Yahoo Tuning Groups Ultimate Backup tuning The grand majestic ultimate well-temperament of all times :)

Dear George, thank you for your evaluation. More in between the lines:

On May 18, 2010, at 9:34 PM, gdsecor wrote:

>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> So, what do you think of it? Is it deserving of its palatial name?
>>
>> Oz.
>
> Well, after looking at the numbers, I'm frankly puzzled about the
> hyperbole. Although it's a low-contrast well temperament with
> reasonably good harmonic balance and no harmonic waste, I could say
> the same about many other well-temperaments. There are simple brat
> numbers for about 2/3 of the major & minor triads, which although
> good, is not exceptional.
>

The exceptional feature is that, you have all that under the hood of
"tunability-by-ear".

> Normally, I would not be so picky, but since you are touting this to
> be a stand-out, I feel obligated to point out what I feel are
> shortcomings that would disqualify it from "all-time-great" status
> in my book:
>
> 1) Fifths D-A and E-B are narrow (-3.9c & -3.5c, respectively),
> while A-E is just. I tend to frown upon a tuning in which tempered
> fifths "zig-zag" in this manner.
>

The optimization here was to secure the A4=440 Hz international
diapason (concert pitch) where the reference note A's relative
frequency (in the Excel worksheet) would yield 220 when multiplied by
either of 131, 131.25, 131.5 or 131 1/3, thus resulting in simple beat
frequencies for fifths when working with a denominator based on them -
which in turn would facilitate setting the temperament on an acoustic
instrument by ear.

Andreas had chosen 131 as the denominator of his tunable-by-ear simple
and proportional beating Well-Temperament ratios. I discovered a
similar temperament based on the same denominator number by myself
through another route. Then I tried 131.25, leading to UWT nr.2.
Lastly I tried 131.3333, leading to UWT nr 3. This is my best try so
far.

I also worked a bit on 131.5 as a multiple for reaching 220, but that
yields a 891 cent A, which falls out of the scope of the Well-
Temperament scheme to that of Modified Meantone. Though Temperament
Ordinaires are a blessing and a breath of fresh air compared to 12-
tone Equal Temperament, I wasn't successful in finding satisfactory
beat ratios using 263 (or its multiples) as the denominator.

> 2) The G-major brat is 2-4/7, which is not as simple as I would like
> to have for a triad this close in the circle of 5ths to C.
>

The chord G-B-D has the 5th/Mrd/mrd ratio as 3:7:18. That means the
brat for M3rd/5th is 7:3, m3rd/5th is 18:3 and m3rd/M3rd is 18:7.

Not the greatest combination perhaps for so close a key as G Major,
but still 7-limit.

I had to make a little sacrifice here to preserve the brat integrity
of other tonalities. I could have chosen 589/394 as the fifth, but
that would only serve to make things worse.

In comparison, your latest revision of 5/23 comma TX, whilst flaunting
very good brats for the spectrum of Majors from Eb-G plus E and B -
along with the Minors spectrum C-B (fabulously at that!), still has an
issue with D Major in the 11-limit (8:22:11) and far-off keys going up
as much as 53-limit!

My temperament's brats are all in the 13-limit range.

And let's do justice to the fact that you are working on a Modified
Meantone, and I at a Well-Temperament scheme. One cannot have the same
type of optimization strategy for both schemes.

> 3) The 5.9-cent error in G-D is rather excessive for a low-contrast
> temperament. I would expect all fifths to have <5 cents error. I'm
> rather puzzled why this was done, considering that it didn't result
> in a simple brat for the G-major triad.
>

By having assigned 330/197 to A in order to acquire simple beat
frequencies for fifths, I had to travel to that ratio thru a cycle-of-
fifths using the same denominator or its multiples. Higher multiples
often mean more complex beat rates. I consigned myself to working with
197 and 394 as the denominators in my ratios.

Still, you are overlooking the 4 cent wide wolf at G#-Eb in your 5/23
comma TX. My previous UWTs feature only 3 cent wide Super-Pythagorean
fifths (if we are to make the issue of 1 cent here).

Besides, all my fifths in UWT nr.3 are veering toward one side, at
most by 6 cents. Your usage of fifths, on the other hand, does make
your temperament a bit "wobbly", no?

But, by all means, you are free to experiment with these UWTs to try
to find better results than I.

> 4) The harmonic balance is skewed toward the flat side of the
> circle: the four major triads with lowest total error are (in order > of increasing error) F, Bb, C, and G.
>

Skewed by one fifth flatwise? Yes. I prioritized the balanced error
spread for the minors centered on A, albeit, at the price of centering
the Majors on F, as you have done in your TX. However, notice, that Eb
Major and A Major are expressed with the same amount of accidentals
and their triad errors are almost equal in my UWT. Why should A Major
have the advantage over Eb Major? Similarly, I have sort of equalized
the triad error for C minor and F# minor. Why rather one to the other?

Yet you have, in your 5/23 comma TX, greatly favoured A Major to Eb
Major and F# minor to C minor, although they are expressed with the
same amount of accidentals. There is no logic to this approach if you
meant to optimize for keys starting on naturals.

> Evidently, we aren't judging temperaments by the same standards.
>

Apparently! My UWT nr.3 seems more balanced in terms of:

1. Lack of harmonic waste.
2. No Super-Pythagorean fifths.
3. Four instances of pure fifths
4. 13-limit brats (E-F# Major and Bb Minor only)
5. Tunability-by-ear
6. Thirds deviations less than a comma.
7. Balanced error spread centered on F Major and A minor.

> But if it's any consolation, I don't happen to have a better
> alternative for a low-contrast well-temperament at the moment. :-)
>

I'm sure you will soon!

> --George
>

Oz.

🔗Ozan Yarman <ozanyarman@...>

Dear Brad, here is a recipe for UWT nr.3:

Make sure you memorize the ticking rhythm of a metronome at 60 beats
per minute. Take a tuning fork or tuner device to calibrate A4 to 440
Hz.

Then, tune the fifth D-A to yield a narrow fifth beating 2 times per
sec.

Next, G'-D (G in the lower octave) again to a narrow fifth 2 times per
sec.

Then, C'-G' (still in the lower octave) to 3 beats per 2 seconds.

Find the pure octaves of C' and G', which are C, c, and G.

Also tune D' from D.

From C, go down a fifth to F', again 3 beats per 2 seconds.

Find the pure octave of F', which is F.

From F, down to Bb', yielding once again 3 beats per 2 seconds.

Check Bb'-D, should make 6 beats per second. If it's hard to count
this way, go an octave down from Bb' and count 3 beats per second
against D.

Tune the octave of Bb' which is Bb.

___________________________

Stop right there. Back to A.

Go down an octave, find A'.

A'-E is pure

E-B is a narrow fifth beating 2 times per sec.

B-f# (upper octave) is a narrow fifth beating once per sec.

Find the octave of f#, which is F#.

Check D' against F# (or D against f#). Should beat 5 times per second.

F#-c# is a narrow fifth beating 3 times every 2 seconds.

Find the octave of c#, which is C#.

C#-G# is pure.

Go down an octave from G# to G#'

G#'-Eb is pure too.

Lastly, Eb-Bb should sound beatless and pure.

_________________________

Voila!

Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 18, 2010, at 6:36 PM, bplehman27 wrote:

> Could somebody please provide instructions to tune it by ear on a
> harpsichord? Can it be done that way in a couple of minutes? If
> not, it's not gonna be tested....
>
> Thanks,
>
> Brad Lehman
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> So, what do you think of it? Is it deserving of its palatial name?
>
>
>
>
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>

🔗Ozan Yarman <ozanyarman@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> Just the ratios:
>
> 1/1 = C unison
> 208/197 = C#
> 441/394 = D
> 234/197 = Eb
> 495/394 = E
> 263/197 = F
> 555/394 = F#
> 295/197 = G
> 312/197 = G#
> 330/197 = A
> 351/197 = Bb
> 741/394 = B
> 2/1 = C' octave
>
Hi Oz, Gene, George & all others,
that's when represented in Werckmeister's terms
an Collatz-Sequence analysis:

[Please view it better under the option: Fixed-wide-Font]

C: 197.
G: (D/3 := 147 294 < ) 295. 590 ( < 591 := C*3 )
D: (A/3 := 55 110 220 440 < ) 441. := 147*3
A: 165 330. := 110*3
E: (B/3 := 247 494 ) 495. := A*3
B: (F#/3 := 185 370 740 < ) 741. := 247*3
F#: 555. := 185*3
C#: 13 26 52 104 208. 416 832 1654 (<1655 := F#*3)
G#: 39 78 156 312. := C#*3
Eb: 117 234. := G#*3 = C#*9
Bb: 351. := Eb*3 = C#*27
F: 263. 526 1053 (< 1053 := Bb*3)
C: 197. 394 788 (<789 := F*3)

fine, that works well,
but but why on earth had you changed now yours choice of the reference
an 5th lower at D:=441
instead of the common ususal standard convention: A=~441 ?

Also attend the recent remark from my Switzerland colleauge:
/clavichord/topicId_unknown.html#9911
"
The idea or notion, yes - but let's look at the practicalities. As a practical temperament, ET probably never or rarely happened, due to the difficulty of tuning the successive fifths accurately. As others have no doubt found, ET is difficult to lay without a little machine (which the 18th century didn't have). I found that the temperament rarely turned out identical, and needed tweaking (these were the days when ET was all I knew about). The result generally slightly favoured one or another key, and not always the same one. So a) it took much longer to tune and b) the end-result was not consistent. I think nevertheless that the goal to tune ET was there in the 18th century, but it was considerably easier and quicker to tune an unequal temperament which came close to ET, and was consistent in its end result. I have to say that Kirnberger, in my experience, is the best solution.

"...wonders whether it is possible to tune nine successive just fifths any more precisely (as in K's scheme)."

I wonder which of Kirnberger's tunings he's talking about? I now use Kirnberger 2 (not 3) by default. This does not entail tuning 9 consecutive 5ths, where the posibilty for error is indeed quite high. One works from C, not A, and the three arms of the temperament all radiate from C; 1) a chain of fifths, almost pure, to F, Bb etc as far as Db (C#) (5 in all). 2) Returning to C the two fifths to D. 3) Again returning to C a pure major third to E, followed by two fifths to F#, and finally; 4) The placing of A exactly between D and E. If an error occurs during the processs, it is easy and quick to find out where. The fact that C/E is pure ensures that C major and surrounding keys still sound the best. The temperament is easy to tune, consistent, and pleasant in all keys. I can tune a five-octave clavichord or harpsichord in 20 minutes. Tantalising and logical though the theory was (is?), I think ET remained just that until the advent of exactly programmed
electronic tuning machines. It is the only temperament, if it is insisted upon, for which I use my Korg.
I have found that modern orchestras will accept Kirnberger 2 in place of ET - the only slight difficulty is getting the overall pitch right - as A is the last note to be tuned this can effect the overall pitch, but with modern electronics the C can be set to produce he required 440 (or 442) A.
"
[So far about that statement of an highly esteemed expert-tuner.]

Just remember my own:
/tuning/topicId_76237.html#76326?var=0&l=1
"
as cycle of a dozen 5ths:

110A_2 220A_3 440Hz=A_4
329 =E_4 (< 330 := 3*A_2)
493.5=B_4 987B_5 := 3*E_4
185F#_3 370 =F#_4 740F#_5...2960F#_7(<2961:=3*B_5)
277.5=C#_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
!
! [Eof]
...
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?
...

In so far mya I do understand Oz's
socalled "majestic-ultimate"
contribution in that way?

See also the already similar precursor of the above ones:
/tuning/topicId_73506.html#73536?var=0&l=1
"
A: 440cps 220 110 (= 330/3 > 329/3)
E: (3*110 = 330 >) 329
H: 987 = 3*E (The German notename 'H' corresponds to English 'B'...)
F#: (3B = 2961>) 2960 1480 740 370 185
C#: 555 = 3*F#
G#: (1665>) 1664 832 416 208 104 52 only-on-real-big-organs: 26 13
Eb: 39
b: 117 (... and also the German 'b' to the English pitch-name 'Bb')
F: 351
C: (3*F = 1053>) 1052 526 263=middle_C4
G: (3*C = 789>) 788 364 197 (>196 98 49=7^2 from Werckmeister VI)
D: (3*G = 591>) 589 (>588 294 147)
A: (441>) 440 cycle ready done!
"

bye
A.S.

🔗bplehman27 <bpl@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Oz,

Charles Lucy has a slew of 12 notes per octave tuning files. If you want to
avoid the (2?) wolfs one must use the correct tuning for your tonality and
avoid far modulations. (Or I guess change Lucy tuning mid-stream but I've
not done that yet.)

As for my Roland GR-20. I have a Fender Mustang guitar with a special pick
up that senses each string individually. The Roland GR-20 can change this to
6 independent midi channels. Using Fractal Tune Smithy I can send this set
of midi signals to my computer and have it re-tuned using on-the-fly relay
tuning and send the new re-tuned midi signal to my Roland GR-20 which has
decent synthesized guitar pre-sets. So in that way I can indeed try your
tuning without re-fretting my guitar.

