The stellar and magnificent Ultimate Well-Temperament nr.4

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Again my usual superfluous style in the message title, but in sooth, a
colossal improvement from the last time, excelling UWT nr.3 in many
ways... Presenting for the first time ever in the chronicles of
mankind... UWT nr.4!

Pay heed O George and Gene and other worthy temperamentors and tuning-
smiths out there!

This one occupied my mind for the past three days unremittingly. I was
fixed on solving the blood-curdling mathematical formulations flying
in the air throughout endless gruelling and sleepless hours.

The result?

Drum roll and fanfare please.

Featuring the splendidly optimized and zero harmonic waste Ultimate
Well-Temperament nr.4!

Like its predecessor, this one is also totally tunable by ear (via
counting and comparing simple beat rates), has again thirds that are
at most 1 syntonic comma deviating from their regular JI counterparts,
and moreover contains no fifths tempered more than 4.9 cents, has no
zig-zags in the fifths cycle, spreads triad errors uniformly fifths-
wise from C Major and E minor, is based on cleaner and simpler ratios
yielding elegant frequencies and beat rates at A=440 Hz, and last but
not the least, flaunts synchronous and proportional-beating Major &
minor triads in every key!

Miniscule Imperfections:

1. Triad error for E Major is 2 cents greater than the one step
further B Major.

2. One cent narrowed fifth between G#-Eb amidst a flurry of pure
fifths in the vicinity.

3. Brats based on prime-limit 37 (two instances). Still better than
George Secor's 53 in his 5/23 comma TX. Only two more at 29, single
one at 23 and two more at 11. The rest of my brats are 5-limit or below.

4. Just a tad more complex brat distribution starting as early as G
Major up (in 5/23 comma TX, the Major key brats start to get more
complex after D Major). I make up for that with clean 5-limit brats
for all centric minor tonalities (F minor all the way down to C# minor).

Let's hear it for brats! (from C up chromatically)

Beats per second
5th M3rd m3rd ratio
5/6 25/6 25/3 1:5:10 -
0 29/2 87/4 0:2:3 -
5/2 55/6 20 3:11:24 *
0 40/3 60 0:2:3 -
5/6 115/6 185/6 1:23:37 *
3/2 15/2 15 1:5:10 -
0 22 33 0:2:3 $
10/3 20/3 55/3 2:4:11 * $
3/4 75/4 30 1:25:40
5/3 10 115/6 2:12:23 *
1 20/3 25/2 6:40:75
0 44/3 22 0:2:3 $

Beats per second
5th m3rd M3rd ratio
5/6 15 40/3 1:18:16
0 115/6 115/6 0:1:1 -
5/2 25/2 15/2 1:5:3 $
0 22 22 0:1:1 -
5/6 25/3 20/3 1:10:8 $
3/2 87/4 75/4 2:29:25 *
0 20 20 0:1:1 $
10/3 20 40/3 1:6:4
3/4 185/6 88/3 9:370:352 *
5/3 15/2 25/6 2:9:5
1 33/2 29/2 2:33:29 *
0 55/6 55/6 0:1:1 $

Those marked with asterisks are above 5-limit. Dash signs indicate
exact same ratios appearing in UWT nr.3. Dollar signs indicate
improvement over the brat counterparts in UWT nr.3. Overall, 7
tonalities (asterisks) traded for 7 (dollars).

Here is the scale:

Ultimate Synchronous Proportional Beating Well-Temperament nr.4 by Dr.
Oz.
|
0: 1/1 0.000 unison, perfect prime
1: 37/35 96.204
2: 353/315 197.180
3: 208/175 299.074
4: 79/63 391.801
5: 701/525 500.516
6: 148/105 594.249
7: 472/315 700.122
8: 111/70 798.159
9: 176/105 894.223
10: 312/175 1001.029
11: 592/315 1092.294
12: 2/1 1200.000 octave

Cycle of fifths:

0: 0.000 cents 0.000 0 0
commas C
7: 700.122 cents -1.833 -56 G
2: 697.058 cents -6.730
-207 D
9: 697.044 cents -11.642
-357 A
4: 697.578 cents -16.019
-492 E
11: 700.493 cents -17.481
-536 B
6: 701.955 cents -17.481
-536 F#
1: 701.955 cents -17.481
-536 C#
8: 701.955 cents -17.481
-536 G#
3: 700.915 cents -18.521 -568 -15/19 Pyth.
commas Eb
10: 701.955 cents -18.521 -568 -15/19 Pyth.
commas Bb
5: 699.487 cents -20.989
-644 F
12: 699.484 cents -23.460 -720 -Pythagorean comma, ditonic
co C
Average absolute difference: 15.6364 cents
Root mean square difference: 17.4252 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 4.9113 cents

Just the ratios:

1/1
37/35
353/315
208/175
79/63
701/525
148/105
472/315
111/70
176/105
312/175
592/315
2/1

Snapshot of Excel worksheet and keyboard tuning recipe to follow.

If all work nicely, Yarman-36 configuration to follow based on UWT nr.4.

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

As promised, Excel worksheet snapshot showing UWT nr.4. Recipe for
setting up a keyboard to this tuning to follow.

Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> Just the ratios:
>
> C 1/1
> # 37/35
> D 353/315
> # 208/175
> E 79/63
> F 701/525
> # 148/105
> G 472/315
> # 111/70
> A 176/105
> # 312/175
> B 592/315
> c 2/1
>

Sayin Bay Ozan,

in order to gain a deeper understanding of that,
I'll try to analyze that ratios, where the ratios did came from:

C 1575 := |0,2,2,1> = 7 * 5^2 * 3^2
# 1665 := |0,2,1> * 37
D 1765 := |0,0,1> * 353
# 1872 := |4,2,0,0,0,1> = |4,2> *13
E 1975 := |0,0,1> *79
F 2103 := |0,1> * 701
# 2200 := |2,1,1> *37
G 2360 := |3,0,1> *594
# 2497.5 := |-1,3,1 > *37
A 2640 := |4,1,1,0,1>
# 2808 := |3,2,0,0,0,1>
B 2960 := |4,0,1> * 37
c 3150 := |1,2,2,1>

or as presumed cycle of sequential 5ths in Werckmeister's terms:

C: _1575_
G: 295 .... _2360_ 4720=5 * 944(<945 * 5=4725 := 3*C)
D:(A/3=820 1760 <) _1765_ 5 * 353(<354 * 5 = 1770 885 = 3*G)
A: 165 330 ...... _2640_
E: _1975_ (< 1980 990 495 := 3*A)
B: 185... _2960_ 5920 (< 5925 := 3*E)
F#: 555 1110 _2200_
C#: (13...1664<) _1655_ := 3*F#
G#: (39...2496<) _2497.5_ 4995 := 3*C# = 9*F# = 27*B
D#: 117...936 _1872_
A#:(701<)702 1404 _2808_
F:(525 1050 2100<) _2103_ := 701*3
C: _1575_ := 525*3

more concise in superparticular ratios, when splitting the keyboard:

1.) Inbetween the lower-case white keys on the piano:
...F 700/701 C 944/945 G 353/354 D 352/353 A 395/396 E 1184/1185 B...

2.) Within the remote upper-case black keys from B F#...A# F :
... B - F# - C# - G# 1664/1665 D# - A# 701/702 F ...

Quests:
1) Do you agree with that analysis of the 5ths?
2) Why do you consider just that superparticu sub-partition
of the PC as "...stellar and magnificent Ultimate..." &ct...

Saygılarımla, Onun içtenlikle ile Sevgiler
Andy

Dear Andy,

Never mind the Westernizing regime that Turkified "Bey" and invented a
make-shift word "Bay" in place of "Mr./Sir". The quotidian practice
went so far as to attach "SAYIN" before any nomer (even when using Bay
or Bey!), which in fact stands for "respected/dignified" but is at
times understood as cynical and derogatory. Everything thus Turkicized
sounds awkward to my ears. No sense in syntheticism in language.
Language should be a natural process, just like music, in the hands of
those who recognize its true worth.

