Yahoo Tuning Groups Ultimate Backup makemicromusic Asking the right questions, Quantizing JI to 106edo

It's said that a rule of good writing is to know your audience.

The key to success, apparently, is not to ask open-ended questions, unless one wants to invite debate. I don't want to invite debate.

Question 1: What are the EDOs between 53 and approximately 250 or so, that have a 3/2 approximation between 701 cents and 704 cents?

Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?

Or, if not an EDO, a temperament?

I've already said that I've tried the temperament finder-thingy many, many, *many* times and, given the specificity of what I want, I can't figure out how to use it to answer this question.

The reason to quantize to an EDO is that one has a nice closed system which can be calculated by reference to one step size. It's really a matter of convenience or mental economy.

Another reason -- less likely -- is that there may be some EDO which, fortuitously, causes some pitches to be even better when they do double or triple duty.

First priority is that the 5ths that are currently fairly accurate remain accurate. The scale must include some approximation of a 13-ratio on /1 or /3 or /5 or /9 -- i.e. 13/8 or 13/12 or 13/10 or 13/9.

*The resulting scale has to have the approximate ratios on the keys on which they currently exist.*

*The fingerings and step-sizes are my own weird preference -- not up for discussion.*

I've chosen 106edo, because it's 2x53. 53EDO has a very accurate 3/2.

After Quantizing with LMSO, I had to tweak a couple of notes.

Q. Would there be anything better than 106?

I've tried maybe 30 different values for EDOs. So far, 106 and the famous 159 have produced good results. (There's a very useful paper by Mr. Yarman on the subject of 159Edo).

No, not 88. I like 87, I like 89 a little less. Maybe the 5ths in 87 are a little too wide for my taste.

Here's the JI scale: Whatever I quantize to, it has to be pretty darn close.

// Scale Pattern: 135/128, 16/15, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 32/27, 6/5, 11/9, 5/4, 81/64, 9/7, 4/3, 11/8, 45/32, 10/7, 13/9, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 15/8, 243/128, 2/1

JI Scala File

JI wiggle4 81-64
36
!
92.17872
111.73129
150.63706
165.00423
182.40371
203.91000
231.17409
266.87091
294.13500
315.64129
347.40794
386.31371
407.82000
435.08410
498.04500
551.31794
590.22372
617.48781
636.61766
648.68206
663.04923
701.95500
764.91590
782.49204
813.68629
852.59206
884.35871
905.86500
933.12909
968.82591
996.09000
1017.59629
1034.99577
1088.26871
1109.77500
1200.00000

Quantized to 106EDO:

!
wiggle4 81-64 106
36
!
90.56604
113.20755
147.16981
169.81132
181.13208
203.77358
226.41509
271.69811
294.33962
316.98113
350.94340
384.90566
407.54717
430.18868
498.11321
554.71698
588.67925
622.64151
633.96226
645.28302
667.92453
701.88679
769.81132
792.45283
815.09434
849.05660
883.01887
905.66038
928.30189
973.58491
996.22642
1018.86792
1030.18868
1086.79245
1109.43396
1200.00000

🔗Caleb Morgan <calebmrgn@...>

94edo works pretty well for this.

Unless I'm struck by lightning, that's what I'm going with.

Retuning the EXS24 instruments is the most work.

Then there's some fairly easy work doing the conversion charts.

I like it more than 87 because of the more accurate 5ths (3/2's) and the other intervals are certainly ok.

I have to decide within a short time because I can't start really composing until I've retuned the Logic instruments.

In this world -- my world -- there's nothing really at stake about committing to 94 edo for a project beyond this slightly tedious work of retuning -- 94edo is just a way of simplifying things within the sound that I want, otherwise it has no palpable reality.

The xenharmonic pages are helpful for this search, most especially when they explicitly give all the cent values for quick reference -- as they do sometimes.

________________________________
From: calebmrgn <calebmrgn@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Friday, March 29, 2013 1:06 PM
Subject: [MMM] Asking the right questions, Quantizing JI to 106edo

It's said that a rule of good writing is to know your audience.

The key to success, apparently, is not to ask open-ended questions, unless one wants to invite debate. I don't want to invite debate.

Question 1: What are the EDOs between 53 and approximately 250 or so, that have a 3/2 approximation between 701 cents and 704 cents?

Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?

Or, if not an EDO, a temperament?