Chris

On Wed, May 19, 2010 at 4:38 PM, Ozan Yarman <ozanyarman@...>wrote:

>
>
> O Chris,
>
> I wouldn't say that my UWT trials are akin to Lucy-Harrison "meantone"
> setting. True meantones circulate at 19, 31, 50, 55 etc... tones. Only
> Modified Meantones and Well-Temperaments can fit into 12 tones (in the
> Western common-practice sense of course). You won't see any wolfs with
> those two schemes (except perhaps thirds?).
>
> Ah, but UWT nr.3 won't suit your guitar with the frets already aligned
> at the 12th root of 2. You need irregularly and strategically placed
> individual frets to accurately capture the intervals of my UWT.
>
> Yes, I find 380-384 cents a most desirably region for the major third.
> This is the crux of the Turkish perde segah, the focal point of many
> delectable maqams such as Segah, Huzzam, Huseyni, Ushshaq, Karjighar,
> etc...
>
> Oz.
>
> ✩ ✩ ✩
> www.ozanyarman.com
>
>
> On May 19, 2010, at 4:54 AM, Chris Vaisvil wrote:
>
> > Hi Oz,
> >
> > Thanks for the reply. So then this is more akin to the Harrison (Lucy
> > Tuned) mean tone - except without a true wolf interval?
> >
> > I will try this tuning on my guitar (via the Roland Gr-20 and fractal
> > tune smithy). I generally like chromaticism and this sounds like a
> > temperament that will be kinder than Lucy tuning - though I *love* the
> > in tune thirds in Lucy tuning.
> >
> > Chris
> >
> > On Tue, May 18, 2010 at 6:09 PM, Ozan Yarman <ozanyarman@...<ozanyarman%40ozanyarman.com>
> > > wrote:
> >>
> >>
> >>
> >> Dear Chris,
> >> Well Temperament is meant to favour natural keys and tonalities
> >> with few accidentals against those with lots of accidentals in
> >> their key signatures.
> >> As I have said to George just a while ago, my UWT nr. 3 has a
> >> balanced triad error spread centered on F Major and A minor.
> >> When you travel to the far side of the circle of fifths from C in
> >> such a setting, the major thirds grow larger and minor thirds grow
> >> smaller. This effect is much more noticable with Modified Meantone
> >> Temperaments.
> >> The result is often referred to as key-colour or tonality-contrast.
> >> I prefer such colour and contrast against the flaccid staleness of
> >> equality at every key with equal division schemes.
> >> Oz.
> >> ✩ ✩ ✩
> >> www.ozanyarman.com
> >> On May 18, 2010, at 5:16 AM, Chris Vaisvil wrote:
> >>
> >>
> >> Hi Oz,
> >>
> >> If I read your screen shot correctly - it seems that you have pure
> >> 5ths at the expense of the 3rds?
> >> Or put in another way - your major 3rds are generally sharp but
> >> have much more variability than the 5ths?
> >>
> >> Thanks,
> >>
> >> Chris
> >>
> >
> >
> > ------------------------------------
>
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>

🔗Ozan Yarman <ozanyarman@...>

Uh, sorry George, nice try but no cigar. My turn to be picky. Here are
my reasons for not finding this GS-WT24d likable in terms of
mathematical purity:

1. 7-limit brats for C Major, D minor, E minor and C minor! And a whopping 17-limit monstrosity right on G minor. You had critized me
earlier for letting a complex brat as 3:7:18 take hold on G Major, and
here you have similar or more complex brats for the starting tonality
and four proximous minors to boot!

2. The triad error spread is not uniform in both directions from
either the focality of C Major or A minor. UWT nr.3 spreads uniformly
in fifths cycle in both the Major (< F >) and minor (< A >) keys.

3. Your focal points with least triad error are at G major and B
minor! At least I did not skew the minor focality sharp-wise in my UWT
nr.3.

4. All my brats are 13-limit, yours go up to 17-limit.

5. Surely, no one can tune this temperament by listening to as complex
a beat frequency as 1.76 beats per sec, unless you want us to count 9
beats every 5 seconds at 450 Hz for A!

6. As you mentioned, the zigzag there between C#-G# in the midst of F#-
C# and G#-Eb. I don't know which one is worse? A pure fifth amongst
two tempered fifths or a tempered fifth among two pure fifths.

___________________

Battle of Temperaments tie-breakers:

-My UWT nr.3 in possession of two 6 2/3 and two 9 1/3 beats per second
Major thirds (F, C, G, A) and two 20 + two 13 1/3 + two 24 beats per
second minor thirds.

-Your WT has 8.8 beat per seconds in four Major thirds, and 21.12 in
two other Major thirds while flaunting four 17.6, two 12.32 and two
31.68 beats per second minor thirds.

No harmonic waste in either temperament.

Doubly skewed tonality focus in GS-WT24d versus -6 cent tempered fifth
in UWT nr.3 even out?

Both equally undeserving of the Lumma Temperament Championship Seal. :)

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 18, 2010, at 11:12 PM, gdsecor wrote:

>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>>
>> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>>>
>>> So, what do you think of it? Is it deserving of its palatial name?
>>>
>>> Oz.
>>
>> Well, after looking at the numbers, I'm frankly puzzled about the
>> hyperbole. Although it's a low-contrast well temperament with
>> reasonably good harmonic balance and no harmonic waste, I could say
>> the same about many other well-temperaments. There are simple brat
>> numbers for about 2/3 of the major & minor triads, which although
>> good, is not exceptional.
>> ...
>> Evidently, we aren't judging temperaments by the same standards.
>>
>> But if it's any consolation, I don't happen to have a better
>> alternative for a low-contrast well-temperament at the moment. :-)
>
> When I said that, I forgot about the following.
>
> For the record, here's a two-year-old temperament that remains my
> best effort to date for a low-contrast well-temperament. This is a
> slight modification (changing Bb from 265/149 to 531/298, as
> suggested by Andreas Sparschuh in msg. #7606) to a (previous flawed)
> well-temperament I posted in April 2008 (msg. #76023):
>
> Secor well-temperament 24d
> 315/298
> 167/149
> 177/149
> 187/149
> 199/149
> 210/149
> 223/149
> 236/149
> 250/149
> 531/298
> 280/149
> 2/1
>
> Summary of features:
> Good harmonic balance, with no harmonic waste;
> Maximum tempered 5th is -4.62 cents (A-E);
> The fifths from F thru E all beat at same rate (1.76 bps);
> The major 3rds on F, G, D, & A all beat at same rate (8.8 bps);
> The major triads on F, G, D, & A all have brat = 2;
> The major triads on Eb, B, F#, & G# all have brat = 1.5 (just 5ths);
> Minor-triad brats are simpler than in most rational temperaments.
>
> Summary of flaws:
>
> "Zig-zag" tempered fifth (C#-G#) is adjacent to two just 5ths (F#-C#
> & Ab-Eb); however it is lightly tempered (1.83 cents);
> Major 3rds on B & F# are tempered by 19.67 cents, which will not win
> the Lumma seal of approval (max allowed ~18.2 cents).
>
> Feature or flaw? -- this has not much relevance for me:
> Prime limit for brats: 17.
>
> There it is; judge for yourself.
>
> --George

🔗Ozan Yarman <ozanyarman@...>

🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

Dear Brad,

Yes, UWT nr.3 flaunts 4 pure fifths and all the others are tempered
differently from each other in the -6 to -1 cent range.

If you work with the ratios I have given for UWT nr.3, which were:

1/1
208/197
441/394
234/197
495/394
263/197
555/394
295/197
312/197
330/197
351/197
741/394
2/1

then you must set A4 to 440 Hz to realize the correct beat
frequencies. Surely, you can choose another diapason, the brats will
remain intact even then, but you shall not be able to set the
temperament by ear via the beat frequencies, because they will be
altered, however slightly.

But if your strategy is to start tuning by listening to the beat
frequencies, you can, by all means, start from an A that is not 440
Hz. You can, for instance, select anywhere from 420 to 460 without the
slightest noticeable deviation in the size of the fifths when
utilizing the same recipe I have provided, save the fact that you have
no way to check against the thirds then and no pure fifth closing the
G#-Eb diminished sixth gap. Also, you will destroy the beat rates in
this fashion for any diapason other than 440 Hz for A. Proportional
brats being the exclusive feature of UWT nr.3, one ought not want to
pursue such a course.

There is, however, an alternative. You can both choose a new diapason,
preserve the brats and beat frequencies... if you discard "second" as
the unit of time measurement and adopt a modified unit proximous to it.

For example, instead of the second, your time increment can be
1.025641025641026 seconds (58.5 beats per minute). Then you can set
the temperament exactly using the same recipe at A=429 Hz. Modify your
time increment to find the diapason you need.

If you have diapasons to suggest in order to historically realize a
proportional beeating Well-Temperament all the while preserving the
second as the unit of time, by all means, I shall labour to find ear-
tunable UWTs based on them.

_________________

The beating frequencies will occur exactly as prescribed no matter
when you strike the notes in a chord, and the brats will take hold
instantly whether you strike them at the same time or arpeggiate. It's
the same with 4:5:6. All the waves will phase-lock independent of
their entry to the scene. This says something musical to us,especially since the relationships between the musical tones depend on
the beating or the lack of beating of their partials!

True, one does not ordinarily or necessarily focus on the beats
between frequencies, one is perhaps not even aware of their presence
under regular circumstances. But they surely register in the brain,
perhaps subconsciously, as we listen to and "calculate" intervals.
What makes a fifth a wolf and what makes it pure is all dependent on
the beating of its harmonics unless you are working with sine waves.

Frankly, I'm surprised that you of all people disregard the role of
beats in the tuning of musical intervals Brad! Especially since you
have procured recipes to tune your Bach Temperament by ear based on
them! Maybe an organ or harpsichord tuner is not searching for exact
beat counts, but there are surely acceptable ranges for which beats
are permissible, and key zones where they are more desirable than
others. A 692 cent fifth would surely sound awful in Classical Western
music simply because it would create awful beats that fall out of the
range of permissability in the proper context. Ditto for the wolf
fifth at G#-Eb for Meantone Temperaments, unless used for effect in
the right setting. How about a 440 cent Major third? Or a 330 cent
minor third? What do you presume is the reason they would resonate
monstrously in a Bach Toccata or a Haydn Sonata? One need not listen
to their beat speeds in order to deprecate them there. They register
in the brain and signal a response. Please correct me if I
misunderstood you here.

In sooth, having tried all the UWTs on my electronic piano using Logic
Pro 8 sampled Steinway piano sounds, I definitely like the moments
when I notice some regularity to the beat patterns in the flow of
music. I appreciate it immensely when the air is filled with beat
rates that resonate to the clock. I am cognizant of the beats more
than the untrained layman here perhaps, but I have an inkling that
commoners are more receptive to slight intervallic modifications than
oft surmised. All mammalian brains will most likely be attuned to
these special beat rates whether or not they are directly cognizant of
them.

____________________

UWT nr.3 is the Layer I of my projected Yarman-36d, unless I discard
the previous c, in which case the 36-tone tuning based on UWT nr.3
will be Yarman-36c. I also want it to work as a replacement for 12-ET.
The rationale is, of course, to faciliate access to beautiful music-
making. What other practical purpose is there for a temperament?

