12- and 41-tone "progressive" well-temperaments revisited

A long time ago I proposed, either here or on tuning-math, a couple of temperaments using sine and cosine functions.

One of them is a classic 12-tone WT centered on re/D using the formula T = -sin (pi * n / 12) / 2 * (531441 / 524288), where n is the number of fifths away from the central note, integer range -6 to +6, and T is the frequency ratio which is multiplied into the note to be tempered (so find the base-2 logarithm and multiply by 1200 for pitch value in cents). Also, I'm using radians for angles; replace pi with 180� if you'd rather think in terms of degrees.

The other is 41-tone, and the formula is T = (1 - cos (pi * n / 41)) / 2 * (2^65 / 3^41). This time, the range of n can either be 0 to +41, -20 to +20 or even -41 to 0 if you prefer.

I'm more concerned with the 41-tone tuning right now. My goal is to come up with some tuning that smoothly approximates a modification of Partch's JI scale (it's his 43-tone scale with 11/10 and 20/11 removed and 81/80 and 160/81 replaced with 64/63 and 63/32). It handles 7- and 11-limit intervals well, as does 41-EDO, and is better for 5-limit, but not as good as 53-EDO or untempered Pythagorean, so it's a compromise.

(Do) 0.00000, 27.36723, 59.47178, 89.50564, 115.07842, 145.92955, 177.67687,
(Re) 204.02629, 232.55233, 264.81148, 293.87399, 319.86240, 351.35756,
(Mi) 382.55340, 408.28243, 437.84853, 470.05605,
(Fa) 498.01588, 524.82581, 556.77665, 587.23970, 612.76030, 643.22335, 675.17419,
(Sol) 701.98412, 729.94395, 762.15147, 791.71757, 817.44660, 848.64244, 880.13760,
(La) 906.12601, 935.18852, 967.44767, 995.97371, 1022.32313, 1054.07045,
(Si) 1084.92158, 1110.49436, 1140.52822, 1172.63277

I still need to do a good in-depth analysis of this tuning again and try to come up with a better representation of JI, so this is my current project.

I also have a more "quick-and-dirty" 41-tone, which is just 17-tone symmetrical Pythagorean with the intervening limmas divided into thirds, 30.075 cents each.

~Danny~

On Mon, 1 May 2006, "Daniel A. Wier" wrote:
>
> A long time ago I proposed, either here or on tuning-math, a couple of
> temperaments using sine and cosine functions.

I remember reading this. Would you mind refreshing
my memory on why you thought of using sinusoids to
temper your tuning?

> One of them is a classic 12-tone WT centered on re/D using the formula T
> = -sin (pi * n / 12) / 2 * (531441 / 524288), where n is the number of
> fifths away from the central note, integer range -6 to +6, and T is the
> frequency ratio which is multiplied into the note to be tempered (so find
> the base-2 logarithm and multiply by 1200 for pitch value
> in cents). Also, I'm using radians for angles; replace pi with
> 180� if you'd rather think in terms of degrees.
>
> The other is 41-tone, and the formula is T = (1 - cos (pi * n /
> 41)) / 2 * (2^65 / 3^41). This time, the range of n can either
> be 0 to +41, -20 to +20 or even -41 to 0 if you prefer.

The prime powers form 2^65 / 3^41 here is
easier to understand than the raw integer form
531441 / 524288 above. ;-)

> I'm more concerned with the 41-tone tuning right now. My
> goal is to come up with some tuning that smoothly approximates
> a modification of Partch's JI scale (it's his 43-tone scale
> with 11/10 and 20/11 removed and 81/80 and 160/81 replaced
> with 64/63 and 63/32). It handles 7- and 11-limit intervals
> well, as does 41-EDO, and is better for 5-limit, but not as good
> as 53-EDO or untempered Pythagorean, so it's a compromise.
>
> (Do) 0.00000, 27.36723, 59.47178, 89.50564, 115.07842, 145.92955,
> 177.67687,
> (Re) 204.02629, 232.55233, 264.81148, 293.87399, 319.86240, 351.35756,
> (Mi) 382.55340, 408.28243, 437.84853, 470.05605,
> (Fa) 498.01588, 524.82581, 556.77665, 587.23970, 612.76030, 643.22335,
> 675.17419,
> (Sol) 701.98412, 729.94395, 762.15147, 791.71757, 817.44660, 848.64244,
> 880.13760,
> (La) 906.12601, 935.18852, 967.44767, 995.97371, 1022.32313, 1054.07045,
> (Si) 1084.92158, 1110.49436, 1140.52822, 1172.63277
>
> I still need to do a good in-depth analysis of this tuning again
> and try to come up with a better representation of JI, so this
> is my current project.