I've already said that I've tried the temperament finder-thingy many, many, *many* times and, given the specificity of what I want, I can't figure out how to use it to answer this question.

The reason to quantize to an EDO is that one has a nice closed system which can be calculated by reference to one step size. It's really a matter of convenience or mental economy.

Another reason -- less likely -- is that there may be some EDO which, fortuitously, causes some pitches to be even better when they do double or triple duty.

*The resulting scale has to have the approximate ratios on the keys on which they currently exist.*

*The fingerings and step-sizes are my own weird preference -- not up for discussion.*

I've chosen 106edo, because it's 2x53. 53EDO has a very accurate 3/2.

After Quantizing with LMSO, I had to tweak a couple of notes.

Q. Would there be anything better than 106?

I've tried maybe 30 different values for EDOs. So far, 106 and the famous 159 have produced good results. (There's a very useful paper by Mr. Yarman on the subject of 159Edo).

No, not 88. I like 87, I like 89 a little less. Maybe the 5ths in 87 are a little too wide for my taste.

Here's the JI scale: Whatever I quantize to, it has to be pretty darn close.

JI Scala File

Quantized to 106EDO:

[Non-text portions of this message have been removed]

🔗Graham Breed <gbreed@...>

"calebmrgn" <calebmrgn@...> wrote:
> It's said that a rule of good writing is to know your audience.
>
> The key to success, apparently, is not to ask open-ended questions, unless one wants to invite debate. I don't want to invite debate.
>
> Question 1: What are the EDOs between 53 and approximately 250 or so, that have a 3/2 approximation between 701 cents and 704 cents?

53 58 65 70 77 82 87 89 94 99 104 106 111 116 118 123 128 130 133 135 140 142 145 147 152 154 157 159 162 164 166 169 171 174 176 178 179 181 183 186 188 190 191 193 195 198 200 203 205 207 208 210 212 215 217 219 220 222 224 227 229 231 232 234 236 237 239 241 243 244 246 248 249

This is with a cutoff of exactly 250. That is
important because the more steps you have, the more likely
it is that an interval in this range will exist, so they
come thick and fast after 250.

Also assuming exact octaves.

> Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?

> // Scale Pattern: 135/128, 16/15, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 32/27, 6/5, 11/9, 5/4, 81/64, 9/7, 4/3, 11/8, 45/32, 10/7, 13/9, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 15/8, 243/128, 2/1

Scored by minimizing the worst error for consistent
temperings of these intervals, and nothing bigger than 250,

224, 248, 183, 241, 212, 217, 229, 159, 236, 246, 207, 193, 176, 130, 152, 200, 243, 171, 234, 231, 205, 188, 181, 118, 222, ...

Scored by minimizing the worst error*complexity badness, or
relative error, or error as a fraction of scale step, these
are best:

224, 183, 130, 159, 94, 118, 152, 248, 212, 89, 111, 176, 241, 217, 193, 171, 99, 140, 229, 207, 87, 58, 106, 53, 236, ...

In neither case is 106 especially good.

Graham

🔗Caleb Morgan <calebmrgn@...>

That was exactly the answer I was hoping for. Thank you.

________________________________
From: Graham Breed <gbreed@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Friday, March 29, 2013 6:02 PM
Subject: Re: [MMM] Asking the right questions, Quantizing JI to 106edo

This is with a cutoff of exactly 250. That is
important because the more steps you have, the more likely
it is that an interval in this range will exist, so they
come thick and fast after 250.

Also assuming exact octaves.

> Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?

Scored by minimizing the worst error for consistent
temperings of these intervals, and nothing bigger than 250,

224, 248, 183, 241, 212, 217, 229, 159, 236, 246, 207, 193, 176, 130, 152, 200, 243, 171, 234, 231, 205, 188, 181, 118, 222, ...

Scored by minimizing the worst error*complexity badness, or
relative error, or error as a fraction of scale step, these
are best:

224, 183, 130, 159, 94, 118, 152, 248, 212, 89, 111, 176, 241, 217, 193, 171, 99, 140, 229, 207, 87, 58, 106, 53, 236, ...

In neither case is 106 especially good.

Graham

[Non-text portions of this message have been removed]

🔗Ozan Yarman <ozanyarman@...>

I did at one time consider 106-EDO as a master tuning resolution, but quickly let go of it due to the impossibility of transition from one fifth size to the other (690.56604 cents was too low). That is when 159-EDO came into the picture, but only after Gene defined my 79 MOS as a subset of it, with the other MOS being 80.