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 19, 2010, at 9:24 PM, bplehman27 wrote:

>
>
>
>
>
> Thanks! So, in summary: this is a temperament that has four pure
> 5ths (A-E, and C#-G#-D#-A#), while all eight of the others are
> tempered differently from one another? And it works only at A=440Hz?
> And all the tempered 5ths (except maybe the leftover Bb-F) are
> supposed to beat at some simple multiple of 1 per second, each in
> some specific octave?
>
> I'll try it out, but on paper I don't see how any of this
> synchronous-beating business is supposed to do anything perceptibly
> desirable (or undesirable) in the music. It just looks to me like a
> mathematical pursuit, for its own ends: to be able to say
> theoretically that some beats are lining up, and that this matters
> somehow if we'd ever strike several notes exactly together -- and
> then hold them for several seconds while nothing else happens.
>
> I know that when I play and listen to harpsichord music, I don't
> listen for beat speeds anywhere, or notice that any triads ever have
> anything synchronized (or unsynchronized) within them. Except for
> final dyads or chords, there's never any time to perceive clearly
> what the beats are doing, or compare them with the beats in other
> dyads or chords. The music invariably has other interesting things
> happening, anyway, more noticeable than trying to count anything out
> of the sound during sustained notes. The beats are just too subtle
> to matter, IMO, next to the higher priorities of hearing harmonic
> quality of an interval, the tension/resolution, any melodic
> movement, and the broader Affekt of the music as it moves.
>
> Overall, please help me understand: is this some attempt to get
> keyboard instruments to project a consistent vibrato when playing at
> various places in the compass? If so, why is such a consistency a> desideratum?
>
> Brad Lehman
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Dear Brad, here is a recipe for UWT nr.3:
>>
>> Make sure you memorize the ticking rhythm of a metronome at 60 beats
>> per minute. Take a tuning fork or tuner device to calibrate A4 to 440
>> Hz.
>>
>> Then, tune the fifth D-A to yield a narrow fifth beating 2 times per
>> sec. (...)
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
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> Yahoo! Groups Links
>
>
>

🔗Ozan Yarman <ozanyarman@...>

Andreas,

✩ ✩ ✩
www.ozanyarman.com

On May 19, 2010, at 6:21 PM, Andreas Sparschuh wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>> Just the ratios:
>>
>> 1/1 = C unison
>> 208/197 = C#
>> 441/394 = D
>> 234/197 = Eb
>> 495/394 = E
>> 263/197 = F
>> 555/394 = F#
>> 295/197 = G
>> 312/197 = G#
>> 330/197 = A
>> 351/197 = Bb
>> 741/394 = B
>> 2/1 = C' octave
>>
> Hi Oz, Gene, George & all others,
> that's when represented in Werckmeister's terms
> an Collatz-Sequence analysis:
>
> [Please view it better under the option: Fixed-wide-Font]
>
> C: 197.
> G: (D/3 := 147 294 < ) 295. 590 ( < 591 := C*3 )
> D: (A/3 := 55 110 220 440 < ) 441. := 147*3
> A: 165 330. := 110*3
> E: (B/3 := 247 494 ) 495. := A*3
> B: (F#/3 := 185 370 740 < ) 741. := 247*3
> F#: 555. := 185*3
> C#: 13 26 52 104 208. 416 832 1654 (<1655 := F#*3)
> G#: 39 78 156 312. := C#*3
> Eb: 117 234. := G#*3 = C#*9
> Bb: 351. := Eb*3 = C#*27
> F: 263. 526 1053 (< 1053 := Bb*3)
> C: 197. 394 788 (<789 := F*3)
>
> fine, that works well,
> but but why on earth had you changed now yours choice of the reference
> an 5th lower at D:=441
> instead of the common ususal standard convention: A=~441 ?
>

This doesn't make sense. A is already 330/197 at 440 Hz. That's the
convention today. UWT nr. 3 is optimized for the concert pitch as it is.

> SNIP
>
> Just remember my own:
> /tuning/topicId_76237.html#76326?var=0&l=1
> "
> as cycle of a dozen 5ths:
>
>
> 110A_2 220A_3 440Hz=A_4
> 329 =E_4 (< 330 := 3*A_2)
> 493.5=B_4 987B_5 := 3*E_4
> 185F#_3 370 =F#_4 740F#_5...2960F#_7(<2961:=3*B_5)
> 277.5=C#_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
> !

Ah, I see you have utilized multiples of 131.5 in the denominators as
I had started to work with. Close, but no cigar. the fifths are -7
cents tempered; you could have used 393/263 for G and 493/263 for B as
I did for better brats and less fifth errors. Though we have the same
uniform triad error spread from the focalities of F Major and A minor,
this temperament, even with my modifications, is a horror of brats.

> ! [Eof]
> ...
> Quests:
> 1. Who in that group dares to try out that one on his/hers own piano?

Not me!

> 2. Has anybody suggestions for better ratios in order to improve it?
> ...
>

Try 393/263 for G and 493/263 for B. But still not good enough.

> In so far mya I do understand Oz's
> socalled "majestic-ultimate"
> contribution in that way?
>

Theatrics aside, I am really confident that my suggestion awaits to be
surpassed just yet!

> See also the already similar precursor of the above ones:
> /tuning/topicId_73506.html#73536?var=0&l=1
> "
> A: 440cps 220 110 (= 330/3 > 329/3)
> E: (3*110 = 330 >) 329
> H: 987 = 3*E (The German notename 'H' corresponds to English 'B'...)
> F#: (3B = 2961>) 2960 1480 740 370 185
> C#: 555 = 3*F#
> G#: (1665>) 1664 832 416 208 104 52 only-on-real-big-organs: 26 13
> Eb: 39
> b: 117 (... and also the German 'b' to the English pitch-name 'Bb')
> F: 351
> C: (3*F = 1053>) 1052 526 263=middle_C4
> G: (3*C = 789>) 788 364 197 (>196 98 49=7^2 from Werckmeister VI)
> D: (3*G = 591>) 589 (>588 294 147)
> A: (441>) 440 cycle ready done!
> "
>

Do we have this in ratios?

> bye
> A.S.
>
>

Cordially,
Oz.

🔗genewardsmith <genewardsmith@...>

🔗bplehman27 <bpl@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> (...)
> Frankly, I'm surprised that you of all people disregard
> the role of beats in the tuning of musical intervals Brad!
> Especially since you have procured recipes to tune your
> Bach Temperament by ear based on them! Maybe an organ or
> harpsichord tuner is not searching for exact beat counts,
> but there are surely acceptable ranges for which beats
> are permissible, and key zones where they are more
> desirable than others. (...)
>

"Disregard the role of beats"? Nay. In tuning my harpsichords, and other people's harpsichords, I use the beats all the time. (And, all these "procured" recipes on my http://www.larips.com site are *by* me, describing the way I set all those various temperaments by ear.)

The crucial thing here is: after setting one or two intervals at the beginning of a temperament bearing (such as the tenor F-A major 3rd of 3 beats/sec in 1/6 comma systems, from a diapason at or near A=440) -- all the other use of beats throughout the temperament *is not* by comparing them against any external standard: as "per second", or in any other fixed tempo, where we know ahead of time what it ought to be. It is all done by comparing the beats of intervals only with *one another*.

How do I know that a note such as G is similarly tempered from both the C and the D below it, in a temperament where both C-G and G-D are supposed to have the same fraction of some comma? I move the G until I find the spot where the speed of the beat in the C-G 5th is 2/3 the speed of the beat in the D-G 4th: duplets against triplets (playing the two intervals separately, of course, not all three notes at once). That is the only thing I need to know, in placing that note. Numeric beat rates are irrelevant.

I have explained all of that here:
http://www-personal.umich.edu/~bpl/larips/tetrasect.html

The quality of the interval (perceived directly) is of course defined by the presence or absence of beats in it -- the more beats, the more "sour" it is, along with the understanding that everything doubles at each octave -- , but again, I'm *not* listening for beat-counting against any external standard of time or rate. I compare the beat speeds of neighboring intervals: "that one is twice as fast as this one", or "that one makes triplets against this one's duplets", or "that one is slightly faster than this", but only to evaluate and finely adjust the intervals that I have already set by perceived quality alone. 90% of the work is done without the beats; they are used only to confirm the results, or make tiny tweaks.

All of that stuff with numerically-measured speeds is *digital*. I work in analog. The harpsichord has to sound "in tune" to a brain that processes analog input (intervals sounding the same as or different from one another, by quality), not a digital brain that sits there counting everything that it encounters and assigning numbers to make value judgments. I can get the quality of C-G and D-G to be the same, or very nearly so, without consciously listening for (or counting, or comparing) any beats. If one of them is too good while the other is too sour, my wrist automatically moves in the right direction to rotate the pin and even these out. I can tune the harpsichord while thinking about other things!

As a listener, I almost never listen for any beats. Music is invariably doing other things more interesting than that.

This is why I don't "get" what people here write about, regarding brats and other synchronous stuff. I fail to see or hear how it is musically important. It looks to me like just a bunch of mathematical overkill, playing with numbers for the amusement of playing with numbers. The beats in intervals are useful to me only to get my tuning task (the physical process) done well and quickly, so I can get on to the more interesting task of practicing or performing the music.

Brad Lehman

🔗Carl Lumma <carl@...>

🔗Ozan Yarman <ozanyarman@...>

Dear Brad,

✩ ✩ ✩
www.ozanyarman.com

On May 20, 2010, at 11:28 PM, bplehman27 wrote:

>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>> (...)
>> Frankly, I'm surprised that you of all people disregard
>> the role of beats in the tuning of musical intervals Brad!
>> Especially since you have procured recipes to tune your
>> Bach Temperament by ear based on them! Maybe an organ or
>> harpsichord tuner is not searching for exact beat counts,
>> but there are surely acceptable ranges for which beats
>> are permissible, and key zones where they are more
>> desirable than others. (...)
>>
>
>
> "Disregard the role of beats"? Nay. In tuning my harpsichords, and
> other people's harpsichords, I use the beats all the time. (And,
> all these "procured" recipes on my http://www.larips.com site are
> *by* me, describing the way I set all those various temperaments by
> ear.)
>

I know that the recipes there are formulated by you. Remember that
English is my second tongue, so I may not always use the right word
for the job. By "procure", I meant in the figurative context: "arrive
at".

> The crucial thing here is: after setting one or two intervals at the
> beginning of a temperament bearing (such as the tenor F-A major 3rd
> of 3 beats/sec in 1/6 comma systems, from a diapason at or near
> A=440) -- all the other use of beats throughout the temperament *is
> not* by comparing them against any external standard: as "per
> second", or in any other fixed tempo, where we know ahead of time
> what it ought to be. It is all done by comparing the beats of
> intervals only with *one another*.
>
> How do I know that a note such as G is similarly tempered from both
> the C and the D below it, in a temperament where both C-G and G-D
> are supposed to have the same fraction of some comma? I move the G
> until I find the spot where the speed of the beat in the C-G 5th is
> 2/3 the speed of the beat in the D-G 4th: duplets against triplets
> (playing the two intervals separately, of course, not all three
> notes at once). That is the only thing I need to know, in placing
> that note. Numeric beat rates are irrelevant.
>
> I have explained all of that here:
> http://www-personal.umich.edu/~bpl/larips/tetrasect.html
>
> The quality of the interval (perceived directly) is of course
> defined by the presence or absence of beats in it -- the more beats,
> the more "sour" it is, along with the understanding that everything
> doubles at each octave -- , but again, I'm *not* listening for beat-
> counting against any external standard of time or rate. I compare
> the beat speeds of neighboring intervals: "that one is twice as fast
> as this one", or "that one makes triplets against this one's
> duplets", or "that one is slightly faster than this", but only to
> evaluate and finely adjust the intervals that I have already set by
> perceived quality alone. 90% of the work is done without the beats;
> they are used only to confirm the results, or make tiny tweaks.
>

I don't know what you conceive from "counting the beats", but what you
describe above is exactly "counting the beats". You merely substitute
the time increment. Above, you apparently use, instead of the second,
the increment which is the least common multiple of the beat speed you
take as reference. So, you count one fifth's beat as 1 unit per
second, and calibrate the other so that it is 3/2 compared to the
previous, etc...

Wallahi! Your recipe for the Bach Temperament depends 100 percent on
listening to and counting the beats. How can you say you do 90 percent
of the work without the beats? How do you tune the pure fifths if not
by eliminating the beats? And how do you compare one fifth's beat to a
third's if not by the beat count in the time increment you have chosen
that yields simple beat frequency fractions for the whole operation?

I'm amazed.

Moreover you have written:

Middle E from tenor A slightly flat, same quality as these other
fifths. It should beat as a major third from middle C at 4.5 times per
second. (4.2 if you started from an A=415.) Also check it as a 10th
from tenor C, the same 4.5 times per second. An excellent test for
this particular F-A and C-E is found at bar 21 of the WTC book 1 C
major prelude. These two major thirds should have exactly the same
character as one another, although the beat rate is different (since
the pitch is different).Coincidentally (?) this is at the golden mean
of this piece.... The A-E fifth should beat exactly half as fast as
our original F-A major 3rd, sharing the same A.

Clearly, you have overlooked the fact that the tenor C cannot beat
against the E an octave above 4.5 times per second while the middle C
beats the same against it. Was this a typographical error? The beat
frequencies are halved octaves down, and doubled octaves up. Moreover,
the first approach yields a 694.5 cent fifth between A and E while
your "golden mean" approach yields 698 cents. I presume the latter is
correct.

There are even more wrongs here:

Copy middle E down to tenor E. For especially careful accuracy, fuss
with both these E's until all four of the following points are true:
(1) Whatever speed the tenor D-A 5th is, the fifth A-E across middle C
should beat in 3/2 ratio from that speed (i.e. triplets vs duplets).

where the beat speed of D-A is clearly not 1.5 times the beat speed of
A-E according to your recipe (1.694702091 Hz at 440 Hz diapason in the
former versus 1.5 Hz in the latter). Forcing it reduces the A-E fifth
size from 698 cents down to 693 cents.

And here:

(2) Likewise, whatever speed the tenor F-A major 3rd is, the middle C-
E major 3rd is triplets against its duplets. [We are up a fifth on the
keyboard, and all frequencies are 3/2 as much.]

