Perfect Fifths Tuning

To the tuning Yahoogroup:A few days ago and for the first time, I imagined a twelft tone of the scale as havinga relative frequency sufficiently higher than 2 to get a set formed by purefifths only. I had conceived that this number makes an arithmetical octave only and thatthe real musical octave could be slightly higher than 2. Such unknown value would permitthe scale development based exclusively on perfect fifths. I also thought that the relativefrequency of note G with respect to C = 1 must be 1,5 since this value is a natural consonance;it was the basic assumption.Starting from C = 1, and since the Pythagorean sequence D, E, F, G ends on 1,5, each toneinterval might be twice than the E/F interval, that is, seven equal semitones from C = 1 to G=1,5.If we call X to each semitone factor, then X(7) = 1,5 where (7) is an exponent.Therefore: 7 log X = log 1,5 = 0,17609125905 and log X = (0,17609125905 / 7) = 0,02515589415Solving for X, we get X = Semitone Factor = 1,05963402267.The twelve frequencies of the perfect fifths scale follow:C = 1C# = 1,05963402267D = 1,12282426199Eb = 1,18978278946E = 1,26073432331F = 1,3359169825F# = 1,41558300612G = 1,5Ab = 1,58945103401A = 1,68423639297Bb = 1,78467418421B = 1,89110148494Musical Octave = 2,00387547376 --------- (1203,35 CENTS)------------------------------------------------------------------------------------------THE 88 PIANO TONE FREQUENCIES ARE RULED BY A CONSTANT SEMITONE FACTOR 0F 1,05963402267 TO PRODUCE PURE FIFTHS (1.5) WHATEVER THE KEY IS PLAYED. SO ANY CONSECUTIVE SEVEN TONES ARE DISTANCED BY A 3/2 PERFECT FIFTH. 01 -- A = 27,28787 Hz 02 -- Bb = 28,91516 03 -- B = 30,63949 04 -- C = 32,46664 40 -- C = 261,24598 76 -- C = 2102,1408805 -- C#= 34,40276 41 -- C#= 276,82513 77 -- C#= 2227,506 -- D = 36,45433 42 -- D = 293,33333 78 -- D = 2360,3347807 -- Eb = 38,6282578874 43 - Eb = 310,82597 79 -- Eb = 2501,0910408 -- E = 40,93181 44 -- E = 329,36178 80 -- E = 2650,2411609 -- F = 43,37274 45 -- F = 349,00295 81 -- F = 2808,2857010 -- F#= 45,95923 46 -- F#= 369,81540 82 -- F#= 2975,7550711 -- G = 48,69997 47 -- G = 391,86898 83 -- G = 3153,2113212 -- Ab = 51,60414 48 -- Ab = 415,23770 84 -- Ab = 3341,2513 -- A = 54,68150 49 -- A = 440 85 -- A = 3540,5021714 -- Bb = 57,9423868311 50 -- Bb = 466,23897 86 -- Bb = 3751,6365615 -- B = 61,39772 51 -- B = 494,04267 87 -- B = 3975,3617416 -- C = 65,05911 52 -- C = 523,50442 88 -- C = 4212,4285517 -- C#= 68,93885 53 -- C#= 554,72310 ---------------------------18 -- D = 73,04995 54 -- D = 587,80347 19 -- Eb = 77,40621 55 -- Eb = 622,85655 20 -- E = 82,02226 56 -- E = 660 21 -- F = 86,9135802467 57 -- F = 699,35845 22 -- F#= 92,09658 58 -- F#= 741,06401 23 -- G = 97,58867 59 -- G = 785,25664 24 -- Ab = 103,40828195 60 -- Ab = 832,08465 25 -- A = 109,57493 61 -- A = 881,70520 26 -- Bb = 116,10932 62 -- Bb = 934,28483 27 -- B = 123,03339 63 -- B = 990 28 -- C = 130,370370370.. 64 -- C = 1049,03768 29 -- C#= 138,14487 65 -- C#= 1111,59601 .30 -- D = 146,38301 66 -- D = 1177,88496 31 -- Eb = 155,11242 67 -- Eb = 1248,12698 32 -- E = 164,36240 68 -- E = 1322,55781 33 -- F = 174,16399 69 -- F = 1401,4272534 -- F#= 184,55009 70 -- F#= 148535 -- G = 195,555555555.. 71 -- G = 1573,5565236 -- Ab = 207,21732 72 -- Ab = 1667,3940237 -- A = 219,57452 73 -- A = 1766,8274438 -- Bb = 232,66863 74 -- Bb = 1872,19047 39 -- B = 246,54360 75 -- B = 1983,83671 Hz Mario Pizarro Lima, September 18, 2008piagui@ec-red.com ------------------------------------------------------------------------------Today, Miss Margo Schulter investigated this matter and told me thatthis stretched scale was already proposed in the past century. The paragraphs given below are written by Margo. I appreciate her kind cooperation and next time I will try to discover subjects that were not discovered before.I would like to see that tuning yahoo group pay attention on this intonation for its brilliant and charming chords according to Margo´s information.