Danny, I'm intrigued by what, in your opinion,
such an analysis entails. What facts do you
need to know about it? Does Scala supply them
or not? (Scala tells me many things about a scale
that I don't even know whether I want to know ...
and *still* don't know how to find out.) Is the
kind of analysis you do purely factual, and capable
of being automated, or is it more active, eg asking:
"How will (such-and-such) sound in this?" and then
playing (such-and-such) to find out?

> I also have a more "quick-and-dirty" 41-tone, which is just 17-tone
> symmetrical Pythagorean with the intervening limmas divided into thirds,
> 30.075 cents each.

I don't quite understand this. More details, please?

I'll show you my reasoning, and hope you can tell me
where I'm going wrong:

Pythagorean limma
= 5 fifths - 3 octaves
= 243:256
~= 90c;
so one-third limma
~= 90/3c
= 30c;
ok.

Is "17-tone symmetrical Pythagorean" a sequence
of 8 perfect 2:3 fifths, and 8 more down, octave-
reduced? If so, starting from C, we have:
C - G - D - A - E - B - F# - C# - G# ascending and
C - F - Bb - Eb - Ab - Db - Gb - Cb - Fb descending;

or a chain of 16 fifths ascending from Fb, which I
show "word-wrapped" every five steps:
Fb Cb Gb Db Ab
Eb Bb F C G
D A E B F#
C# G#.

The above arrangement shows that in this sequence
the Pythagorean limma occurs between every pair
of notes 5 steps apart, viz:
Fb - Eb, Cb - Bb, ... Db - C, ... A - G#.

There are exactly 12 (= 17 - 5) such limmas in the
chain of 17 fifths. (There would be n-5 Pythagorean
limmas in a chain of n perfect fifths, for any n > 4.)

By splitting these 12 limmas into exact thirds,
that's equivalent to interpolating two more rows
between every two rows in the table above, say:
Fb= Cb= Gb= Db= Ab=
Fb- Cb- Gb- Db- Ab-
Eb+ Bb+ F+ C+ G+
Eb= Bb= F= C= G=
Eb- Bb- F- C- G-
D+ A+ E+ B+ F#+
D= A= E= B= F#=
D- A- E- B- F#-
C#+ G#+
C#= G#=

where I'm using -/=/+ to denote -1/0/+1 third-
limma intervals. (The = sign helps to visually space
the notes into columns.)

Each column represents a chain of notes that rises
by third-limmas. This latest table includes:
- a chain of 16 fifths (17 notes using =): Fb= ... G#=
- a chain of 14 fifths (15 notes using -): Fb- ... F#-
- a chain of 11 fifths (12 notes using +): Eb+ ... G#+

In this table I count 44 distinct notes - not 41.
Have I included three notes you didn't? If so,
which ones?

Regards,
Yahya

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Yahya Abdal-Aziz wrote:

> On Mon, 1 May 2006, "Daniel A. Wier" wrote:
>>
>> A long time ago I proposed, either here or on tuning-math, a couple of
>> temperaments using sine and cosine functions.
>
> I remember reading this. Would you mind refreshing
> my memory on why you thought of using sinusoids to
> temper your tuning?

Not really sure why I decided on using a trig function (I got loads of other crackpot ideas). But I wanted a temperament that alters fifths between the most common keys the least, since I like to keep fifths as pure as possible in my own music. I was using good ol' fashioned 53-tone Pythagorean, but found that I didn't need all those notes; nor did I like having, for example, two different neutral seconds, neither of which being a very good approximation of 12/11. I also didn't want 41-EDO because it's weak on 5-limit, and I'd rather not have a 5/4 major third that's 5.83 cents too flat.

So a "progressive" or "accellerating" temperament came to mind. I picked a sine/cosine curve because it's nice and smooth and looks good on paper, though as a basis for a musical tuning, may not be the best. This is just one of other possibilities.

>> The other is 41-tone, and the formula is T = (1 - cos (pi * n /
>> 41)) / 2 * (2^65 / 3^41). This time, the range of n can either
>> be 0 to +41, -20 to +20 or even -41 to 0 if you prefer.
>
> The prime powers form 2^65 / 3^41 here is
> easier to understand than the raw integer form
> 531441 / 524288 above. ;-)

Yeah, I should stated it was the Pythagorean comma, 3^12 / 2^19, the same comma used in calculating classic well-temperaments like Werckmeister, Kirnberger and Valotti-Young.

