Yahoo Tuning Groups Ultimate Backup tuning Straehle's (1743) geometric construction for an rational approximation of 12-edo

Andreas

The Five-Limit tuning on the cover page of the Straehle manuscript at the link you gave is interesting:

http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png

While I don’t read Swedish, my guess would be that it shows a ‘just tuning’ as a precursor to a discussion of temperament. At any rate, Straehle’s ratios for the diatonic notes concur with such authorities as Werckmeister (Musicae Mathematicae Hodegus Curiosus), Robert Smith (Harmonics, or, the Philosophy of Musical Sounds) and Kirnberger (Die Kunst des reinen Satzes in der Musik), and, moreover, his Bb and F# also agrees with Werckmeister. Similar to him, there are 8 ratios of 4:5, but otherwise things are very different: Straehle has 9 ratios of 2:3, to the 8 of Werckmeister, and while Werckmeister attempts to place the 4:5 intervals in semblance to meantone, Strahle opts for simplicity and symmetry, leaving DGb and EAb wide, and achieving, amongst other things, very well tuned Db, Ab and Eb major triads. Possibly Straehle optimises the flats in this manner because he is concerned with organ tuning to accommodate Cornet-Ton to Cammerton / Tief-Cammerton transposition, assuming that is that Swedish organs are tuned at a higher pitch than accompanying instruments in church.

Straehle in terms of powers of 3 and 5 is structured as follows:

-1

Versus Werckmeister:

-1

(I’m hoping the Yahoo server manages to preserve the table format here, as otherwise there will be work to do!)

Kind regards

Charles

From: TUNING@yahoogroups.com [mailto:TUNING@yahoogroups.com] On Behalf Of a_sparschuh@...
Sent: 01 December 2013 18:41
To: TUNING@yahoogroups.com
Subject: [TUNING] Straehle's (1743) geometric construction for an rational approximation of 12-edo

for an facsimle of the first page of the original see:
http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png

🔗Charles Francis <Francis@...>

Correction: should be Bb, of course, in Straehle table.

From: TUNING@yahoogroups.com [mailto:TUNING@yahoogroups.com] On Behalf Of Charles Francis
Sent: 04 December 2013 00:56
To: TUNING@yahoogroups.com
Subject: RE: [TUNING] Straehle's (1743) geometric construction for an rational approximation of 12-edo

Andreas

The Five-Limit tuning on the cover page of the Straehle manuscript at the link you gave is interesting:

http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png

Straehle in terms of powers of 3 and 5 is structured as follows:

-1

Versus Werckmeister:

-1

(I’m hoping the Yahoo server manages to preserve the table format here, as otherwise there will be work to do!)

Kind regards

Charles

for an facsimle of the first page of the original see:
http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png

🔗a_sparschuh@...

🔗Freeman Gilmore <freeman.gilmore@...>

This refers to the URL.

From Barbers:

Table 52. Faggot’s Figures for Strahle’s Temperament

Lengths 10000 9379 8811 8290 7809 7365 6953

Name C x D x E F x

Cents 0 111 219 325 428 529 629

Lengths 6570 6213 5881 5568 5274 5000

Name G x A x B C

Cents 727 824 919 1014 1108 1200

Table 53. Correct Figures for Strahle’s Temperament

Lengths 10000 9432 8899 8400 7931 7490 7073

Name C x D x E F x

Cents 0 101 202 302 401 500 600

Lengths 6676 6308 5955 5621 5303 5000

Name G x A x B C

Cents 699 798 897 997 1098 1200

Barber also solved for the angle QRP to correct for F# using (2^2)MR/2. He
gets PQ = 7.028. “But this is almost exactly Strahle’s figure!” Barbers
words. Barber concludes that Faggot’s error made PQ = 8.605.

ƒg

On Thu, Dec 5, 2013 at 1:26 PM, <a_sparschuh@yahoo.com> wrote:

>
>
> Hi Charles,
>
>
> closer en detail see:
>
> http://mathcs.holycross.edu/~groberts/Courses/Mont1/Handouts/Strahle.pdf
>
>
> bye
>
> Andy
>
>
>
>

🔗Charles Lucy <lucy@...>

Thank you Andy;
That does indeed make sense, and I realised that considering only the straight line nut to bridge distance when calculating fret placement would be a compromise, and ignored a multitude of other considerations; e.g. player pressure, fret height, string thickness, winding consistency, etc.

