CARL LUMMA - Re: 60 NOTE MICRO-TUNED SCALE - exhibits interesting phenomenal

tuning@yahoogroups.com wrote:
2b. Re: 60 NOTE MICRO-TUNED SCALE - exhibits interesting phenomenal
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Sun Aug 6, 2006 4:29 pm (PST)
Hello D,

Carl, followed your instructions as per below regarding Scala.
VERY nice software by the way, thanks very much for the heads up.
Only one problem though, the commands you listed do indeed show
a 60 note scale, but not the actual ratios. It lists all sixty, a tiny number
of the intervals are shown as fractions, ratios, but the rest are all show in
cents. I really need the actual math, fractional ratios for each note. Went through a bunch of the menu options under view, still couldnt get all ratios, any ideas?
Surely the software must have a way to do this?
Here is how scala is currently showing the scale;
-------------------------------
0: 1/1 0.000 unison, perfect prime
1: 3.615 cents 3.615
2: 531441/524288 23.460 Pythagorean comma, ditonic comma
3: 46.920 cents 46.920
4: 70.380 cents 70.380
5: 93.840 cents 93.840
-------------------------------
In an letter from Drew, he tells me of a 53 note tuning instrument and lists all the intervals as ratios;
"Date: Mon, 7 Aug 2006 11:26:08 -0700 (PDT)
From: "Drew Hempel" <druhempel@yahoo.com>
Subject: S 52 -- the new electronic instrument
To: "dar kone" <zarkorgon@yahoo.com>

From "A New Electronic Instrument for Musical Research: the S 52" by Alain Danielou, Claude Cellier, Andre Kudelski in the journal "The World of Music", 1981, No. 2, p. 63. further information see A. Danielou, Semantique Musicale, Editions Hermann, Paris, 1978 "...the instrument offers a range of eight octaves each of which is divided into 54 intervals..... A combination of binary-ternary-quinternary systems of numeration used for the classification of expressive intervals appeared to be the only way to explain their significance, their number and its limits. This theory also explains the relative intensity of the expressive contents and the relative importance of the intervals when the component factors evolve from simple to more complex ratios." "Intervals based on a binary system of numeration appear neutral. The
octave is a typical example... The significance of an interval is determined by the resulting effect of ternary and quinternary factors.... The intensity of the meaning becomes weaker with the multiplication of the ternary factor. In the ascending scale of fifths the expressive character will be stronger for the first fifth than for the second one. The successive intervals determined by a series of fifths have a common expressive character which becomes gradually weaker in the order: (C) G D A+ E+ B+. The fifth degreee is the limit of the analysing mechanism of the human brain with regard to ternary elements. For quinternary elements the second degree is the practical limit although the third degree is recognizable. The factor 25 is therefore commonly used while intervals based on the factor 125 appear somewhat odd...." "Thus the number 16, in binary 10.000, will be transcribed as B to the 4th; the number 17, written 1.000 in ternary, will be
transcribed as T to the 3rd; the number 25, written 100 in quinternary, will be transcribed as Q to the 2nd. The limma 135/128 = 5 x 27/128 will appear as QT to the 3rd/B to the 7th." The expressive value of intervals is determined by the following figures and their combinations: Q to the 2nd/ sad, melancholy Q/ tender, soft T/ brilliant, sunny, glorious /T calm, peace, night /Q agreesive, passionate, erotic /Q to the 2nd hard, cruel Thus, with relation to C as tonic: the harmonic third e (5/4 = Q/B to the 2nd) will be soft, tender; the pythagorean third E+ (81/64 = T to the 4th/B to the 6th), brilliant, radiant. A (5/3 = Q/T), similar to E but calmer A+ (27/16 = T to the 3rd/B to the 4th), similar to E+ Ab- (25/16 = Q to the 2nd / B to the 4th), pathetic Ab (128/81 = B to the 7th/T to the 4th), calm, peaceful Ab+ (8/5 - B to the 3rd/Q), erotic, enterprising Db- (25/24 = Q to the second/TB to the 3rd), desperate, pathetic but calm Db (135/128 =
QT to the 3rd/B to the 7th), tender, confident Db+ (16/15 = B to the 4th/QT), erotic, loving, calm Db++ (27/25 = T to the third/Q to the 2nd), harsh, self asserting "The keyboard covers three octaves with 53 notes in each octave represented by 53 keys. A simple device allows to obtain on the same keyboard higher aor lower octaves so as to cover 8 octaves. "The diapason can be adjusted so that the instrument may be tuned to the tonic chosen by the oriental musician of the A of various orchestras." "A micro-processor (a kind of micro-computer) incorporated in the instrument gives it a great adaptability. It is thus possible to produce automatically chords of up to 8 notes, the amplitude of its components can be controlled separately allwing for an additional synthesis fo the sound-effect desired." NUMERATIonfiltered= 1 to 27 Binary = 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 up to 11011 Ternary = 1 2 10 11 12 30 31 33
100 101 102 120 111 112 120 121 122 200 201 202 210 up to 1000 Quinternary = 1 2 3 4 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 100 101 102 The numerical ratios are brought within the same octave; other octaves are obtained by mulitplication or divison by the factor B (2).... The difference between the sounds of the same name found at the ends of the series is of less than 2 cents (a quarter of a frequency in the middle octave). The difference between Bb++ and B-- is of 2 Savarts or 6 cents. The scale of intervals: 1/1 81/80 128/125 25/24 256/243 16/15 27/25 800/729 10/9 9/8 256/225 125/108 75/64 32/27 6/5 243/200 100/81 5/4 81/64 32/25 125/96 675/512 4/3 27/20 512/375 25/18 45/32 64/45 36/25 375/256 40/27 3/2 243/160 125/81 25/16 128/81 8/5 81/50 400/243 5/3 27/16 128/75 125/72 225/128 16/9 9/5 729/400 50/27 15/8 243/128 48/25 125/64 160/81 2/1
"
If there are 53 fractions for musical notes, there MUST be such for 60?
--------------------------------
CARL, rest of response follows;