>> I still need to do a good in-depth analysis of this tuning again
>> and try to come up with a better representation of JI, so this
>> is my current project.
>
> Danny, I'm intrigued by what, in your opinion,
> such an analysis entails. What facts do you
> need to know about it? Does Scala supply them
> or not? (Scala tells me many things about a scale
> that I don't even know whether I want to know ...
> and *still* don't know how to find out.) Is the
> kind of analysis you do purely factual, and capable
> of being automated, or is it more active, eg asking:
> "How will (such-and-such) sound in this?" and then
> playing (such-and-such) to find out?

Not really sure how to answer that, other than what I'm doing now is basically a lot of math, comparing different tunings to others, etc. Factual right now, I guess.

>> I also have a more "quick-and-dirty" 41-tone, which is just 17-tone
>> symmetrical Pythagorean with the intervening limmas divided into thirds,
>> 30.075 cents each.
>
> I don't quite understand this. More details, please?
>
> I'll show you my reasoning, and hope you can tell me
> where I'm going wrong:
>
> Pythagorean limma
> = 5 fifths - 3 octaves
> = 243:256
> ~= 90c;
> so one-third limma
> ~= 90/3c
> = 30c;
> ok.

That is correct.

> Is "17-tone symmetrical Pythagorean" a sequence
> of 8 perfect 2:3 fifths, and 8 more down, octave-
> reduced? If so, starting from C, we have:
> C - G - D - A - E - B - F# - C# - G# ascending and
> C - F - Bb - Eb - Ab - Db - Gb - Cb - Fb descending;

It's eight fifths up and eight down,in my version, but it'll work as long as you have a chain of 17 fifths.

> The above arrangement shows that in this sequence
> the Pythagorean limma occurs between every pair
> of notes 5 steps apart, viz:
> Fb - Eb, Cb - Bb, ... Db - C, ... A - G#.
>
> There are exactly 12 (= 17 - 5) such limmas in the
> chain of 17 fifths. (There would be n-5 Pythagorean
> limmas in a chain of n perfect fifths, for any n > 4.)

So 12 * 3 = 36, and 36 + 5 = 41.

> By splitting these 12 limmas into exact thirds,
> that's equivalent to interpolating two more rows
> between every two rows in the table above, say:
> Fb= Cb= Gb= Db= Ab=
> Fb- Cb- Gb- Db- Ab-
> Eb+ Bb+ F+ C+ G+
> Eb= Bb= F= C= G=
> Eb- Bb- F- C- G-
> D+ A+ E+ B+ F#+
> D= A= E= B= F#=
> D- A- E- B- F#-
> C#+ G#+
> C#= G#=

> In this table I count 44 distinct notes - not 41.
> Have I included three notes you didn't? If so,
> which ones?

E-, B- and F#- are redundant, mapping to the same pitch in 41-tone as Fb, Cb and Gb.

This is not quite a perfect tuning. There are three "wolves" in the full circle of fifths, +6.615 cents each. But the perfect fifth in 19-EDO is 7.218 cents flat of pure, so they're not necessarily intolerable wolves.

~Danny~

SNIP

>
> Not really sure why I decided on using a trig function (I got loads of
other
> crackpot ideas). But I wanted a temperament that alters fifths between the
> most common keys the least, since I like to keep fifths as pure as
possible
> in my own music. I was using good ol' fashioned 53-tone Pythagorean, but
> found that I didn't need all those notes; nor did I like having, for
> example, two different neutral seconds, neither of which being a very good
> approximation of 12/11. I also didn't want 41-EDO because it's weak on
> 5-limit, and I'd rather not have a 5/4 major third that's 5.83 cents too
> flat.
>
> So a "progressive" or "accellerating" temperament came to mind. I picked a
> sine/cosine curve because it's nice and smooth and looks good on paper,
> though as a basis for a musical tuning, may not be the best. This is just
> one of other possibilities.
>

Danny, have you or anyone else tried the exponential function?