Where the beat speed of F-A is clearly not 1.5 times the beat speed of
C-E according to your recipe (6 Hz at 440 Hz diapason in the former
versus 7.153846154 Hz in the latter). That would make the A-E fifth
701 cents.

And here:

(3) Playing the tenor E as a 4th against tenor A, it should beat exactly half as fast as our original F-A major 3rd. Likewise, the A-E
5th we are testing across middle C should be exactly half as fast as
our original F-A major 3rd.

Where the beat speed of E-A is clearly not 1/2 of the beat speed of F-
A according to your recipe (1.5 Hz at 440 Hz diapason in the former
versus 6 Hz in the latter). That would mean a 694 cent fifth across
middle C.

Unless I have done any miscalculations or misunderstood you, it
appears there is something horribly wrong with this recipe Brad.
Please check your instructions above and consider the math.

> All of that stuff with numerically-measured speeds is *digital*. I
> work in analog.

Nothing is stopping you from working analog with UWT nr.3 if you set
your metronome to 60 beats per minute. Or else, you can set the
metronome to beat anywhere around 60 ticks per minute, take the new
increments as the unit instead of the second, and still set the
temperament by ear using exactly the same recipe I provided. The brats
are always the same if you do it right no matter what the diapason
(determined beforehand from the metronome ticks).

And there is nothing "digital" about the synchronous proportional beat-
rates when you hear them filling the air around you.

> The harpsichord has to sound "in tune" to a brain that processes
> analog input (intervals sounding the same as or different from one
> another, by quality), not a digital brain that sits there counting
> everything that it encounters and assigning numbers to make value
> judgments.

Come on! I have demonstrated that you have made value judgments
instructing the human ear and brain when you proportionalized the beat
of one fifth to another. Howcome this approach is analog, and a
similar approach involving the second (or some other unit) as the time
increment "digital"? There is no conceptual or pragmatical difference
in your approach versus mine, save that, my recipe is simpler and
yields very well calculated beat rates that are in sync.

Not to mention... Having followed your recipe in an Excel worksheet to
the letter, I have ascertained that, between the end result and the
Scala file under your name, there are 3 to 3.6 cent errors on every
tone except G and Bb, and a whopping 7 cents error on Bb! A gross
694.5 cent fifth haunts F-C in the cycle also.

I can send you the Excel file I have worked with, so that you can
cross check the results if you like.

Yet one thing is for sure, you can't go wrong with the recipe of UWT
nr.3.

> I can get the quality of C-G and D-G to be the same, or very nearly
> so, without consciously listening for (or counting, or comparing)
> any beats. If one of them is too good while the other is too sour,
> my wrist automatically moves in the right direction to rotate the
> pin and even these out. I can tune the harpsichord while thinking
> about other things!
>

This is a simple example of comparing the beat frequency of one
interval with that of the other and equalizing them. Once again, it is
a process that requires listening and judging the beat frequencies. In
effect, you are making sure that the former interval's "count" is the
same as the latter's.

The fact that you can do this while "thinking other things" shows
somewhat that the process is geared in the subconscious in your case.

> As a listener, I almost never listen for any beats. Music is
> invariably doing other things more interesting than that.
>

As I had said, you don't need to "concentrate" on the beats during
music. You don't even need to "figure out" the beat rates while
playing or hearing. But if there is some sense to seconds as the unit
time and some sense to obtaining a beautifully resonating temperament
by ear using simple beats, then what argument can you have against it?
Synchronous beat-rates are there not to be "counted", but "enjoyed".
There are strong indications that there exists parallels between
simple integer low-prime beat rates and nicely resonating chords or
polyphonic passages.

> This is why I don't "get" what people here write about, regarding
> brats and other synchronous stuff. I fail to see or hear how it is
> musically important.

That's because you didn't try UWT nr. 3 on your harpsichord using the
instructions you requested and which I provided. Try it first, and
then let's hear your input.

> It looks to me like just a bunch of mathematical overkill, playing
> with numbers for the amusement of playing with numbers.

I already stated that I enjoyed the proportional-beating temperaments
with my sampled piano sounds. The pursuit was not simply mathematical,
but also musical.

So, instead of being so prejudiced and pessimistic about it, why not
give it a try? Let's see how quickly you can set up UWT nr.3, and how
it resonates with you.

> The beats in intervals are useful to me only to get my tuning task
> (the physical process) done well and quickly, so I can get on to the
> more interesting task of practicing or performing the music.
>

Unless you are doing your music in JI, beats will run the scene,
whether you are cognizant of them or not. Still, it is evident that
good music requires at least the subconsious to be satiated with
reasonable beating in every interval under the proper historical-
cultural-social context.

>
> Brad Lehman
>

Cordially,
Dr. Oz.

🔗martinsj013 <martinsj@...>

🔗bplehman27 <bpl@...>

Dr Oz,

I have double-checked and triple-checked all the math. Everything I wrote on my web page that you complained about here was indeed correct, the first time.

(0) Comparing a major 3rd and a major 10th with the same top note (and with a pure octave), the beat rate is the same: 4.5 in the example you singled out. That's always true. "The beat frequencies are halved octaves down" is an accurate observation only if we take *both* notes down an octave, which isn't what I said.

That's a useful general procedure in tuning harpsichords or pianos by ear: if the major 3rd is hard to hear accurately, but you've already got a pure octave in place below the lower note, work on the quality by playing the 10th. Working the other way round, it's also a good test for a pure octave: the major 3rd and the major 10th with the same top note have the same beat rate.

For all the other points here, take a look at the beat-rate table here:
http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
(which is from my
http://www-personal.umich.edu/~bpl/larips/math.html
)

(1) The tenor D-A is 0.99 narrow, and the A-E is 1.49 narrow: a 3/2 ratio, as I said. What's the problem?

(2) The tenor F-A is 2.98, and the middle C-E is 4.46. That's a 3/2 ratio, as I said -- where both intervals are the same size geometrically, and one of them is a 5th higher in pitch than the other, hence the 3/2. What's the problem?

(3) The tenor F-A is 2.98, and the tenor E-A 4th is 1.49: half as fast, as I said. The A-E 5th across middle C is that same 1.49, and it too is half as fast as F-A, as I said. What's the problem?

Apparently, there is nothing "horribly wrong with this recipe" that you were citing from the top section of
http://www-personal.umich.edu/~bpl/larips/practical.html
.

As Steve M correctly pointed out, I provide this beat-rate business on my web site in some of my instructions, only for the benefit of users who are inclined to think in that way by their own habits. Serve the customers! If they have to have fixed tables of beats to be able to conceptualize the temperament, I provide it as reference. Or, if they can tune by ear by SCARCELY EVER COUNTING ANY BEATS against any external standard (which is the main point of my posting yesterday), as I do it on my instruments, even better. I never look at my own generated tables of beat rates when I'm doing the work at the instrument. It's unnecessary information to get the job done.

Cheers,

Brad Lehman

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>(...)
> Moreover you have written:
>
> Middle E from tenor A slightly flat, same quality as these other
> fifths. It should beat as a major third from middle C at 4.5 times per
> second. (4.2 if you started from an A=415.) Also check it as a 10th
> from tenor C, the same 4.5 times per second. An excellent test for
> this particular F-A and C-E is found at bar 21 of the WTC book 1 C
> major prelude. These two major thirds should have exactly the same
> character as one another, although the beat rate is different (since
> the pitch is different).Coincidentally (?) this is at the golden mean
> of this piece.... The A-E fifth should beat exactly half as fast as
> our original F-A major 3rd, sharing the same A.
>
> Clearly, you have overlooked the fact that the tenor C cannot beat
> against the E an octave above 4.5 times per second while the middle C
> beats the same against it. Was this a typographical error? The beat
> frequencies are halved octaves down, and doubled octaves up. Moreover,
> the first approach yields a 694.5 cent fifth between A and E while
> your "golden mean" approach yields 698 cents. I presume the latter is
> correct.
>
> There are even more wrongs here:
>
> Copy middle E down to tenor E. For especially careful accuracy, fuss
> with both these E's until all four of the following points are true:
> (1) Whatever speed the tenor D-A 5th is, the fifth A-E across middle C
> should beat in 3/2 ratio from that speed (i.e. triplets vs duplets).
>
> where the beat speed of D-A is clearly not 1.5 times the beat speed of
> A-E according to your recipe (1.694702091 Hz at 440 Hz diapason in the
> former versus 1.5 Hz in the latter). Forcing it reduces the A-E fifth
> size from 698 cents down to 693 cents.
>
> And here:
>
> (2) Likewise, whatever speed the tenor F-A major 3rd is, the middle C-
> E major 3rd is triplets against its duplets. [We are up a fifth on the
> keyboard, and all frequencies are 3/2 as much.]
>
> Where the beat speed of F-A is clearly not 1.5 times the beat speed of
> C-E according to your recipe (6 Hz at 440 Hz diapason in the former
> versus 7.153846154 Hz in the latter). That would make the A-E fifth
> 701 cents.
>
> And here:
>
> (3) Playing the tenor E as a 4th against tenor A, it should beat
> exactly half as fast as our original F-A major 3rd. Likewise, the A-E
> 5th we are testing across middle C should be exactly half as fast as
> our original F-A major 3rd.
>
> Where the beat speed of E-A is clearly not 1/2 of the beat speed of F-
> A according to your recipe (1.5 Hz at 440 Hz diapason in the former
> versus 6 Hz in the latter). That would mean a 694 cent fifth across
> middle C.
>
> Unless I have done any miscalculations or misunderstood you, it
> appears there is something horribly wrong with this recipe Brad.
> Please check your instructions above and consider the math.

🔗bplehman27 <bpl@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> (...) There is no conceptual or pragmatical difference
> in your approach versus mine, save that, my recipe is simpler and
> yields very well calculated beat rates that are in sync.

Completely different: your approach, holding beat rates constant (when played in some specific octave, cherry-picking!), produces temperaments that have at least half a dozen differently-sized tempered 5ths. Most of the temperaments I work with hold the interval sizes constant GEOMETRICALLY across all or most of the 5ths.

See, for example, my page about various sizes of regular (or "meantone") temps:
http://www-personal.umich.edu/~bpl/larips/meantone.html

>
> Not to mention... Having followed your recipe in an Excel worksheet to
> the letter, I have ascertained that, between the end result and the
> Scala file under your name, there are 3 to 3.6 cent errors on every
> tone except G and Bb, and a whopping 7 cents error on Bb! A gross
> 694.5 cent fifth haunts F-C in the cycle also.
>
> I can send you the Excel file I have worked with, so that you can
> cross check the results if you like.

You can send it if you want, but you've clearly followed the instructions (allegedly "to the letter") incorrectly. My own Excel file is on my very old web page: http://how.to/tune

As I mentioned above, almost every temperament I work with has consistent comma-fraction tempering in most 5ths, if they're not pure. My Excel sheet isn't set up to handle anything that has the constant-beat-rates rule that you take as important.

Whoever produced "the Scala file under [my] name" probably did it correctly, by working with fractions of commas. I wouldn't know, as I haven't seen it recently and I don't have Scala...or any tunable synthesizer, or any burning desire to produce any electronically-generated music. I'm an old-fashioned musician with a doctorate in harpsichord performance. I do all my tuning from either a C or an A tuning fork, with no electronics.

>
> Yet one thing is for sure, you can't go wrong with the recipe of UWT
> nr.3.

Not sure what you mean by "can't go wrong", here. I set it up here on my best harpsichord. It took more than twice as long as tuning by more familiar methods that have consistent-sized 5ths; and I couldn't use my normal checks that go all the way up the treble, proving octaves by testing the intervening 4ths and 5ths for proper consistency (because you have half a dozen differently-sized 5ths....). I gave it my best attempt, and it worked out reasonably well, going over everything twice to be sure it was accurate enough to your specs...which (frankly) seemed like an almost-futile pursuit, to me. Harpsichords don't stay that tightly in tune for more than a couple of hours, even under best conditions, and some octaves and 5ths always have to be corrected every half day or so. But, with it freshly tuned to your method, I tried it out in some of the WTC, especially in the B minor fugue.

My first impressions:

- The E-B and F-C 5ths above middle C are much rougher than I like them to be. 4 per second! Those scream at me, on harpsichord.

- The C#-E# and F#-A# major 3rds are spicier than I like them, but plausible, given some other temps where they are worse than that.

- F-A and C-E are both more active than I like them to be. If we're going to make these downtown major 3rds so unimpressive, why can't we at least pick up some more benefit in C#-E# and F#-A# while we're at it? All of this might be a non-problem on electronic pianos and such, but on a real harpsichord with steel strings, these jump out at me.

- Overall, the temperament sounds basically inoffensive, because it's close enough to 12-equal and doesn't have any glaringly bad spots, beyond the things I mentioned here.

