
> 724/529 ! F#
> 792/529 ! G
> 837/529 ! G#
> 885/523 ! A = 442.5Hz*2 absolute a4
> 1883/1058 ! Bb = 941.5/529
> 992/529 ! B
> 2/1
> !
> !
> the relative deviation of the
> 5ths corresponds to the following epimoric decomposition
>
> F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
> B F# C# G# Eb 7532:7533 Bb 5648:5649 F
>
> into the 8 superparticular subfactorization of the PC=3^12/2^19.
>
> > Either I don't understand this,
> or if you prefer the same distribution of the PC=~23.46cents
> in logarithmically values as Cents approximation,
> about the amounts:
>
> F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
> B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F
>
> correspodning to the above 8 epimoric ratios.
> ...

I believe there are still a few mistakes. From the sizes of the
fifths you give, I think that perhaps you meant this:

558/529 ! C#
592/529 ! D
2511/2116 ! Eb = 627.75/529
1323/1058 ! E = 661.5/529
706/529 ! F
744/529 ! F#
792/529 ! G
837/529 ! G#
875/523 ! A = 442.5Hz*2 absolute a4
1883/1058 ! Bb = 941.5/529
992/529 ! B

This will result in:
F~ -1.636 ~C~ -3.276 ~G~ -5.839 ~D~ -5.784 ~A~ -5.953 ~E~ -0.436 ~B
B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F

--George

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

Hi George,
> > epimoric decomposition
> >
> > F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
> > B F# C# G# Eb 7532:7533 Bb 5648:5649 F
> >
> >Cents approximation,
> > about the amounts:
>
> >F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
> > B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F
> >
>
> I believe there are still a few mistakes.
in deed, even that *.scl-file contained some unfixed bug.

> From the sizes of the
> fifths you give, I think that perhaps you meant this:
>
Now -as far as i can see- those ratios appearto be correct:
> 558/529 ! C#
> 592/529 ! D
> 2511/2116 ! Eb = 627.75/529
> 1323/1058 ! E = 661.5/529
> 706/529 ! F
> 744/529 ! F#
> 792/529 ! G
> 837/529 ! G#
> 875/523 ! A = 442.5Hz*2 absolute a4
> 1883/1058 ! Bb = 941.5/529
> 992/529 ! B
>
Many thanks again for that repair.

> This will result in:
when considering more evaluated digits,
as calculated by "Google"s arithmetics, which yields:

> F~ -1.636 ~C
(1 200 * ln(1 058 / 1 059)) / ln(2) = ~-1.63555425...

> ~C~ -3.276 ~G
(1 200 * ln(528 / 529)) / ln(2) = ~-3.27575131...

>~G~ -5.839 ~D
(1 200 * ln(296 / 297)) / ln(2) = ~-5.83890621...
arises that deviation here due to your's rounding procedere?

~D~ -5.784 ~A
1 200 * ln(295 / 296)) / ln(2) = ~-5.85866566...
arises that deviation here due to your's rounding procedere?

~A~ -5.953 ~E
(1 200 * ln(294 / 295)) / ln(2) = ~-5.8785593...

same question as for D~A?
Or what else could be the reason for the tiny
discrepancy amounting about tiny 1/10 Cents
inbetween ours calculations of the relative deviations
in the tempered 5ths flatnesses?

~E~ -0.436 ~B
(1 200 * ln(3 968 / 3 969)) / ln(2) = ~-0.436243936...

> B F# C# G# all just pure 5ths

Eb~ -0.2298 ~Bb
(1 200 * ln(7 532 / 7 533)) / ln(2) = ~-0.229835254...

Bb~ -0.306 ~F
(1 200 * ln(5 648 / 5 649)) / ln(2) = ~-0.306494477...

at least we both do agree now in all others 5ths except D~A~E.

What do you think about that well-temperement,
with an almost JI the C-major chord:

C:E:G = 4 : 5*(2646/2645) : 6*(529/528)

Yours Sincerely
Andreas