-----------------------------------------------------------------------------------------On Fri, 19 Sep 2008, Mario Pizarro wrote:> Even in the case that the scale crowded with pure fifths (1,5) is about the> equal tempered I think that I should give all the data to the tuning list.> Margo, if you are stating that a similar scale has been done in Europe> (When?) and some for example have found the tone qualities brilliant and> charming, why this enchanter scale is hidden in the past? Dear Mario, Please let me quickly give you some references on this style oftemperament where an octave is slightly stretched to create perfectfifths, or in a milder version perfect twelfths. Serge Cordier haspublished on this style of tuning extensively: search that name if youhave access to Google, and you should get some helpful results. One helpful paper is: <http://arpam.free.fr/hellegouarch.pdf> Here's a version of the scale based on a pure 3:2 from the Scala scalearchive: ! cordier.sclSerge Cordier, piano tuning, 1975 (Piano bien tempéré et justesse orchestrale) 12! 100.27929 200.55857 300.83786 401.11714 501.39643 601.67572 3/2 802.23429 902.51357 1002.79286 1103.07214 1203.35143 I'll also attach some discussions from Usenet, rec.music.theory, in whichI participated in the year 2004.> Hoping to get your news soon Please let me thank you for your patience; my purpose is to shareinformation so that you may best decide how to post to the Tuning list. > Regards> Mario> Lima, Perú Default Recordings using stretched (perfect fifth) tuning on the piano _________________________________________________________________________________________________________________ Anyone know of any available recordings, preferably piano trios, where the piano was tuned, à la Cordier, stretched so that every fifth is a perfect fifth? just ffoulkes [23]Reply With Quote francis muir #[24]2 Old 12-02-2004, 06:01 PM francis muir Guest Default "The Well-tempered Klavier and Orchestral Justesse" by SergeCordier. _________________________________________________________________________________________________________________ On 12/2/04 9:52 AM, in article BDD49651.4AE6%francis.muir.1944@balliol.org, "francis muir" <francis.muir.1944@...> wrote: > Anyone know of any available recordings, preferably piano trios, > where the piano was tuned, à la Cordier, stretched so that every > fifth is a perfect fifth? > > just ffoulkes I may be on to something. In Cordier's book* Yehudi Menuhin has a glowing introduction and towards the end there are a number of "Extracts of Appreciation" including one from Hephzibah Menuhin who played on a Cordier-tuned Steinway with her brother at a recital in Gstaad in 1975. Neither she nor her brother had ever heard the piano sound better &c., &c., and it is just possible that this concert or another was recorded. I'll pursue the matter. Just ffoulkes *Serge Cordier's *Piano Bien Tempéré et Justesse Orchestrale* Buchet/Chastel. 1982. [25]Reply With Quote francis muir Old 12-03-2004, 06:33 AM J. B. Wood Guest ----------------------------------------------------------------------------------------------------ThanksMario PizarroSeptember 19, 2008Lima, Perúpiagui@...

There is of course also Herr Stopper's 'patented' piano tuning which
divides the pure twelfth (3:1) into 19 equal semitones... and his
secret method for achieving it - a bit tricky with any standard
technique, given that most people's hands don't stretch to a twelfth.

The idea that we should take Menuhin's word that any tuning whatsoever
was 'good' is comical. Try searching for "menuhin's intonation" - he
was the one & only classical violinist famous for playing out of tune.
~~~T~~~

What Mario seems to be describing is the 7th root
of 1.5 equal tuning, which has been mentioned here many
times in the past. Unfortunately, if you listen to
its triads, they are noticeably harsher-sounding than
those of 12-ET, or the 19th root of 3 tuning.