- The putative synchronous beating does absolutely nothing for me as a player and listener. As I said yesterday, the music is always doing more interesting things on its own, without any of this allegedly synchronous beating to matter. Dyads and chords usually don't get held that long for it to be perceptible; and the music itself doesn't stick to simple major and minor triads very much! Linear motion, suspensions, dissonances, modulations, etc are all much more important than sitting on elementary triads, trying to groove into any materially (or spiritually?) perceptible beating tricks. In other words, as I said yesterday, I Just Don't Get Why This Is Important.

(...)
> As I had said, you don't need to "concentrate" on the beats during
> music. You don't even need to "figure out" the beat rates while
> playing or hearing. But if there is some sense to seconds as the unit
> time and some sense to obtaining a beautifully resonating temperament
> by ear using simple beats, then what argument can you have against it?
> Synchronous beat-rates are there not to be "counted", but "enjoyed".
> There are strong indications that there exists parallels between
> simple integer low-prime beat rates and nicely resonating chords or
> polyphonic passages.

"Nicely resonating chords" come from having decently-clean 5ths, and from non-horrible major 3rds or minor 3rds where the note has the enharmonically correct spelling. If we're trying to get some synchronized vibrato into things, like garnishing a sundae with whipped cream, that seems like a mildly interesting intellectual pursuit; but I don't get any of that from listening to ordinary music that I'm playing. I focus on the composition, not the consistency (or not) of any vibrato speeds anywhere. If anything draws my attention to the temperament, it's 5ths, 4ths, and 3rds that suddenly sound rough or dissonant within their musical contexts, or it's any especially small or large semitones (which you don't have here). Your Mileage May Vary.

If I want to play some music that sticks to the classic need for Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, and no other notes, I pick whichever regular meantone system sounds most overall pleasing for the instrument I'm on (it differs from harpsichord to harpsichord...), and I'm done. If some music requires alteration of a few notes at one end or the other, I retune them to the correct enharmonic spelling. I only need circulating temperaments if I'm going to play music that uses more than 12 notes at once. Then, I know which compromises I'll have to live with, and my expectation of "nicely resonating chords" changes, accordingly.

I don't have a lot of interest in playing and holding elementary major or minor triads, for their own sake. The music I care about has counterpoint, ornamentation, prepared dissonances, and more interesting chords that provide various tensions.

A "beautifully resonating temperament" to me is one that gives me all the enharmonic notes I need to play the given composition, and have all those notes sound reasonable within ALL the context where they appear. Nothing startlingly far off the chromatic scale that is generated by some series of REGULARLY-sized 5ths. Regular, geometrically.

>
>
>
> > This is why I don't "get" what people here write about, regarding
> > brats and other synchronous stuff. I fail to see or hear how it is
> > musically important.
>
>
> That's because you didn't try UWT nr. 3 on your harpsichord using the
> instructions you requested and which I provided. Try it first, and
> then let's hear your input.

I did. See my input, above.

> (...)
> Unless you are doing your music in JI, beats will run the scene,
> whether you are cognizant of them or not. Still, it is evident that
> good music requires at least the subconsious to be satiated with
> reasonable beating in every interval under the proper historical-
> cultural-social context.

I'll agree with that. I'd agree with it even more if it said "satiated with reasonable intervallic quality" rather than "reasonable beating". When the notes are too far out of tune vis-a-vis the way they're spelled, within scales and the harmonies derived from those scales, they stick out individually or collectively as sounding wrong. The "proper historical-cultural-social context" is certainly key, in that regard. If we're talking about the context of 17th and 18th century music, which I care about most, the notion of equal-beating temperaments is (I believe) moot and pointless. If we're talking about the context of experimental or electronic music for 21st century listeners, grooving to synchronous beating that is supposedly important, that's an entirely different h-c-s context.

Brad Lehman

🔗Ozan Yarman <ozanyarman@...>

Dear Brad, in between the lines:

✩ ✩ ✩
www.ozanyarman.com

On May 21, 2010, at 4:04 PM, bplehman27 wrote:

>
>
> Dr Oz,
>
> I have double-checked and triple-checked all the math. Everything I
> wrote on my web page that you complained about here was indeed
> correct, the first time.
>
> (0) Comparing a major 3rd and a major 10th with the same top note
> (and with a pure octave), the beat rate is the same: 4.5 in the
> example you singled out. That's always true. "The beat frequencies
> are halved octaves down" is an accurate observation only if we take
> *both* notes down an octave, which isn't what I said.
>

261.9692308 Hz is the frequency of middle C as I arrived at as per
your recipe (256860830769231/215710000000000), 130.9846154 Hz the C a
pure octave below it, and 329.25 Hz is the frequency of E (1317/880).

(329.25 x 4) - (261.9692308 x 5) = 7.153846 beats per second Major
third.
(329.25 x 2) - (130.9846154 x 5) = 3.576923 beats per second Major tenth

You certainly can see that they are not the same.

> That's a useful general procedure in tuning harpsichords or pianos
> by ear: if the major 3rd is hard to hear accurately, but you've
> already got a pure octave in place below the lower note, work on the
> quality by playing the 10th. Working the other way round, it's also
> a good test for a pure octave: the major 3rd and the major 10th with
> the same top note have the same beat rate.
>
> For all the other points here, take a look at the beat-rate table
> here:
> http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
> (which is from my
> http://www-personal.umich.edu/~bpl/larips/math.html
> )
>

From your table:

(329.256 x 4) - (262.513 x 5) = 4.459 beats per second Major third
(329.256 x 2) - (131.2565 x 5) = 2.2295 beats per second Major tenth

They don't beat the same here either. The latter beat frequency is
halved. Clearly, there is something wrong with your math unless I
misunderstand you.

> (1) The tenor D-A is 0.99 narrow, and the A-E is 1.49 narrow: a 3/2
> ratio, as I said. What's the problem?
>

293.898234 Hz being the frequency of the final D
(78440000000000/58716923076923) and 146.949117 Hz its tenor the way I
calculated it from your recipe, beats -0.8473510455 times per second
against A. A-E beats -1.5 times per second. The ratio is not exactly
3/2.

> (2) The tenor F-A is 2.98, and the middle C-E is 4.46. That's a 3/2
> ratio, as I said -- where both intervals are the same size
> geometrically, and one of them is a 5th higher in pitch than the
> other, hence the 3/2. What's the problem?
>

Tenor F being 877/1100 from tenor A at 220 Hz, its frequency is 175.4
Hz as per your recipe, and the beating is 3 times per second. C-E
(final tuning) in the tenor region beats 3.576923077 times per second.
The brat is not 3/2.

> (3) The tenor F-A is 2.98, and the tenor E-A 4th is 1.49: half as
> fast, as I said. The A-E 5th across middle C is that same 1.49, and
> it too is half as fast as F-A, as I said. What's the problem?

My mistake here. Apologies. Octave halving skipped. The E-A fourth in
the tenor region, where E is 1317/1760 away from A at 220 Hz according
to your recipe, beats -1.5 times per second. F-A in the tenor region
was 3 times per second. Here only do we see the proportions you
describe.

>
> Apparently, there is nothing "horribly wrong with this recipe" that
> you were citing from the top section of
> http://www-personal.umich.edu/~bpl/larips/practical.html

Well, I can still send you my calculations if you like. Here are the
cents starting from C (with previous Bb calculation mistake corrected):

0: 1/1 0.000 unison, perfect prime
1: 101.608 cents 101.608
2: 199.103 cents 199.103
3: 301.592 cents 301.592
4: 395.743 cents 395.743
5: 505.502 cents 505.502
6: 599.653 cents 599.653
7: 698.631 cents 698.631
8: 801.208 cents 801.208
9: 897.727 cents 897.727
10: 1001.449 cents 1001.449
11: 1097.698 cents 1097.698
12: 1200.000 cents 1200.000

Cycle of fifths:
|
0: 0.000 cents 0.000 0 0 commas
7: 698.631 cents -3.324 -102
2: 700.473 cents -4.807 -148
9: 698.624 cents -8.138 -250
4: 698.016 cents -12.077 -371
11: 701.955 cents -12.077 -371
6: 701.955 cents -12.077 -371
1: 701.955 cents -12.077 -371
8: 699.600 cents -14.432 -443
3: 700.383 cents -16.003 -491
10: 699.857 cents -18.101 -556
5: 704.053 cents -16.003 -491
12: 694.498 cents -23.460 -720 -1 Pyth. commas
Average absolute difference: 12.7146 cents
Root mean square difference: 14.4042 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 7.4567 cents

Comparing with lehman-bach.scl:

1: 1: -3.563 cents -3.563280 0.5712 Hertz, 34.2719
cycles/min.
2: 2: -3.013 cents -3.013180 0.5111 Hertz, 30.6648
cycles/min.
3: 3: -3.547 cents -3.546650 0.6382 Hertz, 38.2892
cycles/min.
4: 4: -3.563 cents -3.563280 0.6770 Hertz, 40.6185
cycles/min.
5: 5: -3.547 cents -3.546650 0.7179 Hertz, 43.0753
cycles/min.
6: 6: -3.563 cents -3.563280 0.7616 Hertz, 45.6959
cycles/min.
7: 7: -0.586 cents -0.585590 0.1326 Hertz, 7.9584
cycles/min.
8: 8: -3.163 cents -3.163370 0.7597 Hertz, 45.5815
cycles/min.
9: 9: -3.592 cents -3.592380 0.9121 Hertz, 54.7243
cycles/min.
10: 10: -3.404 cents -3.403790 0.9176 Hertz, 55.0558 cycles/min.
11: 11: -3.563 cents -3.563280 1.0155 Hertz, 60.9278
cycles/min.
12: 12: 1/1 0.000000 0.0000 Hertz, 0.0000
cycles/min.
Mode: 1 1 1 1 1 1 1 1 1 1 1 1 Twelve-tone Chromatic
Total absolute difference : 35.1047 cents
Average absolute difference: 2.9254 cents
Root mean square difference: 3.1605 cents
Highest absolute difference: 3.5924 cents
Number of notes different: 11

If it is me who is doing a miscalculation, I surmise it's about the G-
D fifth. Let's see if you can find where the discrepancy is.

Ratios starting from middle C, cycling in fifths and fourths up to F
within the 220-440 Hz tenor A-middle A region:

256860830769231/215710000000000
1961/1100
78440000000000/58716923076923
1/1
1317/880
3951/3520
11853/7040
35559/28160
1479754324905530/782298000000000
3508/2475
546273819459862/514506666666663
877/550

Corresponding pitch frequencies:

261.9692308
392.2
293.898234
220
329.25
246.9375
370.40625
277.8046875
416.1405903
311.8222222
233.5834462
350.8

> .
>
> As Steve M correctly pointed out, I provide this beat-rate business
> on my web site in some of my instructions, only for the benefit of
> users who are inclined to think in that way by their own habits.
> Serve the customers! If they have to have fixed tables of beats to
> be able to conceptualize the temperament, I provide it as
> reference. Or, if they can tune by ear by SCARCELY EVER COUNTING
> ANY BEATS against any external standard (which is the main point of
> my posting yesterday), as I do it on my instruments, even better. I
> never look at my own generated tables of beat rates when I'm doing
> the work at the instrument. It's unnecessary information to get the
> job done.
>

Well!

>
> Cheers,
>
> Brad Lehman
>

Cordially,
Oz.

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>> (...)
>> Moreover you have written:
>>
>> Middle E from tenor A slightly flat, same quality as these other
>> fifths. It should beat as a major third from middle C at 4.5 times
>> per
>> second. (4.2 if you started from an A=415.) Also check it as a 10th
>> from tenor C, the same 4.5 times per second. An excellent test for
>> this particular F-A and C-E is found at bar 21 of the WTC book 1 C
>> major prelude. These two major thirds should have exactly the same
>> character as one another, although the beat rate is different (since
>> the pitch is different).Coincidentally (?) this is at the golden mean
>> of this piece.... The A-E fifth should beat exactly half as fast as
>> our original F-A major 3rd, sharing the same A.
>>
>> Clearly, you have overlooked the fact that the tenor C cannot beat
>> against the E an octave above 4.5 times per second while the middle C
>> beats the same against it. Was this a typographical error? The beat
>> frequencies are halved octaves down, and doubled octaves up.
>> Moreover,
>> the first approach yields a 694.5 cent fifth between A and E while
>> your "golden mean" approach yields 698 cents. I presume the latter is
>> correct.
>>
>> There are even more wrongs here:
>>
>> Copy middle E down to tenor E. For especially careful accuracy, fuss
>> with both these E's until all four of the following points are true:
>> (1) Whatever speed the tenor D-A 5th is, the fifth A-E across
>> middle C
>> should beat in 3/2 ratio from that speed (i.e. triplets vs duplets).
>>
>> where the beat speed of D-A is clearly not 1.5 times the beat speed
>> of
>> A-E according to your recipe (1.694702091 Hz at 440 Hz diapason in
>> the
>> former versus 1.5 Hz in the latter). Forcing it reduces the A-E fifth
>> size from 698 cents down to 693 cents.
>>
>> And here:
>>
>> (2) Likewise, whatever speed the tenor F-A major 3rd is, the middle
>> C-
>> E major 3rd is triplets against its duplets. [We are up a fifth on
>> the
>> keyboard, and all frequencies are 3/2 as much.]
>>
>> Where the beat speed of F-A is clearly not 1.5 times the beat speed
>> of
>> C-E according to your recipe (6 Hz at 440 Hz diapason in the former
>> versus 7.153846154 Hz in the latter). That would make the A-E fifth
>> 701 cents.
>>
>> And here:
>>
>> (3) Playing the tenor E as a 4th against tenor A, it should beat
>> exactly half as fast as our original F-A major 3rd. Likewise, the A-E
>> 5th we are testing across middle C should be exactly half as fast as
>> our original F-A major 3rd.
>>
>> Where the beat speed of E-A is clearly not 1/2 of the beat speed of
>> F-
>> A according to your recipe (1.5 Hz at 440 Hz diapason in the former
>> versus 6 Hz in the latter). That would mean a 694 cent fifth across
>> middle C.
>>
>> Unless I have done any miscalculations or misunderstood you, it
>> appears there is something horribly wrong with this recipe Brad.
>> Please check your instructions above and consider the math.
>
>
>
>
>

🔗Ozan Yarman <ozanyarman@...>

🔗bplehman27 <bpl@...>

Oz,

From your "cycle of fifths" table copied below, I can't see which notes are supposed to be which. However, I can see that you have too many different sizes of 5ths in there, and it looks like more than rounding errors.