> >> I also have a more "quick-and-dirty" 41-tone, which is just 17-tone
> >> symmetrical Pythagorean with the intervening limmas divided into
thirds,
> >> 30.075 cents each.
> >
> > I don't quite understand this. More details, please?
> >
> > I'll show you my reasoning, and hope you can tell me
> > where I'm going wrong:
> >
> > Pythagorean limma
> > = 5 fifths - 3 octaves
> > = 243:256
> > ~= 90c;
> > so one-third limma
> > ~= 90/3c
> > = 30c;
> > ok.
>
> That is correct.
>
> > Is "17-tone symmetrical Pythagorean" a sequence
> > of 8 perfect 2:3 fifths, and 8 more down, octave-
> > reduced? If so, starting from C, we have:
> > C - G - D - A - E - B - F# - C# - G# ascending and
> > C - F - Bb - Eb - Ab - Db - Gb - Cb - Fb descending;
>
> It's eight fifths up and eight down,in my version, but it'll work as long
as
> you have a chain of 17 fifths.
>
> > The above arrangement shows that in this sequence
> > the Pythagorean limma occurs between every pair
> > of notes 5 steps apart, viz:
> > Fb - Eb, Cb - Bb, ... Db - C, ... A - G#.
> >
> > There are exactly 12 (= 17 - 5) such limmas in the
> > chain of 17 fifths. (There would be n-5 Pythagorean
> > limmas in a chain of n perfect fifths, for any n > 4.)
>
> So 12 * 3 = 36, and 36 + 5 = 41.
>
> > By splitting these 12 limmas into exact thirds,
> > that's equivalent to interpolating two more rows
> > between every two rows in the table above, say:
> > Fb= Cb= Gb= Db= Ab=
> > Fb- Cb- Gb- Db- Ab-
> > Eb+ Bb+ F+ C+ G+
> > Eb= Bb= F= C= G=
> > Eb- Bb- F- C- G-
> > D+ A+ E+ B+ F#+
> > D= A= E= B= F#=
> > D- A- E- B- F#-
> > C#+ G#+
> > C#= G#=
>
> > In this table I count 44 distinct notes - not 41.
> > Have I included three notes you didn't? If so,
> > which ones?
>
> E-, B- and F#- are redundant, mapping to the same pitch in 41-tone as Fb,
Cb
> and Gb.
>
> This is not quite a perfect tuning. There are three "wolves" in the full
> circle of fifths, +6.615 cents each. But the perfect fifth in 19-EDO is
> 7.218 cents flat of pure, so they're not necessarily intolerable wolves.
>
> ~Danny~
>
>
>

Can we have the cent values or a SCL file for this?

Cordially,
Oz.

Ozan wrote to me:

> > Not really sure why I decided on using a trig function (I got loads of
other
> > crackpot ideas). But I wanted a temperament that alters fifths between > > the
> > most common keys the least, since I like to keep fifths as pure as
possible
> > in my own music. I was using good ol' fashioned 53-tone Pythagorean, but
> > found that I didn't need all those notes; nor did I like having, for
> > example, two different neutral seconds, neither of which being a very > > good
> > approximation of 12/11. I also didn't want 41-EDO because it's weak on
> 5-limit, and I'd rather not have a 5/4 major third that's 5.83 cents too
> > flat.
> >
> > So a "progressive" or "accellerating" temperament came to mind. I picked > > a
> > sine/cosine curve because it's nice and smooth and looks good on paper,
> > though as a basis for a musical tuning, may not be the best. This is > > just
> > one of other possibilities.
>
> Danny, have you or anyone else tried the exponential function?

I've thought about it, but I don't know how I'll work that in. I don't remember exactly how a parabolic curve would deviate from a sine curve; that's a future project.

> > This is not quite a perfect tuning. There are three "wolves" in the full
> > circle of fifths, +6.615 cents each. But the perfect fifth in 19-EDO is
> > 7.218 cents flat of pure, so they're not necessarily intolerable wolves.
>
> Can we have the cent values or a SCL file for this?

I didn't post it already? Thought I did... but I'll do it again in Scala format.

! C:\Scala22\41cosine.scl
!
41-tone cosine-function well temperament
41
!
27.36723
59.47178
89.50564
115.07842
145.92955
177.67687
204.02629
232.55233
264.81148
293.87399
319.86240
351.35756
382.55340
408.28243
437.84853
470.05605
498.01588
524.82581
556.77665
587.23970
612.76030
643.22335
675.17419
701.98412
729.94395
762.15147
791.71757
817.44660
848.64244
880.13760
906.12601
935.18852
967.44767
995.97371
1022.32313
1054.07045
1084.92158
1110.49436
1140.52822
1172.63277
2/1

And to answer your other question, on how I derived this tuning: I'm working on that. I'll have to be brief on this list; a more in-depth description I'll just link to when I get it done. Hope you don't mind a no-frills HTML or RTF document.

~Danny~

Correction:

>> Danny, have you or anyone else tried the exponential function?
>
> I've thought about it, but I don't know how I'll work that in. I don't
> remember exactly how a parabolic curve would deviate from a sine curve;
> that's a future project.

I meant exponential, which is not the same as parabolic (i.e. n�).

~Danny~