Nevertheless, 106-EDO happens to be the master tuning grid for several proposed Turkish maqam tone-systems as indicated in my said article.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Mar 29, 2013, at 7:06 PM, calebmrgn wrote:

> It's said that a rule of good writing is to know your audience.
>
> The key to success, apparently, is not to ask open-ended questions, unless one wants to invite debate. I don't want to invite debate.
>
> Question 1: What are the EDOs between 53 and approximately 250 or so, that have a 3/2 approximation between 701 cents and 704 cents?
>
> Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?
>
> Or, if not an EDO, a temperament?
>
> I've already said that I've tried the temperament finder-thingy many, many, *many* times and, given the specificity of what I want, I can't figure out how to use it to answer this question.
>
> The reason to quantize to an EDO is that one has a nice closed system which can be calculated by reference to one step size. It's really a matter of convenience or mental economy.
>
> Another reason -- less likely -- is that there may be some EDO which, fortuitously, causes some pitches to be even better when they do double or triple duty.
>
> First priority is that the 5ths that are currently fairly accurate remain accurate. The scale must include some approximation of a 13-ratio on /1 or /3 or /5 or /9 -- i.e. 13/8 or 13/12 or 13/10 or 13/9.
>
> *The resulting scale has to have the approximate ratios on the keys on which they currently exist.*
>
> *The fingerings and step-sizes are my own weird preference -- not up for discussion.*
>
> I've chosen 106edo, because it's 2x53. 53EDO has a very accurate 3/2.
>
> After Quantizing with LMSO, I had to tweak a couple of notes.
>
> Q. Would there be anything better than 106?
>
> I've tried maybe 30 different values for EDOs. So far, 106 and the famous 159 have produced good results. (There's a very useful paper by Mr. Yarman on the subject of 159Edo).
>
> No, not 88. I like 87, I like 89 a little less. Maybe the 5ths in 87 are a little too wide for my taste.
>
> Here's the JI scale: Whatever I quantize to, it has to be pretty darn close.
>
> // Scale Pattern: 135/128, 16/15, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 32/27, 6/5, 11/9, 5/4, 81/64, 9/7, 4/3, 11/8, 45/32, 10/7, 13/9, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 15/8, 243/128, 2/1
>
> JI Scala File
>
> JI wiggle4 81-64
> 36
> !
> 92.17872
> 111.73129
> 150.63706
> 165.00423
> 182.40371
> 203.91000
> 231.17409
> 266.87091
> 294.13500
> 315.64129
> 347.40794
> 386.31371
> 407.82000
> 435.08410
> 498.04500
> 551.31794
> 590.22372
> 617.48781
> 636.61766
> 648.68206
> 663.04923
> 701.95500
> 764.91590
> 782.49204
> 813.68629
> 852.59206
> 884.35871
> 905.86500
> 933.12909
> 968.82591
> 996.09000
> 1017.59629
> 1034.99577
> 1088.26871
> 1109.77500
> 1200.00000
>
>
> Quantized to 106EDO:
>
>
> !
> wiggle4 81-64 106
> 36
> !
> 90.56604
> 113.20755
> 147.16981
> 169.81132
> 181.13208
> 203.77358
> 226.41509
> 271.69811
> 294.33962
> 316.98113
> 350.94340
> 384.90566
> 407.54717
> 430.18868
> 498.11321
> 554.71698
> 588.67925
> 622.64151
> 633.96226
> 645.28302
> 667.92453
> 701.88679
> 769.81132
> 792.45283
> 815.09434
> 849.05660
> 883.01887
> 905.66038
> 928.30189
> 973.58491
> 996.22642
> 1018.86792
> 1030.18868
> 1086.79245
> 1109.43396
> 1200.00000
>
>
>
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>

🔗Caleb Morgan <calebmrgn@...>

Thanks!

I agree that 690.5 is too low as an alternate 5th.

Yes, what I'm after here is best termed "master tuning resolution".

--------------------

For now, I'm happy as the proverbial clam.

If one is careful to ask the right question, the real experts -- guys like you and Graham -- can narrow the search down to the best candidates.

Now I have just the right size search-space of EDOs to consider, without having to try them all.

In this range of EDOs (between 87 and 248) it becomes *very* hard to choose.

In no case am I really going to be using the whole EDO.