The recipe is supposed to be, plain and simple:
- F-C-G-D-A-E all consistently 1/6 PC narrow
- E-B-F#-C# pure
- C#-G#-D#-A# 1/12 PC narrow
- Leftover A# back to F (the diminished 6th) is 1/12 PC wide

See how simple that is? I don't know why any of it needs to be made incomprehensibly difficult by introducing more numbers. :)

So, there are some pure 5ths, two sizes of tempered 5ths, and then the one leftover diminished 6th is different (slightly wide). Grand total, 4 different sizes of "5ths". Not 10 sizes, as in your "cycle of fifths" table below.

Scala files and tables are basically illegible to me.

Brad Lehman

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> > Apparently, there is nothing "horribly wrong with this recipe" that
> > you were citing from the top section of
> > http://www-personal.umich.edu/~bpl/larips/practical.html
>
>
> Well, I can still send you my calculations if you like. Here are the
> cents starting from C (with previous Bb calculation mistake corrected):
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 101.608 cents 101.608
> 2: 199.103 cents 199.103
> 3: 301.592 cents 301.592
> 4: 395.743 cents 395.743
> 5: 505.502 cents 505.502
> 6: 599.653 cents 599.653
> 7: 698.631 cents 698.631
> 8: 801.208 cents 801.208
> 9: 897.727 cents 897.727
> 10: 1001.449 cents 1001.449
> 11: 1097.698 cents 1097.698
> 12: 1200.000 cents 1200.000
>
>
> Cycle of fifths:
> |
> 0: 0.000 cents 0.000 0 0 commas
> 7: 698.631 cents -3.324 -102
> 2: 700.473 cents -4.807 -148
> 9: 698.624 cents -8.138 -250
> 4: 698.016 cents -12.077 -371
> 11: 701.955 cents -12.077 -371
> 6: 701.955 cents -12.077 -371
> 1: 701.955 cents -12.077 -371
> 8: 699.600 cents -14.432 -443
> 3: 700.383 cents -16.003 -491
> 10: 699.857 cents -18.101 -556
> 5: 704.053 cents -16.003 -491
> 12: 694.498 cents -23.460 -720 -1 Pyth. commas
> Average absolute difference: 12.7146 cents
> Root mean square difference: 14.4042 cents
> Maximum absolute difference: 23.4600 cents
> Maximum formal fifth difference: 7.4567 cents
>
>
> Comparing with lehman-bach.scl:
>
> 1: 1: -3.563 cents -3.563280 0.5712 Hertz, 34.2719
> cycles/min.
> 2: 2: -3.013 cents -3.013180 0.5111 Hertz, 30.6648
> cycles/min.
> 3: 3: -3.547 cents -3.546650 0.6382 Hertz, 38.2892
> cycles/min.
> 4: 4: -3.563 cents -3.563280 0.6770 Hertz, 40.6185
> cycles/min.
> 5: 5: -3.547 cents -3.546650 0.7179 Hertz, 43.0753
> cycles/min.
> 6: 6: -3.563 cents -3.563280 0.7616 Hertz, 45.6959
> cycles/min.
> 7: 7: -0.586 cents -0.585590 0.1326 Hertz, 7.9584
> cycles/min.
> 8: 8: -3.163 cents -3.163370 0.7597 Hertz, 45.5815
> cycles/min.
> 9: 9: -3.592 cents -3.592380 0.9121 Hertz, 54.7243
> cycles/min.
> 10: 10: -3.404 cents -3.403790 0.9176 Hertz, 55.0558
> cycles/min.
> 11: 11: -3.563 cents -3.563280 1.0155 Hertz, 60.9278
> cycles/min.
> 12: 12: 1/1 0.000000 0.0000 Hertz, 0.0000
> cycles/min.
> Mode: 1 1 1 1 1 1 1 1 1 1 1 1 Twelve-tone Chromatic
> Total absolute difference : 35.1047 cents
> Average absolute difference: 2.9254 cents
> Root mean square difference: 3.1605 cents
> Highest absolute difference: 3.5924 cents
> Number of notes different: 11
>
>
> If it is me who is doing a miscalculation, I surmise it's about the G-
> D fifth. Let's see if you can find where the discrepancy is.

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

Dear Brad Lehman,

You seem to use some beat proportions to precise your tunings, that's not very different from others who create tunings entirely based on those from the beginning.
You say you do not especially listen to beats when you hear or play music, and this is not fundamentally different either from others who, as Ozan said, enjoy them, that is let the music breath with them and eventually flow with their various cadences.
Anyway in the most heaviest equibattant tuning the beats would only be slightly more harmonised, at best could comfort the resolution of some chords, but will always be transparent enough to let you be able to forget them when you perform music, as we all do.
Acoustic choices are, and always will be, a subjective matter.
It is not something that can be discussed intellectually with much profit and as you say, the final thing is performing music.
Only one thing I think important to add is that integrating beat ratios does not concerns only pulsations, but also the musicality of specific chords in a temperament and their interelations, and in my experience, this will have an effect on its global timbre, and I will even say, on its emotional palette.
Just an example : specific simple proportional beat rates provides fine tuning, that can occasionnally turn some intervals to become differentially coherent. In meantone and well-temperaments, this applies to seconds, minor thirds, major thirds, tritones, minor sixths and major sixths.
In other terms, this (and other multiple approaches) is not only using beats because they're here or because they can help interval corrections or compromises, it is also harmonising the whole sound spectrum through open relations between tempo, beats, virtual fondamentals, difference tones, fondamentals, harmonics - and perhaps even colors, at least in the listener's poetical imagination !

- - - - - - - - - - - - -
Jacques Dudon

Brad Lehman wrote :

"Disregard the role of beats"? Nay. In tuning my harpsichords, and other
people's harpsichords, I use the beats all the time. (And, all these "procured"
recipes on my http://www.larips.com site are *by* me, describing the way I set
all those various temperaments by ear.)

The crucial thing here is: after setting one or two intervals at the beginning
of a temperament bearing (such as the tenor F-A major 3rd of 3 beats/sec in 1/6
comma systems, from a diapason at or near A=440) -- all the other use of beats
throughout the temperament *is not* by comparing them against any external
standard: as "per second", or in any other fixed tempo, where we know ahead of
time what it ought to be. It is all done by comparing the beats of intervals
only with *one another*.

How do I know that a note such as G is similarly tempered from both the C and
the D below it, in a temperament where both C-G and G-D are supposed to have the
same fraction of some comma? I move the G until I find the spot where the speed
of the beat in the C-G 5th is 2/3 the speed of the beat in the D-G 4th: duplets
against triplets (playing the two intervals separately, of course, not all three
notes at once). That is the only thing I need to know, in placing that note.
Numeric beat rates are irrelevant.

I have explained all of that here:
http://www-personal.umich.edu/~bpl/larips/tetrasect.html

The quality of the interval (perceived directly) is of course defined by the
presence or absence of beats in it -- the more beats, the more "sour" it is,
along with the understanding that everything doubles at each octave -- , but
again, I'm *not* listening for beat-counting against any external standard of
time or rate. I compare the beat speeds of neighboring intervals: "that one is
twice as fast as this one", or "that one makes triplets against this one's
duplets", or "that one is slightly faster than this", but only to evaluate and
finely adjust the intervals that I have already set by perceived quality alone.
90% of the work is done without the beats; they are used only to confirm the
results, or make tiny tweaks.

All of that stuff with numerically-measured speeds is *digital*. I work in
analog. The harpsichord has to sound "in tune" to a brain that processes analog
input (intervals sounding the same as or different from one another, by
quality), not a digital brain that sits there counting everything that it
encounters and assigning numbers to make value judgments. I can get the quality
of C-G and D-G to be the same, or very nearly so, without consciously listening
for (or counting, or comparing) any beats. If one of them is too good while the
other is too sour, my wrist automatically moves in the right direction to rotate
the pin and even these out. I can tune the harpsichord while thinking about
other things!

As a listener, I almost never listen for any beats. Music is invariably doing
other things more interesting than that.

This is why I don't "get" what people here write about, regarding brats and
other synchronous stuff. I fail to see or hear how it is musically important.
It looks to me like just a bunch of mathematical overkill, playing with numbers
for the amusement of playing with numbers. The beats in intervals are useful to
me only to get my tuning task (the physical process) done well and quickly, so I
can get on to the more interesting task of practicing or performing the music.

Brad Lehman

🔗john777music <jfos777@...>

🔗Ozan Yarman <ozanyarman@...>

O Bradley,

✩ ✩ ✩
www.ozanyarman.com

On May 21, 2010, at 5:36 PM, bplehman27 wrote:

>
>
>
>
>
> --- In tuning@...m, Ozan Yarman <ozanyarman@...> wrote:
>> (...) There is no conceptual or pragmatical difference
>> in your approach versus mine, save that, my recipe is simpler and
>> yields very well calculated beat rates that are in sync.
>
> Completely different: your approach, holding beat rates constant
> (when played in some specific octave, cherry-picking!)

Are you serious? Don't you know that beat ratios for chords are the
same at every octave? Only beat frequencies double or halve by
octaves, not beat ratios. Brats stay constant throughout. Dividing the
half of one beat freq. by the half of another beat freq. is the same
as dividing the double of the former by the double of the latter.

> , produces temperaments that have at least half a dozen differently-
> sized tempered 5ths. Most of the temperaments I work with hold the
> interval sizes constant GEOMETRICALLY across all or most of the 5ths.
>

My dear colleague, what you profess to describe via your Excel
worksheet is the same as the lehman-bach.scl file and utterly
different from your instructions if you follow them to the letter as I did.

Your values look neat on paper, because you have equally partitioned
the comma among the fifths by some mathematical formula. Now THAT's
digital. How you propose to set those fifths up by ear preserving
CONSTANT SIZES is an altogether different story.

Either I am doing something wrong when following your recipe on www.larips.com
, or you are not following what you describe, or else, you are
describing what you are not doing!

> See, for example, my page about various sizes of regular (or
> "meantone") temps:
> http://www-personal.umich.edu/~bpl/larips/meantone.html
>
>

Ok.

>>
>> Not to mention... Having followed your recipe in an Excel worksheet
>> to
>> the letter, I have ascertained that, between the end result and the
>> Scala file under your name, there are 3 to 3.6 cent errors on every
>> tone except G and Bb, and a whopping 7 cents error on Bb! A gross
>> 694.5 cent fifth haunts F-C in the cycle also.
>>
>> I can send you the Excel file I have worked with, so that you can
>> cross check the results if you like.
>
> You can send it if you want, but you've clearly followed the
> instructions (allegedly "to the letter") incorrectly. My own Excel
> file is on my very old web page: http://how.to/tune
>

My dear Bradley, on the contrary, I am now more confident that ever
after your recent slip of claiming 4.5 beats per second for both
middle C-E and tenor C-middle E that it is you who should attempt to
quadruple, or yet, quintuple to check the math behind your instructions.

> As I mentioned above, almost every temperament I work with has
> consistent comma-fraction tempering in most 5ths, if they're not
> pure. My Excel sheet isn't set up to handle anything that has the
> constant-beat-rates rule that you take as important.
>

The issue is not your magical ability to distribute the comma
fractions uniformly in an Excel worksheet. The issue is:

1. Your claim to acquire fifths of the same size with your recipe on
an acoustical instrument without the aid of any gadgets to underline
the superiority of your scheme over mine,

2. Your claim that my recipe is "digital", while yours is "analog.

Both of which appear to be grossly false, quod erat demonstrandum.