I like his solution, although it is about thirty years too late to do much about it it now.

Maybe you would like to modify the xls spreadsheet which I used and link to from the bottom of this url:

http://www.lucytune.com/guitars_and_frets/frets.html

Although there are a few hundred LucyTuned guitars, that I know of, there are probably many more which have been produced. It seems that the most popular application of LucyTuning has been by using scala tuning codes with Logic/CuBase/Melodyne etc. as playing guitars with so many unfamiliar frets is quite difficult for your unpracticed fret virtuoso.

Thanks for the new (to me) info.

best wishes

On 5 Dec 2013, at 18:26, <a_sparschuh@...> <a_sparschuh@...> wrote:

>
> Hi Charles,
>
>
>
> closer en detail see:
>
> http://mathcs.holycross.edu/~groberts/Courses/Mont1/Handouts/Strahle.pdf
>
>
>
> bye
>
> Andy
>
>
>
>
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can be found at:

http://www.lullabies.co.uk

🔗manuel.op.de.coul@...

That's interesting. Dan Stearns also invented this method with regularly varied
numerators and denominators. So it was actually invented much earlier. I had already built it into Scala but I have now made an extra function to automatically take the best coefficients for Stråhle's method in general. It's called EQUALTEMP/FRACTIONAL. You can do any division, for example 19 gives:
0: 1/1 0.000000 unison, perfect prime
1: 524/505 0.782213
2: 535/497 1.235858
3: 182/163 1.406099
4: 557/481 1.336941
5: 568/473 1.071421
6: 193/155 0.651790
7: 590/457 0.119683
8: 601/449 -0.483702
9: 68/49 -1.117456
10: 623/433 -1.740781
11: 634/425 -2.312843
12: 215/139 -2.792604
13: 656/409 -3.138665
14: 667/401 -3.309096
15: 226/131 -3.261268
16: 689/385 -2.951671
17: 700/377 -2.335732
18: 79/41 -1.367613
19: 2/1 0.000000 octave

The formal octave can also be different from 2/1, but to guarantee a result it has to be rational and superparticular. Other values like 3/1 for example has only a result for even divisions. Download the latest Windows version to try it. The dialog is File:New:Varied interval scale, then select Most equal over/undertone series.

Manuel

🔗a_sparschuh@...

---In tuning@yahoogroups.com, <Francis@...> wrote:

>...ratios for the diatonic notes concur with such authorities as

> Werckmeister (Musicae Mathematicae Hodegus Curiosus),....,

> ...moreover, his Bb and F# also agrees with Werckmeister.

see also for better Werckmeister ratios in his "Septenarius" tuning:
http://archive.is/8sbit
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
public-domain available as as facsimile
http://imslp.org/wiki/Musicalische_Temperatur_%28Werckmeister,_Andreas%29
for details examine the momochord on p.126

especially attend here the epimoric ratio:

https://xenharmonic.wikispaces.com/Comma
"441 / 440 = |-3 2 -1 2 -1>: werckisma https://xenharmonic.wikispaces.com/werckismic+chords (3.930 cents)"

Personally I use similar epimorics alike Werckmeister
to tune my own piano since some years at the absolute pitches:

Cycle of 5ths:

A : (E/3 = 55 110 220 440 <) 441 Hz normal tuning-fork arised by +1Hz

E : 165 330
B : 247 494 (< 495 = E*3)
F#: 185 370 740 (< 741 = B*3)
C#: 277.5 555
G#: 13 26 52 104 208 416 832 1664 (< 1665 = C#*3)
Eb: 39 78 156
Bb: 117 234
F : 351
C : 263 526 1052 (< 1053 = F*3)
G : 197 394 788 (< 789 = C*3)
D : (A/3 = 147 294 <) 295 590 (<591 = G*3)
A : 441 Hz

or more concise, only that fractional temperings inside the 5ths relative:

A 440/441 E 494/495 B 740/741 F#C# 1664/1665 G#EbBbF 1052/1053 C 788/789 G 590/591 D 294/295 A

that yields in cents-units approximation:

441 Hz = A ~-3.9 E ~-3.5 B ~-2.3 F# C# ~-1.04 G# Eb Bb F ~-1.64 C ~-2.2 G ~-2.9 D ~-5.9 A = 441 Hz

which results, when recombined in chromatically ascending order inside the 'middle-octave':

C 263 middle_C4 unison
# 277.5
D 295
# 312
E 330 tierce
F 351 quarte
# 370 tritone
G 394 quinte
# 416
A 441 Hz
# 468
B 494
c 526 octave

or compiled as
http://www.huygens-fokker.org/scala/scl_format.html

! Werckisma.scl
Sparschuh's (1991) tuning, that contains 441/440 from Werckmeister's 'Septenarius" monochord
12
555/526 ! C#/C = 277.5/263
295/263 ! D/C
312/263 ! Eb/C
330/263 ! E/C
351/263 ! F/C
370/263 ! F#/C
394/263 ! G/C
416/263 ! G#/C
441/263 ! A/C
468/263 ! Bb/C
494/263 ! B/C
2/1
!
![eof]

bye
Andy

🔗gdsecor@...

🔗manuel.op.de.coul@...

There is no searching, it's straightforward arithmetic. If we don't simplify the ratio's you will see the pattern, look at the differences between the numerators and denominators, they are an arithmetic series:

0: 513/513 0.000000 unison, perfect prime
1: 524/505 63.940109
2: 535/497 127.551648
3: 546/489 190.879783
4: 557/481 253.968520
5: 568/473 316.860895
6: 579/465 379.599158
7: 590/457 442.224947
8: 601/449 504.779455
9: 612/441 567.303597
10: 623/433 629.838166
11: 634/425 692.423999
12: 645/417 755.102132
13: 656/409 817.913966
14: 667/401 880.901430
15: 678/393 944.107153
16: 689/385 1007.574645
17: 700/377 1071.348478
18: 711/369 1135.474492
19: 722/361 1200.000000 octave

Also note that 11+8=19=the number of notes.

Manuel

---In TUNING@yahoogroups.com, <gdsecor@...> wrote:

Why do 5deg19 and 14deg19 have 568/473 and 667/401, respectively, when 6/5 and 5/3 are much closer?

--George

🔗a_sparschuh@...

🔗manuel.op.de.coul@...

🔗a_sparschuh@...

🔗glenn.leider@...

http://www.mathcs.holycross.edu/slides/musicintervalsHC.pdf‎ gives a "File not found" error, and the department web site is now listed as http://academics.holycross.edu/mathcs for now. It's not found there, either, so I guess they redid the web page structure. :(

---In TUNING@yahoogroups.com, <a_sparschuh@...> wrote:

> the 0.15% maximum error he mentions comes from. Could it be Barbour?

Hi Manuel,

compare that against his corresponding article:

M. BARBOUR,
"A Geometrical approximation to the Roots of Numbers"

(American Mathematical Monthly vol. 64, 1957. p.1-9)

or see online:
http://www.mathcs.holycross.edu/slides/musicintervalsHC.pdf‎
there on page 60 for Barbour's analysis of Strähle's deviations against 12-edo in percent units.

bye
Andy

🔗Freeman Gilmore <freeman.gilmore@...>

Manuel:

I have not read, mentioned as further reading: Barbour, "A geometrical
approximation to the roots of numbers".

In his book, Barbour rounds off his cent values to the nearest cent.

He did not mention any % error. The notes with the largest errors are A
at 897¢ and A# at 997¢. That would give an error of 0.25% not 0.15%.
Barbour did not deal in cents.

My guess is that Stewart came up with 0.15%.

ƒg

On Fri, Dec 20, 2013 at 9:30 AM, <manuel.op.de.coul@...> wrote:

>
>
> Nice find Andreas. It's written by Ian Stewart though. Martin Gardner
> wrote the book's introduction.
>
> What I don't understand is where the 0.15% maximum error he mentions comes
> from.
> Could it be Barbour? The A has the highest error, 2.624428 cents. If we
> convert this into a percentage of log octave it becomes 2.624428 / 12 =
> 0.2187%
>
> Manuel
>