> An writer of alternative music related ideas recently expressed
> that a 60 note scale could be both exponential and lograithmic
> in its structure and that it would have interesting properties.
Who was that?

Carl, a fellow named Drew Hempel

>72-tone equal temperament has been discussed extensively on
>this mailing list. Especially owing to the fact that it is
>a tuning of the MIRACLE temperament, which makes the comma
>225/224 vanish (among others).
Is there a list of the intervals for this expressed as fractions, ratios?

>Also, Byzantine music theory divides the octave into 72 parts.
>60 is less often discussed, but you can find it here and there.
Am very interested in seeing a 72 note scale as a chain of 5ths....
>> note. The modal structure can therefore be ...

>Whoa, you lost me. Who are you quoting?

Hightower, in the link in my first post... interesting guy.
But he lost me as well, dont quite 'get' what he means....
> I understand further, however, that a 60 note scale can
> accomodate both EQUAL LOGARITHMIC AND PYTHAGOREAN JUST TUNING

-------------
If you download Scala
http://www.xs4all.nl/~huygensf/scala/
and type "pythag" in the box at the bottom, followed by
the following answers to the prompts:
size= 60
formal octave= [return]
fifth degree= [return]
formal fifth= 3/2
count downwards= [return]
and then type "show", I think you'll see what you're
looking for.
> Alternatively, How would a purely logarithmic scale for a 22, 60,
> and 72 note scale appear?
Just type "equal 22", etc. into that box, then "show" and
I think you'll see what you're looking for.
----------------------------

---------------------------------
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Hi Dar,

In the future, why not reply on the Tuning list? That way others
may have their questions answered, or have something to contribute.
Anyway,

> Carl, followed your instructions as per below regarding Scala.
> VERY nice software by the way, thanks very much for the heads up.
> Only one problem though, the commands you listed do indeed show
> a 60 note scale, but not the actual ratios. It lists all sixty,
> a tiny number of the intervals are shown as fractions, ratios,
> but the rest are all show in cents.

Ah yes, I hadn't thought of that. Most programming languages have
limited ability to handle large rationals, and 3^60 is getting up
there. I have Scheme code that can do this, but the output would
fill your screen with numbers.

> I really need the actual math, fractional ratios for each note.

Why's that?

> In an letter from Drew, he tells me of a 53 note tuning
> instrument and lists all the intervals as ratios;

//

> The scale of intervals: 1/1 81/80 128/125
> 25/24 256/243 16/15 27/25 800/729 10/9 9/8 256/225 125/108 75/64
> 32/27 6/5 243/200 100/81 5/4 81/64 32/25 125/96 675/512 4/3 27/20
> 512/375 25/18 45/32 64/45 36/25 375/256 40/27 3/2 243/160 125/81
> 25/16 128/81 8/5 81/50 400/243 5/3 27/16 128/75 125/72 225/128
> 16/9 9/5 729/400 50/27 15/8 243/128 48/25 125/64 160/81 2/1
> "
> If there are 53 fractions for musical notes, there MUST be such
> for 60?

These are 5-limit numbers. The additional prime (5) makes the
fractions MUCH smaller.

> > An writer of alternative music related ideas recently expressed
> > that a 60 note scale could be both exponential and lograithmic
> > in its structure and that it would have interesting properties.
> Who was that?
>
> Carl, a fellow named Drew Hempel

Hm, sounds vaguely familiar, but maybe not.

> >72-tone equal temperament has been discussed extensively on
> >this mailing list. Especially owing to the fact that it is
> >a tuning of the MIRACLE temperament, which makes the comma
> >225/224 vanish (among others).
> Is there a list of the intervals for this expressed as
> fractions, ratios?

72-tone equal temperament can be interpreted as ratios in
a lot of different ways. For one thing, any two ratios that
differ by a 225/224 will be approximated by the same pitch
in the system.

> Am very interested in seeing a 72 note scale as a chain
> of 5ths...

Why's that?

>>> note. The modal structure can therefore be ...
>>
>>Whoa, you lost me. Who are you quoting?
>
> Hightower, in the link in my first post... interesting guy.
> But he lost me as well, dont quite 'get' what he means....

Yeah, sounded a bit like numerology to me. But then again,
I didn't actually read it. :)

-Carl