> Whoever produced "the Scala file under [my] name" probably did it
> correctly, by working with fractions of commas. I wouldn't know, as
> I haven't seen it recently and I don't have Scala...or any tunable
> synthesizer, or any burning desire to produce any electronically-
> generated music. I'm an old-fashioned musician with a doctorate in
> harpsichord performance. I do all my tuning from either a C or an A
> tuning fork, with no electronics.
>
>

Figures. My dear colleague, how then are we to ascertain the validity
or precision of your claims if you will not accept to evaluate some
electronic or digital verification of your instructions? Are we to
take them at face value?

>>
>> Yet one thing is for sure, you can't go wrong with the recipe of UWT
>> nr.3.
>
> Not sure what you mean by "can't go wrong", here.

I mean obviously: Although my tuning scheme was digitally calculated
on paper, just as your 1/6 comma distributions on an Excel worksheet,
my recipe provides exactly the tuning desired, unless, by some far-off
chance, I made a mistake somewhere. I don't claim to be above
mistakes... But really! I am not the one whose claims are hanging by
the hook right now.

> I set it up here on my best harpsichord.

I trust that you did it according to your perception and understanding
of beats.

> It took more than twice as long as tuning by more familiar methods
> that have consistent-sized 5ths;

My dear colleague, by all that is good and holy, there is a glaring
700.5 cent fifth right there between G-D among the more or less equal
sized 1/6 comma tempered fifths (if you aren't picky about 0.6 cents
deviations) when you set up your Bach tuning per your instructions.
There is also a terrible 694.5 cent fifth between F-C. Please
scrutinize the Excel worksheet to follow.

> and I couldn't use my normal checks that go all the way up the
> treble, proving octaves by testing the intervening 4ths and 5ths for
> proper consistency (because you have half a dozen differently-sized
> 5ths....)

You need to test the purity of octaves via the addition of a fourth to
a fifth or vice versa? Your complaint is strange, since the octaves
are not to beat, but to be tuned purely. Why would there be a need to
involve fifths and fourths when the octaves are not to be meddled with?

> . I gave it my best attempt, and it worked out reasonably well,
> going over everything twice to be sure it was accurate enough to
> your specs...which (frankly) seemed like an almost-futile pursuit,
> to me.

How unfortunate. Still, I thank you for finding the time to evaluate
UWT nr.3.

> Harpsichords don't stay that tightly in tune for more than a couple
> of hours, even under best conditions, and some octaves and 5ths
> always have to be corrected every half day or so.

And you say this because it helps you promote the notion of equal
sized fifths in a 1/6 comma Well-Temperament setting above other
tunings?

Now I wonder, what is the margin of error for your Bach tuning? would
you accept its validity even though it is distorted out of proportion
to the berth of a Modified Meantone after a week of playing?

> But, with it freshly tuned to your method, I tried it out in some
> of the WTC, especially in the B minor fugue.
>
> My first impressions:
>
> - The E-B and F-C 5ths above middle C are much rougher than I like
> them to be. 4 per second! Those scream at me, on harpsichord.
>

My dear colleague, I'm almost certain that you have set it up wrong.
My E-B is 698.454 cents, my F-C is 699.759 cents. The former is
supposed to beat 2 times per second, the latter (from tenor F), 3
times every 2 seconds. Mayhap you are confused about the counts?

> - The C#-E# and F#-A# major 3rds are spicier than I like them, but
> plausible, given some other temps where they are worse than that.
>

My C#-E# beat 16 times per second, and F#-A# 22 times per second. The
former is 406 cents, the latter is 407 cents. Versus your 404 and 402.
If you can differentiate 2-3 cents spiciness, then bravo!

> - F-A and C-E are both more active than I like them to be. If we're
> going to make these downtown major 3rds so unimpressive, why can't
> we at least pick up some more benefit in C#-E# and F#-A# while we're
> at it? All of this might be a non-problem on electronic pianos and
> such, but on a real harpsichord with steel strings, these jump out
> at me.
>

My dear colleague, you must have really done something wrong, for see,
my F-A is 393 cents, only 1 cent larger than yours, and C-E is 395
cents versus your 392 cents. Are you certain that 1-3 cents difference
"jump out at you"?

> - Overall, the temperament sounds basically inoffensive, because
> it's close enough to 12-equal and doesn't have any glaringly bad
> spots, beyond the things I mentioned here.
>

If you would humour me and labour the cross-check with a tuning device
after you set it up one more time, and give a scholarly evaluation
that corresponds to my tuning, and not some fancy set up, I would be
glad.

> - The putative synchronous beating does absolutely nothing for me as
> a player and listener.

Figures, since you must not have set it up right from the recipe I
provided. Or else, there is the remote chance that I made an error
somewhere, which I don't think I did!

> As I said yesterday, the music is always doing more interesting
> things on its own, without any of this allegedly synchronous beating
> to matter. Dyads and chords usually don't get held that long for it> to be perceptible; and the music itself doesn't stick to simple
> major and minor triads very much! Linear motion, suspensions,
> dissonances, modulations, etc are all much more important than
> sitting on elementary triads, trying to groove into any materially
> (or spiritually?) perceptible beating tricks. In other words, as I
> said yesterday, I Just Don't Get Why This Is Important.
>

You just still might my dear fellow, if what you claim to do actually
matches what the data is telling us.

> (...)
>> As I had said, you don't need to "concentrate" on the beats during
>> music. You don't even need to "figure out" the beat rates while
>> playing or hearing. But if there is some sense to seconds as the unit
>> time and some sense to obtaining a beautifully resonating temperament
>> by ear using simple beats, then what argument can you have against
>> it?
>> Synchronous beat-rates are there not to be "counted", but "enjoyed".
>> There are strong indications that there exists parallels between
>> simple integer low-prime beat rates and nicely resonating chords or
>> polyphonic passages.
>
> "Nicely resonating chords" come from having decently-clean 5ths, and
> from non-horrible major 3rds or minor 3rds where the note has the
> enharmonically correct spelling.

All of which have to do with beating.

> If we're trying to get some synchronized vibrato into things, like
> garnishing a sundae with whipped cream, that seems like a mildly
> interesting intellectual pursuit;

It's not just an intellectual pursuit as I told you before. It has
musical significance, which awaits your evaluation once you set the
tuning right.

> but I don't get any of that from listening to ordinary music that
> I'm playing. I focus on the composition, not the consistency (or
> not) of any vibrato speeds anywhere. If anything draws my attention
> to the temperament, it's 5ths, 4ths, and 3rds that suddenly sound
> rough or dissonant within their musical contexts, or it's any
> especially small or large semitones (which you don't have here).
> Your Mileage May Vary.
>

Ok. This point has been emphasized by me also, save the notion that
beats are registered in the subconsious and constitute one pillar upon
which harmonious music rests.

> If I want to play some music that sticks to the classic need for Eb-
> Bb-F-C-G-D-A-E-B-F#-C#-G#, and no other notes, I pick whichever
> regular meantone system sounds most overall pleasing for the
> instrument I'm on (it differs from harpsichord to harpsichord...),
> and I'm done. If some music requires alteration of a few notes at
> one end or the other, I retune them to the correct enharmonic
> spelling. I only need circulating temperaments if I'm going to play
> music that uses more than 12 notes at once. Then, I know which
> compromises I'll have to live with, and my expectation of "nicely> resonating chords" changes, accordingly.
>

Ok.

> I don't have a lot of interest in playing and holding elementary
> major or minor triads, for their own sake. The music I care about
> has counterpoint, ornamentation, prepared dissonances, and more
> interesting chords that provide various tensions.
>

All of which exhibit beats, or in my UWT case, proportional beat rates
whether you hold any chords for a prolonged period or not.

> A "beautifully resonating temperament" to me is one that gives me
> all the enharmonic notes I need to play the given composition, and
> have all those notes sound reasonable within ALL the context where
> they appear.

Agreed, if you do not underestimate the role of beats in the
formulation of your "reasonable sounding notes".

> Nothing startlingly far off the chromatic scale that is generated
> by some series of REGULARLY-sized 5ths. Regular, geometrically.
>

So you claim on paper, so it remains to be seen in practice!

>
>>
>>
>>
>>> This is why I don't "get" what people here write about, regarding
>>> brats and other synchronous stuff. I fail to see or hear how it is
>>> musically important.
>>
>>
>> That's because you didn't try UWT nr. 3 on your harpsichord using the
>> instructions you requested and which I provided. Try it first, and
>> then let's hear your input.
>
> I did. See my input, above.
>

And I refuted that you apparently did some things wrong. See also above.

>> (...)
>> Unless you are doing your music in JI, beats will run the scene,
>> whether you are cognizant of them or not. Still, it is evident that
>> good music requires at least the subconsious to be satiated with
>> reasonable beating in every interval under the proper historical-
>> cultural-social context.
>
> I'll agree with that. I'd agree with it even more if it said
> "satiated with reasonable intervallic quality" rather than
> "reasonable beating".

But they surely are the same. You say it like a gem-stone dealer, I
say it like a researcher.

> When the notes are too far out of tune vis-a-vis the way they're
> spelled, within scales and the harmonies derived from those scales,
> they stick out individually or collectively as sounding wrong. The
> "proper historical-cultural-social context" is certainly key, in
> that regard. If we're talking about the context of 17th and 18th
> century music, which I care about most, the notion of equal-beating
> temperaments is (I believe) moot and pointless.

Ok, I understand you think that way.

> If we're talking about the context of experimental or electronic
> music for 21st century listeners, grooving to synchronous beating
> that is supposedly important, that's an entirely different h-c-s
> context.
>

Well!

>
> Brad Lehman
>
>

Cordially,
Oz.

🔗Ozan Yarman <ozanyarman@...>

O Bradley,

I sent you the Excel worksheet that involves the calculations I
acquired from your recipe. You can check for yourself and tell me any
mistakes you find!

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 21, 2010, at 7:37 PM, bplehman27 wrote:

>
>
>
>
>
> Oz,
>
> From your "cycle of fifths" table copied below, I can't see which
> notes are supposed to be which. However, I can see that you have
> too many different sizes of 5ths in there, and it looks like more
> than rounding errors.
>
> The recipe is supposed to be, plain and simple:
> - F-C-G-D-A-E all consistently 1/6 PC narrow
> - E-B-F#-C# pure
> - C#-G#-D#-A# 1/12 PC narrow
> - Leftover A# back to F (the diminished 6th) is 1/12 PC wide
>
> See how simple that is? I don't know why any of it needs to be made
> incomprehensibly difficult by introducing more numbers. :)
>
> So, there are some pure 5ths, two sizes of tempered 5ths, and then
> the one leftover diminished 6th is different (slightly wide). Grand
> total, 4 different sizes of "5ths". Not 10 sizes, as in your "cycle
> of fifths" table below.
>
> Scala files and tables are basically illegible to me.
>
> Brad Lehman
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>> Apparently, there is nothing "horribly wrong with this recipe" that
>>> you were citing from the top section of
>>> http://www-personal.umich.edu/~bpl/larips/practical.html
>>
>>
>> Well, I can still send you my calculations if you like. Here are the
>> cents starting from C (with previous Bb calculation mistake
>> corrected):
>>
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 101.608 cents 101.608
>> 2: 199.103 cents 199.103
>> 3: 301.592 cents 301.592
>> 4: 395.743 cents 395.743
>> 5: 505.502 cents 505.502
>> 6: 599.653 cents 599.653
>> 7: 698.631 cents 698.631
>> 8: 801.208 cents 801.208
>> 9: 897.727 cents 897.727
>> 10: 1001.449 cents 1001.449
>> 11: 1097.698 cents 1097.698
>> 12: 1200.000 cents 1200.000
>>
>>
>> Cycle of fifths:
>> |
>> 0: 0.000 cents 0.000 0 0 commas
>> 7: 698.631 cents -3.324 -102
>> 2: 700.473 cents -4.807 -148
>> 9: 698.624 cents -8.138 -250
>> 4: 698.016 cents -12.077 -371
>> 11: 701.955 cents -12.077 -371
>> 6: 701.955 cents -12.077 -371
>> 1: 701.955 cents -12.077 -371
>> 8: 699.600 cents -14.432 -443
>> 3: 700.383 cents -16.003 -491
>> 10: 699.857 cents -18.101 -556
>> 5: 704.053 cents -16.003 -491
>> 12: 694.498 cents -23.460 -720 -1 Pyth. commas
>> Average absolute difference: 12.7146 cents
>> Root mean square difference: 14.4042 cents
>> Maximum absolute difference: 23.4600 cents
>> Maximum formal fifth difference: 7.4567 cents
>>
>>
>> Comparing with lehman-bach.scl:
>>
>> 1: 1: -3.563 cents -3.563280 0.5712 Hertz, 34.2719
>> cycles/min.
>> 2: 2: -3.013 cents -3.013180 0.5111 Hertz, 30.6648
>> cycles/min.
>> 3: 3: -3.547 cents -3.546650 0.6382 Hertz, 38.2892
>> cycles/min.
>> 4: 4: -3.563 cents -3.563280 0.6770 Hertz, 40.6185
>> cycles/min.
>> 5: 5: -3.547 cents -3.546650 0.7179 Hertz, 43.0753
>> cycles/min.
>> 6: 6: -3.563 cents -3.563280 0.7616 Hertz, 45.6959
>> cycles/min.
>> 7: 7: -0.586 cents -0.585590 0.1326 Hertz, 7.9584
>> cycles/min.
>> 8: 8: -3.163 cents -3.163370 0.7597 Hertz, 45.5815
>> cycles/min.
>> 9: 9: -3.592 cents -3.592380 0.9121 Hertz, 54.7243
>> cycles/min.
>> 10: 10: -3.404 cents -3.403790 0.9176 Hertz, 55.0558
>> cycles/min.
>> 11: 11: -3.563 cents -3.563280 1.0155 Hertz, 60.9278
>> cycles/min.
>> 12: 12: 1/1 0.000000 0.0000 Hertz, 0.0000
>> cycles/min.
>> Mode: 1 1 1 1 1 1 1 1 1 1 1 1 Twelve-tone Chromatic
>> Total absolute difference : 35.1047 cents
>> Average absolute difference: 2.9254 cents
>> Root mean square difference: 3.1605 cents
>> Highest absolute difference: 3.5924 cents
>> Number of notes different: 11
>>
>>
>> If it is me who is doing a miscalculation, I surmise it's about the
>> G-
>> D fifth. Let's see if you can find where the discrepancy is.
>
>

