Lil' Miss Scale Oven is already changing my life

I really feel that my understanding of tuning will soon reach another level with the help of LMSO.

Scala will also be a big help, when I can install a Linux version on a spare Mac laptop--fingers crossed.

The big revelation came when I watched the effect of having small changes in a simple generator around 3:2 inside a modulus of 2:1. This is perhaps the most basic tuning schema to experiment with.

You can make the generator or the modulus a little bigger or a little smaller, and make the number of pitches larger or smaller, and observe the results in a few seconds.

The difference between a generator of --for example--702 and 703.6, becomes a major difference in result when you multiply that difference out dozens of times--it results in an entirely different scale.

Contrast this with using a hand-calculator to do each experiment in a half-hour or so, with resulting tendonitis.

So, as a learning exercise, I'm spending today--or as long as it takes--to explore the possibilities when you make small variations in the modulus/generator sizes.

I'm starting with 3:2 gen and 2:1 modulus, and giving both of them 5 cents leeway. Should I give each more leeway? I don't want *too* much honkey-tonk.

Each result will be tabulated with numbers of pitches between, perhaps, 12 to 48, hitting the common EDOs or useful combinations. There are two lists of interest--step sizes in cents (relative) and pitch locations in cents (absolute). Like so:

========================================================================
3:2 exact in 2:1 exact, 17 tones:

0., 90.225, 113.685, 203.91, 294.135, 384.36, 407.82, 498.045, 588.27, 611.73, 701.955, 792.18, 815.64, 905.865, 996.09, 1086.315, 1109.775

90.225, 23.46, 90.225, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 90.225, 23.46, 90.225

==========================================================================
3:2 exact in 2:1 exact, 19 tones:

0., 90.225, 113.685, 203.91, 294.135, 317.595, 384.36, 407.82, 498.045, 588.27, 611.73, 701.955, 792.18, 815.64, 882.405, 905.865, 996.09, 1086.315, 1109.775

90.225, 23.46, 90.225, 90.225, 23.46, 66.765, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 66.765, 23.46, 90.225, 90.225, 23.46, 90.225

After this, I intend to try other modulus/generator combinations.

Armed with the insight I hope to achieve, I can come back and perhaps understand all the work that people have already done on this list, which of course already incorporates all this knowledge.

If anyone has suggestions for how far I should stretch the octave and fifth (I want to be fairly reasonable or conservative) and other generator/modulus combinations to map out, let me know.

What a difference a computer can make. Math is--in the hands of someone like me who is not mathematically knowledgeable--a matter of doing, of trial and error, of simple procedures done over and over with small variations. It's not abstract.

I don't expect I'll find anything really new.

Anyone who wants the results or has ideas for range or completeness--what to include, what to leave out, what not to miss, let me know.

Here's the values for fifths and EDOs I'm going to be looking at as candidates for "stretching":

720 (20EDO)
711.1111 (27EDO)
709.09 (44EDO)
707.6923 (39EDO)
705.8823529 (17&34EDO)
704.347826 (46EDO)
703.448 (29EDO)
702.439 (41EDO)
--------------------------------------------------
700 (12&36&48EDO)
696 (50EDO)
694.736842 (19&38EDO)
690.909 (33EDO)

with more detail:

EDOs: 12,17,19,(24),27,29,31,[33],(34),(36),(38),39,41,44,46,(48),50

3/2= 701.95500 cents, is the "5th" we are trying to approximate

in 12EDO, 5th is 700 cents (7 steps)

in 17EDO, 5th is 705.8823529 cents (10 steps)

in 19EDO, (63.157 cents), 5th is 694.736842 cents (11 steps)

[in 20EDO, (60 cents), 5th is 720 cents (12 steps)]-------------------probably too far off.

in 27EDO, (44.444...) 5th is 9 cents sharp @ (711.1111), (16 steps) --so make ox small!

in 29EDO, (41.3793), 5th is 703.448 (17 steps),

in 31EDO, (38.709677), 5th is 696.774 (18 steps)

[in 33EDO (36.363636), 5th is 690.909... (19 steps)] -----------------this might be too far off.

in 34EDO, (35.294117), 5th is 705.8823 (20 steps) --see 17EDO.

