Yahoo Tuning Groups Ultimate Backup tuning Re: [tuning] Digest Number 3454 - WNR creationists

In this posting, I shall attempt to answer three postings from Digest Number 3454, two from Gene "W(iz)ard" Smith and one from Monz.

Dr. Robert Smith, (in Harmonics 1749? ), Jorgensen, and apparently many of the naive subscribers to the tuning list seem to have missed a subtle, yet fundamentally important concept, which John 'Longitude' Harrison clearly and consistently stated in all his writings about musical tuning.
This is the same misunderstanding, which contributed to the emnity between Harrison and Smith nearly 250 years ago.

Harrison developed an entirely novel basis for his calculations of music intervals.
He calculated them from pi.
As far as I am aware we have yet to discover the underlying reason or logic which he used to come to this extraordinary conclusion. He merely states that this is the case. Any of his experimental or theoretical data to substantiate his claim has yet to be found or understood, which is why I am continuing to research his recently discovered last "missing" manuscript.

Harrison was a particularly original (and eccentric) character and thinker, as anyone who has read the book or seen the film of "Longitude" will appreciate; yet he did successfully devise at least three fundamentally important inventions, one of which changed the course of world history in the years following his discoveries.

(So John Harrison does fairly deserve a due level of respect.)

Now to the nitty-gritty of the significance of Harrison's writings and thoughts about musical tuning ......

1. The only integer ratio which Harrison used in his system is 2:1 for the octave ratio.

2. He refused to accept the traditional thinking which states that "Perfect" intervals should only be at small integer frequency ratios.

3. Harrison's system can generate any number of intervals in an octave. i.e. pi is an irrational, transcendental number, (as far as we know at present).

4. The beat rates seem to be important in biological and perceptive ways which we have yet to fully understand.

Now for the problems with the systems which it is compared to by Monz, Smith and others.

88 Equal

Although it approximates Harrison's results, 88 tET (edo) is derived by splitting the octave (ratio 2:1) into 88 equal intervals. Eighty eight equal intervals provides a lower limit of tuning granularity (1200/88)/2 = 13.63636/2 = 6.818182 cents.
So any 88 edo interval is + or - 6.818182 cents from any chosen interval.
I suppose this may seem satisfactory for the late 20th century microtonalists, who are still using pitchbend with a resolution of (1200/12)/64 = 1.5625 cents or YAMS at (1200/1024) at 1.171875 cents.

3/10 comma meantone

see:

http://www.lucytune.com/tuning/mean_tone.html for details.

3/10 comma meantone avoids the same resolution limitation of 88edo as it has the same a "limited spiral" mapping system as do all meantones, which are derived from integer ratios.
The values for 1/3 comma meantone are:
derived from 6/5
Fifth is at: 694.77 cents
Large interval is at: 189.58 cents
small interval is at: 126.15 cents
It uses a generator which is derived 6/5 and will eventually become an equal temperament if you extend the steps for enough intervals.
I am sure someone on the list can run a program to tell us the granularity limit.

Harrison used 695.49 for his fifth;

So each step of fourths or fifth from the starting point will produce an extra error of 695.49 - 694.77 = .72 cents per step.

So after 12 steps we have an error of 12 * .72 = 8.64 cents.
As you continue the error increases.

If anyone tells you that LucyTuning (derived from Harrison's writings) is for all practical purposes the same as 88edo and 1/3 comma meantone, examine their hidden aganda.
Scratch a little deeper and I suspect that you will find that these same individuals are still advocating only tunings derived from integer frequency ratios. i.e. the same misguided logic that antagonised Harrison against Dr. Robert Smith 250-odd years ago.

Some people still argue against the whole principle of Charles Darwin's theory of evolution, and get labelled "creationists" or worse.
Maybe someone on the tuning lsit can come up with a suitable name for those who insist that all musical intervals and harmonics should only be at integer frequency ratios.

Please post suggestions!!!!

I have put a transcription of the musical parts of "Concerning Such Mechanism ......." at:
http://www.lucytune.com/academic/csm_transcription.html

Information about the "missing" last manuscript is at:
http://www.lucytune.com/academic/manuscript_search.html

I have made digital images of all the photocopies of the 180-odd pages of the "missing" manuscript from the Library of Congress, so if anyone wants to plough through them please email me for instructions and I'll get them to you somehow.