🔗martinsj013 <martinsj@...>

🔗Ozan Yarman <ozanyarman@...>

🔗bplehman27 <bpl@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> >> (...) There is no conceptual or pragmatical difference
> >> in your approach versus mine, save that, my recipe is simpler and
> >> yields very well calculated beat rates that are in sync.
> >
> > Completely different: your approach, holding beat rates constant
> > (when played in some specific octave, cherry-picking!)
>
> Are you serious? Don't you know that beat ratios for chords are the
> same at every octave? Only beat frequencies double or halve by
> octaves, not beat ratios. Brats stay constant throughout. Dividing the
> half of one beat freq. by the half of another beat freq. is the same
> as dividing the double of the former by the double of the latter.

Oz,

I wrote, very clearly: "beat rates".

I did not write "beat ratios".

Of course I know that beat *ratios* stay constant across octave transpositions, presuming pure octaves!

Your temperaments here are based on beat *rates*, beat *speeds* being held constant. Nothing about "ratios" there in my quoted phrase, "your approach, holding beat rates constant....".

Your beat rates are:

D-A of 2 per second; G'-D of 2 per second; E-B of 2 per second....

C'-G' of 3 per 2 seconds; F'-C of 3 per 2 seconds; Bb'-F of 3 per 2 seconds....

Etc., etc., etc., directly from the instructions you kindly provided on 5/18.

Playing all those intervals an octave higher, the beat rates double, because the frequencies of both notes are twice as fast. Playing them all an octave lower, they halve. Obviously. The ratio of the frequencies stays the same, if we double both frequencies together. Also, obviously; anyone who knows basic algebra knows that. Intervals beat twice as fast when we play both notes up an octave, on a keyboard where octaves are accurately pure. Once again, obviously; anyone who has ever listened closely for that phenomenon on an in-tune keyboard instrument knows that, in any reasonable temperament.

Now:

When you're making up these temperaments that keep some *beat rate* constant (as I said, and not "beat ratio"!!), it always matters which octave you're playing the notes in. It's a series of arbitrary choices: this 5th here should match that other 5th over there in a different octave, with the same beat speed, and both of them happen to line up neatly with ticking seconds on some timekeeping device. What process informs and justifies the selection of the octave where you're going to temper pitch-class X from pitch-class Y? Why didn't you pick the higher or lower octave, in any given instance? What determines the spots in the instructions where you're supposed to go up or down an octave before continuing on to the next pitch-class?

It all looks to me like a "cherry-picking" process of choosing the octave haphazardly, just so the whole thing will work out neatly. There are a handful of isolated 5ths beating 2/sec, and a handful of isolated 5ths beating 1.5/sec, and some major 3rds beating 5/sec or 6/sec, but all these forced coincidences don't mean anything! Listening to it, who except the guy who set it up is ever going to know which ones are supposed to beat at the same rate as which other ones? Would the guy who set it up be able to recognize these coincidences himself, in a double-blind listening test, knowing if all the nicely-synchronized beat ratios are actually happening...or if they have all been replaced with Folger's Crystals?

In what music does any of this matter as a desideratum, except perhaps some experimental music designed to sit there on static chords, in some timbre where the beats and their interaction are decently audible, long enough to "get" that something special is allegedly happening in there? Intellectually, it's all a nifty pastime to get these things to work out, but really, how could anyone go about inventing such temperaments without computers (or long pages of tedious arithmetic operations)? And, are these expected to be useful on ordinary acoustic instruments?

Suppose we've got some nice temperament all set up perfectly, according to somebody's recipe that delivers all these synchronized brats that are alleged to be important. Somebody else comes along and secretly knocks every G on the keyboard flatward by 1 cent, accurately. Now, we get to hear samples of C major triads, C minor triads, G major and minor, Eb major, and E minor. Is there anybody who can listen to all those isolated triads and diagnose that the G was wrong, "Aw, dang, somebody came along and messed up all these brats!" ? Or know that *anything* with the nicely-organized brats has ceased to happen?

Hence, my skepticism; and I've spent more than enough time this week saying so.

Also, let's point out, once more: temperaments generated by a bunch of integers and ratios have NOTHING to do with temperaments from accurate geometrical splits of commas. It's a completely different approach to selecting the frequencies. Ratios can never accurately represent irrational numbers, obviously (from the definition of irrational numbers). So, anything reverse-engineered USING RATIOS from a set of by-ear tuning instructions is never going to hit the spot, in trying to reproduce a comma-splitting temperament. Ratios can never do more than deliver a bunch of approximations.

Brad Lehman

🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

O Brad,

✩ ✩ ✩
www.ozanyarman.com

On May 22, 2010, at 1:00 AM, bplehman27 wrote:

>
>
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>>>> (...) There is no conceptual or pragmatical difference
>>>> in your approach versus mine, save that, my recipe is simpler and
>>>> yields very well calculated beat rates that are in sync.
>>>
>>> Completely different: your approach, holding beat rates constant
>>> (when played in some specific octave, cherry-picking!)
>>
>> Are you serious? Don't you know that beat ratios for chords are the
>> same at every octave? Only beat frequencies double or halve by
>> octaves, not beat ratios. Brats stay constant throughout. Dividing
>> the
>> half of one beat freq. by the half of another beat freq. is the same
>> as dividing the double of the former by the double of the latter.
>
> Oz,
>
> I wrote, very clearly: "beat rates".
>
> I did not write "beat ratios".
>

Ok.

> Of course I know that beat *ratios* stay constant across octave
> transpositions, presuming pure octaves!
>
> Your temperaments here are based on beat *rates*, beat *speeds*
> being held constant. Nothing about "ratios" there in my quoted
> phrase, "your approach, holding beat rates constant....".
>
> Your beat rates are:
>
> D-A of 2 per second; G'-D of 2 per second; E-B of 2 per second....
>
> C'-G' of 3 per 2 seconds; F'-C of 3 per 2 seconds; Bb'-F of 3 per 2
> seconds....
>
> Etc., etc., etc., directly from the instructions you kindly provided
> on 5/18.
>

They are as constant as the second as an incremental unit of time is
constant. I have demonstrated how you can set up the exact same
temperament with the exact same brats using another diapason if you
modified your time increment accordingly. Now how is that "constant"?

> Playing all those intervals an octave higher, the beat rates double,
> because the frequencies of both notes are twice as fast. Playing
> them all an octave lower, they halve. Obviously. The ratio of the
> frequencies stays the same, if we double both frequencies together.
> Also, obviously; anyone who knows basic algebra knows that.
> Intervals beat twice as fast when we play both notes up an octave,
> on a keyboard where octaves are accurately pure. Once again,
> obviously; anyone who has ever listened closely for that phenomenon
> on an in-tune keyboard instrument knows that, in any reasonable
> temperament.
>

OK. You also knew that a 4.5 per second beating 5:4 meant a 4.5 per
second beating 5:2. Ok, wonderful.

> Now:
>
> When you're making up these temperaments that keep some *beat rate*
> constant (as I said, and not "beat ratio"!!), it always matters
> which octave you're playing the notes in. It's a series of
> arbitrary choices: this 5th here should match that other 5th over
> there in a different octave, with the same beat speed, and both of
> them happen to line up neatly with ticking seconds on some
> timekeeping device. What process informs and justifies the
> selection of the octave where you're going to temper pitch-class X
> from pitch-class Y? Why didn't you pick the higher or lower octave,
> in any given instance? What determines the spots in the
> instructions where you're supposed to go up or down an octave before
> continuing on to the next pitch-class?
>
> It all looks to me like a "cherry-picking" process of choosing the
> octave haphazardly, just so the whole thing will work out neatly.
> There are a handful of isolated 5ths beating 2/sec, and a handful of
> isolated 5ths beating 1.5/sec, and some major 3rds beating 5/sec or
> 6/sec, but all these forced coincidences don't mean anything!
> Listening to it, who except the guy who set it up is ever going to
> know which ones are supposed to beat at the same rate as which other
> ones? Would the guy who set it up be able to recognize these
> coincidences himself, in a double-blind listening test, knowing if
> all the nicely-synchronized beat ratios are actually happening...or
> if they have all been replaced with Folger's Crystals?
>

Am I to be accused of cherry-picking when you yourself rely on
pragmatically no different beat rates and beat ratios in setting up
your Bach Temperament?

It just seems to me, that you are seeking new territory for complaint
every time it is shown to you that UWT nr.3's recipe is as analog as
your apparently incorrect recipe plus the proportional beat-ratios
feature to boot.

> In what music does any of this matter as a desideratum, except
> perhaps some experimental music designed to sit there on static
> chords, in some timbre where the beats and their interaction are
> decently audible, long enough to "get" that something special is
> allegedly happening in there?

Have I not said that the flow of music filling the air need not
feature any particularly sustained chords for the brats to take hold?
As long as there is some continuous polyphony and harmony at work, the
beat ratios will instantaneously govern the medium.

> Intellectually, it's all a nifty pastime to get these things to > work out, but really, how could anyone go about inventing such
> temperaments without computers (or long pages of tedious arithmetic
> operations)?

Why not also shun geometric divisions and arithmetical operations
while we are at it? Is labouring to temper an interval using some
logarithmic fraction of a commatic interval philosophically any
different from labouring over even more complex calculations using (or
not using) computers?

Why this tendency to get stuck in an 18th Century mechanical clockwork
setting? Why the deprecation of the benefits and speed of modern
digital calculators?

> And, are these expected to be useful on ordinary acoustic instruments?
>

Yes, I think.

> Suppose we've got some nice temperament all set up perfectly,
> according to somebody's recipe that delivers all these synchronized
> brats that are alleged to be important. Somebody else comes along
> and secretly knocks every G on the keyboard flatward by 1 cent,
> accurately. Now, we get to hear samples of C major triads, C minor
> triads, G major and minor, Eb major, and E minor. Is there anybody
> who can listen to all those isolated triads and diagnose that the G
> was wrong, "Aw, dang, somebody came along and messed up all these
> brats!" ? Or know that *anything* with the nicely-organized brats
> has ceased to happen?
>

That would perhaps rather mean that the tolerance for deviations are
higher than previously thought? And yet, the trained listener could
perhaps sense something amiss?

> Hence, my skepticism; and I've spent more than enough time this week
> saying so.
>

Ditto my counterarguments.

> Also, let's point out, once more: temperaments generated by a bunch
> of integers and ratios have NOTHING to do with temperaments from
> accurate geometrical splits of commas.

That is because logarithmic divisions on parchment are infinitely more
superior to tuning fifths by listening to beats as you have consigned
to do in your own recipe at www.larips.com?

> It's a completely different approach to selecting the frequencies.
> Ratios can never accurately represent irrational numbers, obviously
> (from the definition of irrational numbers).

That is because irrational is infinitely more superior to rational?

> So, anything reverse-engineered USING RATIOS from a set of by-ear
> tuning instructions is never going to hit the spot, in trying to
> reproduce a comma-splitting temperament. Ratios can never do more
> than deliver a bunch of approximations.
>

A bunch of approximations that defy the discerning ability of the
human ear? How wonderful if you are the sole human being in the entire
planet able to identify the discrepancy to the infinite remainer of a
fifth tempered by a logarithmic fraction of a Syntonic comma.

>
> Brad Lehman
>

Cordially,
Dr. Oz.

The grand majestic ultimate well-temperament of all times :)

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