More equal divisions of the octave, or more accuracy? Accuracy for which intervals? Stretch the 3/2s but get the 11s and 13s just right?

What do I want, now or in the future?

Even 128edo gives me most of the accuracy I want, and is highly divisible, which can be a good thing.

224EDO is frighteningly accurate. Too many steps to really simplify my thinking?

159 is excellent.

I need to consider the issue of alternate sizes of 5ths available in big EDOs.

With the PianoTech piano sound, I hear subtle differences, but these all sound good.

Because these are all limited to my 36-note format with most the pitches staying where I'm used to them, I can play them all equally easily.

(This isn't even raising the issue of other kinds of timbres, just relatively harmonic ones.)

More later.

________________________________
From: Ozan Yarman <ozanyarman@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Saturday, March 30, 2013 8:46 AM
Subject: Re: [MMM] Asking the right questions, Quantizing JI to 106edo

Nevertheless, 106-EDO happens to be the master tuning grid for several proposed Turkish maqam tone-systems as indicated in my said article.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Mar 29, 2013, at 7:06 PM, calebmrgn wrote:

[Non-text portions of this message have been removed]

🔗Caleb Morgan <calebmrgn@...>

It amazes *me*, but perhaps will not amaze the more experienced people here, that I never fully understood certain commas.

94 edo tempers out 32805/32768 and 225/224! Is *Miracle*!! (No, not Miracle, but just really, really good.)

Now, what other EDOs under consideration temper out that comma?

Because I'm approximating a chain of 4ths with

9/5
6/5
8/5
16/15
10/7
243/128
81/64
27/16
9/8
3/2
1/1
4/3
16/9
32/27
405/256
135/128
45/32
15/8
5/4
5/3
10/9

And, 94edo does this perfectly, because it tempers out those commas, which I never understood 'til I did the simple math.

I see on the xenharmonic page that 94edo in fact does this, and I was able to get the Temperament Finder online to give me this answer as well. I feel almost competent!

What I'm not grokking is how to find other EDOs that do this.

I see that 106 does, and 82. Is that correct?

Are those all the EDOs in that vicinity that do?

This is with a cutoff of exactly 250. That is
important because the more steps you have, the more likely
it is that an interval in this range will exist, so they
come thick and fast after 250.

Also assuming exact octaves.

> Question 2 (more opinion involved): Given that my wiggle-room is actually *very small*, what would be the best EDO to try to quantize the following JI scale to?

Scored by minimizing the worst error for consistent
temperings of these intervals, and nothing bigger than 250,

224, 248, 183, 241, 212, 217, 229, 159, 236, 246, 207, 193, 176, 130, 152, 200, 243, 171, 234, 231, 205, 188, 181, 118, 222, ...

Scored by minimizing the worst error*complexity badness, or
relative error, or error as a fraction of scale step, these
are best:

224, 183, 130, 159, 94, 118, 152, 248, 212, 89, 111, 176, 241, 217, 193, 171, 99, 140, 229, 207, 87, 58, 106, 53, 236, ...

In neither case is 106 especially good.

Graham

[Non-text portions of this message have been removed]

🔗Graham Breed <gbreed@...>

🔗Caleb Morgan <calebmrgn@...>

After much exploration, the winner is 318 for my master tuning resolution.

This is because I'm half the man Mr. Yarman is. (Joke.)

This does represent a commitment of sorts, as I hope to get familiar with the numbers of 318 (159).

318 is 6x53.

I wanted a master tuning resolution that would have increments at 400, 600, and 800 cents, as well as a lot of accuracy in the 11's and 13's, and an accurate 3/2 that would support a consistent chain of 3/2's without different size 3/2's.

53 is the smallest prime that gives me a 3/2 approximation accurate enough for such a chain that can be multiplied by 6 without being a humungous number.

There are excellent tables at 270edo and 130edo.

But I wanted a composite number, and Ozan Yarman has already blazed the trail, so it's practical.

Now I have to do the laborious work of retuning my orchestra, so I have about 2 hours before I could really start to regret my decision. (Caleb makes wry face.)

318 (159) simplifies the JI world without sacrificing anything, and allows for some symmetry, also.

There's a nice little modular multiplier applet online for quickly checking tunings, in case anyone is interested. It saved me a lot of time.

I appreciate everyone's help.