🔗monz <monz@tonalsoft.com>

hi Charles,

--- In tuning@yahoogroups.com, Charles Lucy <lucy@h...> wrote:

> Harrison developed an entirely novel basis for his
> calculations of music intervals.
> He calculated them from pi.
> As far as I am aware we have yet to discover the
> underlying reason or logic which he used to come to
> this extraordinary conclusion. He merely states that
> this is the case. Any of his experimental or theoretical
> data to substantiate his claim has yet to be found or
> understood, which is why I am continuing to research his
> recently discovered last "missing" manuscript.
>
> <snip>
>
> Now for the problems with the systems which it is
> compared to by Monz, Smith and others.
>
> 88 Equal
>
> Although it approximates Harrison's results, 88 tET (edo)
> is derived by splitting the octave (ratio 2:1) into 88 equal
> intervals. Eighty eight equal intervals provides a lower
> limit of tuning granularity (1200/88)/2 = 13.63636/2 =
> 6.818182 cents.
> So any 88 edo interval is + or - 6.818182 cents from any
> chosen interval.
> I suppose this may seem satisfactory for the late
> 20th century microtonalists, who are still using pitchbend
> with a resolution of (1200/12)/64 = 1.5625 cents or YAMS
> at (1200/1024) at 1.171875 cents.

OK, since we're nitpicking over miniscule differences
of tuning, let's *really* get to the nitty-gritty ...

1. *exactly* how did Harrison specify his tuning?
did he give a measurement for the "5th", or for the
semitone, or exactly what?

2. and did he use something that looks like a ratio,
which is what i have on my webpage (which i believe
i got from you, Charles)?

2^( (2pi + 1) / 4pi )
or
2^(1/2 + 1/4pi)

or did he actually say that the "5th" is 600+300/pi cents?

it actually doesn't matter, because as far as my calculations
can tell me, all three representations are exactly the same.
but i'm curious as to exactly how Harrison described it himself.

so, with that out of the way ... now i must address
the fact that your calculations of the "granularity"
of 88-edo misrepresent how closely it approximates
LucyTuning.

the *maximum* amount of error of "any 88 edo interval
is + or - 6.818182 cents from any chosen interval" --
but it's totally incorrect to state that "any 88edo
interval *is* + or - 6.818182 cents from any chosen
interval".

in fact, even a 101-tone LucyTuning chain of 5ths will
*still* present less than 2 cents error from 88edo.
i'm not trying to make any enemies here, Charles, but
i've said it before, and apparently you find it hard
to believe, so here's the (long) proof:

generator . LucyTuning ... 88-edo ......... error

.. 50 .. 1174.648293 .. 1172.727273 ... - 1.92102003
.. 49 ... 479.1553269 .. 477.2727273 .. - 1.882599629
.. 48 ... 983.662361 ... 981.8181818 .. - 1.844179228
.. 47 ... 288.1693952 .. 286.3636364 .. - 1.805758828
.. 46 ... 792.6764293 .. 790.9090909 .. - 1.767338427
.. 45 .... 97.18346348 .. 95.45454545 . - 1.728918027
.. 44 ... 601.6904976 .. 600.0000000 .. - 1.690497626
.. 43 .. 1106.197532 .. 1104.545455 ... - 1.652077225
.. 42 ... 410.7045659 .. 409.0909091 .. - 1.613656825
.. 41 ... 915.2116001 .. 913.6363636 .. - 1.575236424
.. 40 ... 219.7186342 .. 218.1818182 .. - 1.536816024
.. 39 ... 724.2256684 .. 722.7272727 .. - 1.498395623
.. 38 .... 28.7327025 ... 27.27272727 . - 1.459975222
.. 37 ... 533.2397366 .. 531.8181818 .. - 1.421554822
.. 36 .. 1037.746771 .. 1036.363636 ... - 1.383134421
.. 35 ... 342.2538049 .. 340.9090909 .. - 1.344714021
.. 34 ... 846.7608391 .. 845.4545455 .. - 1.30629362
.. 33 ... 151.2678732 .. 150.0000000 .. - 1.26787322
.. 32 ... 655.7749074 .. 654.5454545 .. - 1.229452819
.. 31 .. 1160.281942 .. 1159.090909 ... - 1.191032418
.. 30 ... 464.7889757 .. 463.6363636 .. - 1.152612018
.. 29 ... 969.2960098 .. 968.1818182 .. - 1.114191617
.. 28 ... 273.8030439 .. 272.7272727 .. - 1.075771217
.. 27 ... 778.3100781 .. 777.2727273 .. - 1.037350816
.. 26 .... 82.81711223 .. 81.81818182 . - 0.998930415
.. 25 ... 587.3241464 .. 586.3636364 .. - 0.960510015
.. 24 .. 1091.831181 .. 1090.909091 ... - 0.922089614
.. 23 ... 396.3382147 .. 395.4545455 .. - 0.883669214
.. 22 ... 900.8452488 .. 900.0000000 .. - 0.845248813
.. 21 ... 205.352283 ... 204.5454545 .. - 0.806828412
.. 20 ... 709.8593171 .. 709.0909091 .. - 0.768408012
.. 19 .... 14.36635125 .. 13.63636364 . - 0.729987611
.. 18 ... 518.8733854 .. 518.1818182 .. - 0.691567211
.. 17 .. 1023.38042 ... 1022.727273 ... - 0.65314681
.. 16 ... 327.8874537 .. 327.2727273 .. - 0.614726409
.. 15 ... 832.3944878 .. 831.8181818 .. - 0.576306009
.. 14 ... 136.901522 ... 136.3636364 .. - 0.537885608
.. 13 ... 641.4085561 .. 640.9090909 .. - 0.499465208
.. 12 .. 1145.91559 ... 1145.454545 ... - 0.461044807
.. 11 ... 450.4226244 .. 450.0000000 .. - 0.422624407
.. 10 ... 954.9296586 .. 954.5454545 .. - 0.384204006
... 9 ... 259.4366927 .. 259.0909091 .. - 0.345783605
... 8 ... 763.9437268 .. 763.6363636 .. - 0.307363205
... 7 .... 68.45076099 .. 68.18181818 . - 0.268942804
... 6 ... 572.9577951 .. 572.7272727 .. - 0.230522404
... 5 .. 1077.464829 .. 1077.272727 ... - 0.192102003
... 4 ... 381.9718634 .. 381.8181818 .. - 0.153681602
... 3 ... 886.4788976 .. 886.3636364 .. - 0.115261202
... 2 ... 190.9859317 .. 190.9090909 .. - 0.076840801
... 1 ... 695.4929659 .. 695.4545455 .. - 0.038420401
... 0 ..... 0.000000000 .. 0.000000000 .. 0.000000000
.. -1 ... 504.5070341 .. 504.5454545 .. + 0.038420401
.. -2 .. 1009.014068 .. 1009.090909 ... + 0.076840801
.. -3 ... 313.5211024 .. 313.6363636 .. + 0.115261202
.. -4 ... 818.0281366 .. 818.1818182 .. + 0.153681602
.. -5 ... 122.5351707 .. 122.7272727 .. + 0.192102003
.. -6 ... 627.0422049 .. 627.2727273 .. + 0.230522404
.. -7 .. 1131.549239 .. 1131.818182 ... + 0.268942804
.. -8 ... 436.0562732 .. 436.3636364 .. + 0.307363205
.. -9 ... 940.5633073 .. 940.9090909 .. + 0.345783605
. -10 ... 245.0703414 .. 245.4545455 .. + 0.384204006
. -11 ... 749.5773756 .. 750.0000000 .. + 0.422624407
. -12 .... 54.08440974 .. 54.54545455 . + 0.461044807
. -13 .. .558.5914439 .. 559.0909091 .. + 0.499465208
. -14 .. 1063.098478 .. 1063.636364 ... + 0.537885608
. -15 ... 367.6055122 .. 368.1818182 .. + 0.576306009
. -16 ... 872.1125463 .. 872.7272727 .. + 0.614726409
. -17 ... 176.6195805 .. 177.2727273 .. + 0.65314681
. -18 ... 681.1266146 .. 681.8181818 .. + 0.691567211
. -19 .. 1185.633649 .. 1186.363636 ... + 0.729987611
. -20 ... 490.1406829 .. 490.9090909 .. + 0.768408012
. -21 ... 994.647717 ... 995.4545455 .. + 0.806828412
. -22 ... 299.1547512 .. 300.0000000 .. + 0.845248813
. -23 ... 803.6617853 .. 804.5454545 .. + 0.883669214
. -24 ... 108.1688195 .. 109.0909091 .. + 0.922089614
. -25 ... 612.6758536 .. 613.6363636 .. + 0.960510015
. -26 .. 1117.182888 .. 1118.181818 ... + 0.998930415
. -27 ... 421.6899219 .. 422.7272727 .. + 1.037350816
. -28 ... 926.1969561 .. 927.2727273 .. + 1.075771217
. -29 ... 230.7039902 .. 231.8181818 .. + 1.114191617
. -30 ... 735.2110243 .. 736.3636364 .. + 1.152612018
. -31 .... 39.71805849 .. 40.90909091 . + 1.191032418
. -32 ... 544.2250926 .. 545.4545455 .. + 1.229452819
. -33 .. 1048.732127 .. 1050.000000 ... + 1.26787322
. -34 ... 353.2391609 .. 354.5454545 .. + 1.30629362
. -35 ... 857.7461951 .. 859.0909091 .. + 1.344714021
. -36 ... 162.2532292 .. 163.6363636 .. + 1.383134421
. -37 ... 666.7602634 .. 668.1818182 .. + 1.421554822
. -38 .. 1171.267298 .. 1172.727273 ... + 1.459975222
. -39 ... 475.7743316 .. 477.2727273 .. + 1.498395623
. -40 ... 980.2813658 .. 981.8181818 .. + 1.536816024
. -41 ... 284.7883999 .. 286.3636364 .. + 1.575236424
. -42 ... 789.2954341 .. 790.9090909 .. + 1.613656825
. -43 .... 93.80246823 .. 95.45454545 . + 1.652077225
. -44 ... 598.3095024 .. 600.0000000 .. + 1.690497626
. -45 .. 1102.816537 .. 1104.545455 ... + 1.728918027
. -46 ... 407.3235707 .. 409.0909091 .. + 1.767338427
. -47 ... 911.8306048 .. 913.6363636 .. + 1.805758828
. -48 ... 216.337639 ... 218.1818182 .. + 1.844179228
. -49 ... 720.8446731 .. 722.7272727 .. + 1.882599629
. -50 .... 25.35170724 .. 27.27272727 . + 1.92102003

