the best of the best ultimate proportional synchronous beating rational WT

Dear members,

Probably never before presented, here is the best of the best,
ultimate proportional synchronous beating rational Well-Temperament nr.
2 by (yours truly) Dr. Oz.:

0: 1/1 0.000 unison, perfect prime
1: 37/35 96.204
2: 28/25 196.198 middle second
3: 25/21 301.847 BP second, quasi-tempered
minor third
4: 44/35 396.178
5: 234/175 502.984
6: 148/105 594.249
7: 157/105 696.450
8: 278/175 801.276
9: 176/105 894.223
10: 312/175 1001.029
11: 988/525 1094.632
12: 2/1 1200.000 octave

Cycle of fifths:

0: 0.000 cents 0.000 0 0
commas C
7: 696.450 cents -5.505
-169 G
2: 699.748 cents -7.712 -237 -septimal
kleisma D
9: 698.025 cents -11.642
-357 A
4: 701.955 cents -11.642
-357 E
11: 698.454 cents -15.143
-465 B
6: 699.617 cents -17.481
-536 F#
1: 701.955 cents -17.481
-536 C#
8: 705.072 cents -14.364
-441 G#
3: 700.571 cents -15.748
-483 Eb
10: 699.183 cents -18.521 -568 -15/19 Pyth.
commas Bb
5: 701.955 cents -18.521 -568 -15/19 Pyth.
commas F
12: 697.016 cents -23.460 -720 -Pythagorean comma, ditonic
co C
Average absolute difference: 14.7681 cents
Root mean square difference: 16.2145 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 5.5047 cents

Just the ratios:

1/1
37/35
28/25
25/21
44/35
234/175
148/105
157/105
278/175
176/105
312/175
988/525
2/1

The harmonic waste has been reduced by 2 cents compared to the
previous UWT version and all the brats are now 11-limit. Actually,
they are all 7-limit except two (F#-A#-C# and Eb-Gb-Bb). It's a
beautiful sonorous well-temperament that gives the listener pure joy.
Snapshot of Excel worksheet to follow.

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

Correction from previous message: There is only 1 brat with 11-limit
proportionality (Eb-Gb-Bb). There are three brats that are 7-limit,
eleven that are 5-limit, five that are 3-limit and three that are 1-
limit.

Following is the snapshot from the Excel worksheet.

This temperament is tunable by ear alone, just like the previous UWT
version. All the beat frequencies are simple integers at the higher
octave.

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

 A question that will no doubt show my ignorance.

Why doesn't the (any of these equal beat ) tunings have octaves beating?

And related to that - then do / should pianos using this ( or any of
these equal beating ) tuning have:

1. a beating unison?

2. And is the intent here not to have stretched octaves?

Thanks,

Chris

On Wed, May 12, 2010 at 10:49 PM, Ozan Yarman <ozanyarman@...> wrote:
>
>
>
> Dear members,
>
> Probably never before presented, here is the best of the best,
> ultimate proportional synchronous beating rational Well-Temperament nr.
> 2 by (yours truly) Dr. Oz.:
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
>  A question that will no doubt show my ignorance.
>
> Why doesn't the (any of these equal beat ) tunings have octaves beating?

We've done that also, but I didn't think there was any interest this round.

Thanks Gene.

Does the "error" get bad then when you start stacking octaves when they beat
as well?
(In the context of the slow beating I think we are talking here)

(And I did mean the beating unison in the context of a typical piano string
set.. of 3 I think it is.)

I have been interested - but never tried - writing music that ignored
tempering out the "errors".
That is - what happens when one lets go of octave equivalency - in other
words where every note is unique.

I'd be surprised that someone hasn't done it.

Chris

On Wed, May 12, 2010 at 11:32 PM, genewardsmith <genewardsmith@sbcglobal.net
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > A question that will no doubt show my ignorance.
> >
> > Why doesn't the (any of these equal beat ) tunings have octaves beating?
>
> We've done that also, but I didn't think there was any interest this round.
>
>
>

Chris,

The octaves are taken as 2/1, so ordinarily there is no beating there
against the triads or their inversions (rememer division by zero?).
Triads involving pure fifths have no instances of thirds forming beat
ratios with said fifths. They just form beat ratios among themselves
(since they are the ones being impure).

In case of inharmonicity (as in octave-stretched piano strings),
modify 2/1 (and all other intervals of course) to account for the non-
integer partials. If you preserve 2/1 with inharmonicity, you will get
beating against octaves then, and the whole tuning scheme will be
ruined. You wouldn't want to do that of course.

Almost all other proportional beating solutions have complex beat
frequencies that deter a tuner from setting them by ear on an
instrument with inharmonicity (and which instrument doesn't have it?).
You need the proper algorithm in your tuning device to account for
octave stretching to combat the effects of inharmonicity when setting
such temperaments. Naturally, it is next to impossible to do that
without relying on some feedback method.

My UWT solutions are possible to set with only listening to the simple
integer beats per second of the fifths. Because you are listening to
the beating by ear, you already account for the effects of
inharmonicity. No creeping errors there if you do the tuning right.

As for the other question, you won't have a beating unison unless you
purposefully modify the tricordi (or whatnot) components away from
each other. Piano or qanun courses are tuned by eliminating the beats
between the "same pitched" strings. Even then, some instruments,
especially pianos, feature rigid strings with strangely behaving
overtones that waver (so I assume), yielding what is known as a "false-
beat" - although there is no beating against the partials of the
neighbouring unison pitch. Thanks to that that quality pianos actually
"sing"!

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 13, 2010, at 6:03 AM, Chris Vaisvil wrote:

> A question that will no doubt show my ignorance.
>
> Why doesn't the (any of these equal beat ) tunings have octaves
> beating?
>
> And related to that - then do / should pianos using this ( or any of
> these equal beating ) tuning have:
>
> 1. a beating unison?
>
> 2. And is the intent here not to have stretched octaves?
>
> Thanks,
>
> Chris
>
> On Wed, May 12, 2010 at 10:49 PM, Ozan Yarman <ozanyarman@...
> > wrote:
>>
>>
>>
>> Dear members,
>>
>> Probably never before presented, here is the best of the best,
>> ultimate proportional synchronous beating rational Well-Temperament
>> nr.
>> 2 by (yours truly) Dr. Oz.:
>>
>