Special thanks to the LMSO guy for, well, he-knows-what.

caleb

________________________________
From: Graham Breed <gbreed@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Saturday, March 30, 2013 3:13 PM
Subject: Re: [MMM] Asking the right questions, Quantizing JI to 94EDO and others

Caleb Morgan <calebmrgn@yahoo.com> wrote:

> 94 edo tempers out 32805/32768 and 225/224! Is *Miracle*!! (No, not Miracle, but just really, really good.)
>
> Now, what other EDOs under consideration temper out that comma?

The temperament finder should have told you that tempering
out these two commas gives Garibaldi temperament.

http://x31eq.com/cgi-bin/uv.cgi?uvs=[-15%2C8%2C1%2C0>+225%3A224

It also tells you that other EDOs are 41, 12, 53, and 29.

It doesn't tell you directly, but you may notice that
32805/32768 is the schisma. This leads to a historically important temperament class currently called "Helmholtz" because the great man worked out optimal tunings for it. The EDOs it supports are those that approximate Pythagorean intonation well. They can all be twisted to support Garibaldi but the optimal Garibaldi diverts a little from Pythagorean, to be more like 94-EDO.

Graham

[Non-text portions of this message have been removed]

🔗Ozan Yarman <ozanyarman@...>

Dear Caleb,

Multiples of 53-EDO have a certain charm to them, so it is perfectly fine to prefer a huge multiple of a workable medium-voluminous equal temperament as a master tuning grid. My tuning solution for the 79-tone qanun turned out to be a subset of 159-EDO, which helped things get across to my Turkish audiences. In our famous JNMR article:

http://www.ozanyarman.com/files/theoryVSpractice.pdf

which I co-authored, the tunings compared with histogram figures were quantized according to 1/12th the Holderian comma resolution, which equates to 636-EDO.

Multiples of 41-EDO may also work, and also of 171-EDO, which is ninefold 19-tET that yields a near pure fifth.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Apr 3, 2013, at 1:50 PM, Caleb Morgan wrote:

> After much exploration, the winner is 318 for my master tuning resolution.
>
> This is because I'm half the man Mr. Yarman is. (Joke.)
>
> This does represent a commitment of sorts, as I hope to get familiar with the numbers of 318 (159).
>
> 318 is 6x53.
>
> I wanted a master tuning resolution that would have increments at 400, 600, and 800 cents, as well as a lot of accuracy in the 11's and 13's, and an accurate 3/2 that would support a consistent chain of 3/2's without different size 3/2's.
>
> 53 is the smallest prime that gives me a 3/2 approximation accurate enough for such a chain that can be multiplied by 6 without being a humungous number.
>
> There are excellent tables at 270edo and 130edo.
>
> But I wanted a composite number, and Ozan Yarman has already blazed the trail, so it's practical.
>
> Now I have to do the laborious work of retuning my orchestra, so I have about 2 hours before I could really start to regret my decision. (Caleb makes wry face.)
>
> 318 (159) simplifies the JI world without sacrificing anything, and allows for some symmetry, also.
>
> There's a nice little modular multiplier applet online for quickly checking tunings, in case anyone is interested. It saved me a lot of time.
>
> I appreciate everyone's help.
>
> Special thanks to the LMSO guy for, well, he-knows-what.
>
> caleb
>
>
>
>
> ________________________________
> From: Graham Breed <gbreed@...>
> To: MakeMicroMusic@yahoogroups.com
> Sent: Saturday, March 30, 2013 3:13 PM
> Subject: Re: [MMM] Asking the right questions, Quantizing JI to 94EDO and others
>
>
>
> Caleb Morgan <calebmrgn@...> wrote:
>
>> 94 edo tempers out 32805/32768 and 225/224! Is *Miracle*!! (No, not Miracle, but just really, really good.)
>>
>> Now, what other EDOs under consideration temper out that comma?
>
> The temperament finder should have told you that tempering
> out these two commas gives Garibaldi temperament.
>
> http://x31eq.com/cgi-bin/uv.cgi?uvs=[-15%2C8%2C1%2C0>+225%3A224
>
> It also tells you that other EDOs are 41, 12, 53, and 29.
>
> It doesn't tell you directly, but you may notice that
> 32805/32768 is the schisma. This leads to a historically important temperament class currently called "Helmholtz" because the great man worked out optimal tunings for it. The EDOs it supports are those that approximate Pythagorean intonation well. They can all be twisted to support Garibaldi but the optimal Garibaldi diverts a little from Pythagorean, to be more like 94-EDO.
>
> Graham
>
>
>
> [Non-text portions of this message have been removed]
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>

🔗Caleb Morgan <calebmrgn@...>

Hah!