Returning to the matter at hand, I delve in between the lines:

✩ ✩ ✩
www.ozanyarman.com

On May 31, 2010, at 8:37 PM, Andy wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Just the ratios:
>>
>> C 1/1
>> # 37/35
>> D 353/315
>> # 208/175
>> E 79/63
>> F 701/525
>> # 148/105
>> G 472/315
>> # 111/70
>> A 176/105
>> # 312/175
>> B 592/315
>> c 2/1
>>
>
> Sayin Bay Ozan,
>
> in order to gain a deeper understanding of that,
> I'll try to analyze that ratios, where the ratios did came from:
>
> C 1575 := |0,2,2,1> = 7 * 5^2 * 3^2
> # 1665 := |0,2,1> * 37
> D 1765 := |0,0,1> * 353
> # 1872 := |4,2,0,0,0,1> = |4,2> *13
> E 1975 := |0,0,1> *79
> F 2103 := |0,1> * 701
> # 2200 := |2,1,1> *37
> G 2360 := |3,0,1> *594
> # 2497.5 := |-1,3,1 > *37
> A 2640 := |4,1,1,0,1>
> # 2808 := |3,2,0,0,0,1>
> B 2960 := |4,0,1> * 37
> c 3150 := |1,2,2,1>
>

Ok, so far so good. But why keep the remainder at G# when you couldutilize 6300 as the denominator for whole integer numbers?

> or as presumed cycle of sequential 5ths in Werckmeister's terms:
>
> C: _1575_
> G: 295 .... _2360_ 4720=5 * 944(<945 * 5=4725 := 3*C)
> D:(A/3=820 1760 <) _1765_ 5 * 353(<354 * 5 = 1770 885 = 3*G)
> A: 165 330 ...... _2640_
> E: _1975_ (< 1980 990 495 := 3*A)
> B: 185... _2960_ 5920 (< 5925 := 3*E)
> F#: 555 1110 _2200_
> C#: (13...1664<) _1655_ := 3*F#
> G#: (39...2496<) _2497.5_ 4995 := 3*C# = 9*F# = 27*B
> D#: 117...936 _1872_
> A#:(701<)702 1404 _2808_
> F:(525 1050 2100<) _2103_ := 701*3
> C: _1575_ := 525*3
>
>

Again, one can take 6300 to calculate without a remainder.

> more concise in superparticular ratios, when splitting the keyboard:
>
> 1.) Inbetween the lower-case white keys on the piano:
> ...F 700/701 C 944/945 G 353/354 D 352/353 A 395/396 E 1184/1185 B...
>

These ratios have no bearing to the fifth sizes between the said keys.
The actual fifth ratios are:

F 1050/701 C 472/315 G 353/236 D 528/353 A 395/264 E 592/395 B

> 2.) Within the remote upper-case black keys from B F#...A# F :
> ... B - F# - C# - G# 1664/1665 D# - A# 701/702 F ...
>

B 3/2 F# 3/2 C# 3/2 G# 832/555 Eb 3/2 Bb 701/468 F

>
> Quests:
> 1) Do you agree with that analysis of the 5ths?

The actual ratios for the fifths are the ones I gave above.

> 2) Why do you consider just that superparticu sub-partition
> of the PC as "...stellar and magnificent Ultimate..." &ct...
>

Ah, those were Pythagorean comma temperings? I did not regard UWT nr.
4's highest virtue as conformity to a series of superparticular
divisions of 531441/524288. The goal was rather to improve on UWT nr.3
by overcoming the criticisms against it all the while finding simple proportional beat rates and a temperament tunable by ear on a keyboard
instrument... as a basis for the future Yarman-36 variant setting of
course.

One can still try to insert 28/25 as D and 234/175 as F for even
cleaner-looking brats at the cost of uniform triad error spread from C
Major and at the peril of a -6 cent tempered fifth in a zig-zag
configuration. The replacements yield UWT nr.4b, but I do not think
elegant looking numbers on paper is justification enough for the
shortcomings.

>
> Saygılarımla, Onun içtenlikle ile Sevgiler
> Andy
>

Cordially,
Oz.