36 EDO is a multiple of 12, (21 steps)

in 38EDO (31.5785), 5th is 694.736 cents (22 steps) --see 19EDO.

in 39EDO, (30.7692307), 5th is 707.6923 (23 steps)

in 41EDO, (29.26829268), 5th is 702.439 (24 steps)

in 44EDO, (27.27272727), 5th is 709.09 (26 steps)---------------this is the outer limit, perhaps

in 46EDO, (26.086956521), 5th is 704.347826 (27 steps)

in 50EDO, (24 cents), 5th is 696 cents (29 steps)

On Sep 3, 2010, at 7:45 AM, calebmrgn wrote:

> I really feel that my understanding of tuning will soon reach another level with the help of LMSO.
>
> Scala will also be a big help, when I can install a Linux version on a spare Mac laptop--fingers crossed.
>
> The big revelation came when I watched the effect of having small changes in a simple generator around 3:2 inside a modulus of 2:1. This is perhaps the most basic tuning schema to experiment with.
>
> You can make the generator or the modulus a little bigger or a little smaller, and make the number of pitches larger or smaller, and observe the results in a few seconds.
>
> The difference between a generator of --for example--702 and 703.6, becomes a major difference in result when you multiply that difference out dozens of times--it results in an entirely different scale.
>
> Contrast this with using a hand-calculator to do each experiment in a half-hour or so, with resulting tendonitis.
>
> So, as a learning exercise, I'm spending today--or as long as it takes--to explore the possibilities when you make small variations in the modulus/generator sizes.
>
> I'm starting with 3:2 gen and 2:1 modulus, and giving both of them 5 cents leeway. Should I give each more leeway? I don't want *too* much honkey-tonk.
>
> Each result will be tabulated with numbers of pitches between, perhaps, 12 to 48, hitting the common EDOs or useful combinations. There are two lists of interest--step sizes in cents (relative) and pitch locations in cents (absolute). Like so:
>
> ========================================================================
> 3:2 exact in 2:1 exact, 17 tones:
>
> 0., 90.225, 113.685, 203.91, 294.135, 384.36, 407.82, 498.045, 588.27, 611.73, 701.955, 792.18, 815.64, 905.865, 996.09, 1086.315, 1109.775
>
> 90.225, 23.46, 90.225, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 90.225, 23.46, 90.225
>
> ==========================================================================
> 3:2 exact in 2:1 exact, 19 tones:
>
> 0., 90.225, 113.685, 203.91, 294.135, 317.595, 384.36, 407.82, 498.045, 588.27, 611.73, 701.955, 792.18, 815.64, 882.405, 905.865, 996.09, 1086.315, 1109.775
>
> 90.225, 23.46, 90.225, 90.225, 23.46, 66.765, 23.46, 90.225, 90.225, 23.46, 90.225, 90.225, 23.46, 66.765, 23.46, 90.225, 90.225, 23.46, 90.225
>
> After this, I intend to try other modulus/generator combinations.
>
> Armed with the insight I hope to achieve, I can come back and perhaps understand all the work that people have already done on this list, which of course already incorporates all this knowledge.
>
> If anyone has suggestions for how far I should stretch the octave and fifth (I want to be fairly reasonable or conservative) and other generator/modulus combinations to map out, let me know.
>
> What a difference a computer can make. Math is--in the hands of someone like me who is not mathematically knowledgeable--a matter of doing, of trial and error, of simple procedures done over and over with small variations. It's not abstract.
>
> I don't expect I'll find anything really new.
>
> Anyone who wants the results or has ideas for range or completeness--what to include, what to leave out, what not to miss, let me know.
>
>