BTW ... at this point i consider myself to be an
early 21st century microtonalist. :-P

> 3/10 comma meantone

i could do the same kind of thing for this too,
but don't have the time right now. you get the idea.

PS -- it is a good idea to change the subject line
when a thread veers off in a different direction,
but in this thread i think you're needlessly making
it harder for readers to follow the thread by whimsically
changing the subject line while the content of your
responses is staying on-topic. couldn't we have just
left the subject line as i had put it: "LucyTuning and 88-edo
(was: History of 50 equal)"?

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Charles Lucy <lucy@h...> wrote:

> Harrison developed an entirely novel basis for his calculations of
> music intervals.
> He calculated them from pi.
> As far as I am aware we have yet to discover the underlying reason or
> logic which he used to come to this extraordinary conclusion.

If no one knows the reason or logic, how can we assume there is one?

> 3. Harrison's system can generate any number of intervals in an octave.
> i.e. pi is an irrational, transcendental number, (as far as we know at
> present).

Pi is definately transcendental; this is Lindemann's theorem. It was
not known at the time of Harrison, so it doesn't seem likely Harrison
chose pi because of it.

It is not required that either the fifth or its logarithm be a
rational number in order for the cicle of fifths not to close; all
that is required is that the base 2 logarithm of the fifth be a
rational number. If a rational approximation to pi is chosen, the
corresponding fifth defined in the manner of Harrison will close; 88
equal is what you get if you use the well-known approximation
pi ~ 22/7. Pi ~ 3 would give you 12-equal.

> Although it approximates Harrison's results, 88 tET (edo) is derived
> by splitting the octave (ratio 2:1) into 88 equal intervals. Eighty
> eight equal intervals provides a lower limit of tuning granularity
> (1200/88)/2 = 13.63636/2 = 6.818182 cents.
> So any 88 edo interval is + or - 6.818182 cents from any chosen
> interval.
> I suppose this may seem satisfactory for the late 20th century
> microtonalists, who are still using pitchbend with a resolution of
> (1200/12)/64 = 1.5625 cents or YAMS at (1200/1024) at 1.171875 cents.

Since Harrison was not proposing to extend the circle of fifths even
as far as 88 fifths, this isn't relevant.

> 3/10 comma meantone

> It uses a generator which is derived 6/5 and will eventually become an
> equal temperament if you extend the steps for enough intervals.

This is true to exactly the same extent, and in the same way, as it is
also true of Harrison's fifth. If you extend either of them to 88
notes to an octave it is already getting pretty regular. If you were
to extend the Harrison fifth to 1420 notes to the octave, it would be
extremely regular, since it would have one fifth which was a mere
0.0115 cents flatter than all the rest. If you want to avoid, as far
as possible, this sort of thing using pi is not a good plan; you
should use phi and consider the golden meantone. However, there is no
reason to think this had anything to do with Harrison's tuning ideas.

> I am sure someone on the list can run a program to tell us the
> granularity limit.
>
> Harrison used 695.49 for his fifth;
>
> So each step of fourths or fifth from the starting point will produce
> an extra error of 695.49 - 694.77 = .72 cents per step.

No, because the 3/10-comma fifth is 695.50, only 0.01 cents sharper.

> If anyone tells you that LucyTuning (derived from Harrison's writings)
> is for all practical purposes the same as 88edo and 1/3 comma meantone,
> examine their hidden aganda.

You might also examine their math. If it is good, then they are
correct. Moreover, even if 0.01 cent is not close enough for your
superhuman ears, there will always be some approximation which will be
The 1420-equal system, which gets within .0000081 cent, should suffice.

Re: [tuning] Digest Number 3454 - WNR creationists - Wake up and hear the coffee !!!!!!

🔗Charles Lucy <lucy@harmonics.com>

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>