For music that doesn't use the 5-limit triads, it is
still an interesting tuning.

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> There is of course also Herr Stopper's 'patented' piano tuning
> which divides the pure twelfth (3:1) into 19 equal semitones...
> and his secret method for achieving it - a bit tricky with any
> standard technique, given that most people's hands don't stretch
> to a twelfth.
>
> The idea that we should take Menuhin's word that any tuning
> whatsoever was 'good' is comical. Try searching for "menuhin's
> intonation" - he was the one & only classical violinist famous
> for playing out of tune.
> ~~~T~~~
>

Carl,

The 19th root of 3 tuning establishes a semitone factor of 1.05952606474 while the Perfect Fifths Tuning does with 1,05963402267. Such very slight difference cannot make any trace of superiority of one to the other I believe.
We can also take the fifth root of 4/3 and get almost the same result: 1,05922384105.
Since 2 = (3/2) (4/3), the first seven tones of another scale without including C = 1, whose twelft tone frequency equals 2, are derived from 1,05963402267 and the second group of five, including 2 Do = 2 come from the semifactor 1,05922384105, which, as I said before, is the fifth root of 4/3. I think that for noticing audible difference the third decimal digit should be different. For example: 1,059.... and 1,058.....Therefore the harsher-sounding you mentioned is not having much certainty.

It is important to continue the study of the perfect fifths tuning. I am sure that you are considering that 1,203.35 cents referred to 2C = 2 correspond to only one octave, however this figure is higher if we go up for example about three octaves where the e. tempered tone might show some degree of frequency discrepance. I will verify this matter.

Regards

Mario Pizarro

Lima, September 20
---------------------------------------------------------------------------------------------------------------------------------

----- Original Message ----- From: "Carl Lumma" <carl@...>
To: <tuning@yahoogroups.com>
Sent: Saturday, September 20, 2008 1:10 PM
Subject: [tuning] Re: Perfect Fifths Tuning

> What Mario seems to be describing is the 7th root
> of 1.5 equal tuning, which has been mentioned here many
> times in the past. Unfortunately, if you listen to
> its triads, they are noticeably harsher-sounding than
> those of 12-ET, or the 19th root of 3 tuning.
>
> For music that doesn't use the 5-limit triads, it is
> still an interesting tuning.
>
> -Carl
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>>
>> There is of course also Herr Stopper's 'patented' piano tuning
>> which divides the pure twelfth (3:1) into 19 equal semitones...
>> and his secret method for achieving it - a bit tricky with any
>> standard technique, given that most people's hands don't stretch
>> to a twelfth.
>>
>> The idea that we should take Menuhin's word that any tuning
>> whatsoever was 'good' is comical. Try searching for "menuhin's
>> intonation" - he was the one & only classical violinist famous
>> for playing out of tune.
>> ~~~T~~~
>> ------------------------------------

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>
> Carl,
>
> The 19th root of 3 tuning establishes a semitone factor of
> 1.05952606474 while the Perfect Fifths Tuning does with
> 1,05963402267. Such very slight difference cannot make any
> trace of superiority of one to the other I believe.

I don't believe I said that 19th rt. 3 was superior to
7th rt. 1.5. However the 1604.47 cent 10ths of the 7th rt.
tuning are noticeably more harsh than the 1601.65 10ths
of the 19th rt. tuning.

-Carl

Carl,

You are right. Unfortunately I overlooked the main point you had been talking about. Sorry.

Thanks

Mario Pizarro

Lima, September 22
piagui@...
---------------------------------------------------------------------------------
----- Original Message ----- From: "Carl Lumma" <carl@...>
To: <tuning@yahoogroups.com>
Sent: Saturday, September 20, 2008 4:49 PM
Subject: [tuning] Re: Perfect Fifths Tuning

> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>>
>> Carl,
>>
>> The 19th root of 3 tuning establishes a semitone factor of
>> 1.05952606474 while the Perfect Fifths Tuning does with
>> 1,05963402267. Such very slight difference cannot make any
>> trace of superiority of one to the other I believe.
>
> I don't believe I said that 19th rt. 3 was superior to
> 7th rt. 1.5. However the 1604.47 cent 10ths of the 7th rt.
> tuning are noticeably more harsh than the 1601.65 10ths
> of the 19th rt. tuning.
>
> -Carl
>
>
> ------------------------------------
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