You anticipated my solution. 171edo.

It was there, all along.

I feel like a fool, because I've gone around and around, and put more thought into this than any other decision in my life. More than what house to live in, who to marry.

I chose 171 edo.

I went back to this:

16/15, 13/12, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 13/11, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 15/11, 11/8, 7/5, 10/7, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 15/8, 2/1

changed it to this:

135/128, 16/15, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 13/11, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 15/11, 11/8, 7/5, 10/7, 16/11, 22/15, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 15/8, 2/1

and quantized it to 171edo, with a few tweaks of notes to make a more consistent chain of 4/3's or 3/2's

0., 91.228, 112.281, 147.368, 168.421, 182.456, 203.509, 231.579, 266.667, 294.737, 315.789, 350.877, 385.965, 407.018, 435.088, 498.246, 540.351, 554.386, 589.474, 617.544, 645.614, 659.649, 701.754, 764.912, 792.982, 814.035, 849.123, 884.211, 905.263, 933.333, 968.421, 996.491, 1017.544, 1031.579, 1052.632, 1087.719

steps absolute:
0, 13, 16, 21, 24, 26, 29, 33, 38, 42, 45, 50, 55, 58, 62, 71, 77, 79, 84, 88, 92, 94, 100, 109, 113, 116, 121, 126, 129, 133, 138, 142, 145, 147, 150, 155, 171

steps relative:
13, 3, 5, 3, 2, 3, 4, 5, 4, 3, 5, 5, 3, 4, 9, 6, 2, 5, 4, 4, 2, 6, 9, 4, 3, 5, 5, 3, 4, 5, 4, 3, 2, 3, 5, 16

Finally, I think I'm satisfied.

As hopeless as I am at many of the math concepts I've been reading about, I'm pretty confident that this is what I've been after all this time.

One nice thing is that a 5th (3/2) in this system is 100 steps out of 171, and 100 steps of 171 generates every other interval in mod171.

Phew.

It's been a very enjoyable few days of thinking about it and trying one edo after another.

Caleb

________________________________
From: Ozan Yarman <ozanyarman@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Thursday, April 4, 2013 7:22 AM
Subject: Re: [MMM] Asking the right questions, multiples of 53

Dear Caleb,

http://www.ozanyarman.com/files/theoryVSpractice.pdf

which I co-authored, the tunings compared with histogram figures were quantized according to 1/12th the Holderian comma resolution, which equates to 636-EDO.

Multiples of 41-EDO may also work, and also of 171-EDO, which is ninefold 19-tET that yields a near pure fifth.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Apr 3, 2013, at 1:50 PM, Caleb Morgan wrote:

[Non-text portions of this message have been removed]

🔗Andy <a_sparschuh@...>

--- In MakeMicroMusic@yahoogroups.com, Caleb Morgan <calebmrgn@...> wrote:
> I chose 171 edo.

Hi Caleb,

for 171-edo see:
http://tech.dir.groups.yahoo.com/group/tuning-math/message/3158
"
Right! 171-EDO has been strongly advocated by prolific
German tuning theorist Martin Vogel, who has even had
a keyboard built for it for use in his classes.
"

There exist some pics,
on M.Vogel's realization of his 171-tone keyboard:

http://eufonia.de/index.php/projekte/104-die-enharmonische-pfeifenorgel

also attend the example-videos in youtube,
how to play on that 171-tone pipe-organ:

http://www.youtube.com/watch?v=rBMUwJdvfDw
http://www.youtube.com/watch?v=_oHk9wocuGA

and note an rewiev:
/tuning/topicId_71212.html#71234
Monz remarks:
"
... i heard Stamm's performances on that organ,
and loved them. But i'm pretty sure it was tuned to 171-edo
as an approximation of 7-limit JI, at which 171-edo does
indeed excel. IIRC, Vogel considered the approximation to be
so good that he felt he could just call it "just" and not
be concerned about the error.
"

Further original references of that author
can be found in his curriculum vitae:
http://de.wikipedia.org/wiki/Martin_Vogel
(Literature mostly in German language)

bye
Andy

Asking the right questions, Quantizing JI to 106edo

🔗calebmrgn <calebmrgn@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Andy <a_sparschuh@...>