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Message: 8750 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 15:55:43

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi Dante and paul, > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> > wrote: >>>>>
>>>>> As far as tonal theory being a science, you only >>>>> have to look at or try to analyze some Brahms passages, >>>>> or Wagner et al to see that it is far from >>>>> being so (IMO). >>>>
>>>> It won't be any more scientific simply to look at sets >>>> of equivalences class when analyzing Brahms, will it? >>>> Or are you saying Brahms wrote unscientific music? >>>
>>> No, I'm saying Brahms wrote music that, at times, >>> exhibits ambiguity when subjected to traditional harmonic >>> analysis. And no, I'm not saying Fortean analysis will >>> tell you anything here. My point was that the ambiguity >>> demonstrates that harmonic analysis is more of an art >>> than a science. >>> >>> Dante >>
>> Or it might just demonstrate that Bramhs's music exhibits >> ambiguity -- maybe because he wanted it to! Anyway, I don't >> think any of these modalities of musical analysis are >> anywhere near a "science", but certainly ambiguity is >> something that can be understood, described, and predicted >> in a scientific way. For example, the pitch of an inharmonic >> spectrum, as I've been discussing with Kurt on the tuning >> list lately. > > >
> i think the main reason harmonic analysis would be > characterised as an "art" is precisely *because* of > the ambiguity available to a composer like Brahms, > whether his intended tuning is 12edo or a meantone > (the only two likely possibilities for Brahms IMO). > > my point: that *temperament* allows composers to play > the kinds of games ("punning") that aren't possible > in JI.
Irrelevant -- Dante and I were talking about 'conventional' tonal harmonic analysis, which never distinguishes any 81:80s anyway.
> and of course JI is the tuning which offers > the straightforward "scientific" approach to harmonic > analysis. BS.
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Message: 8751 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 15:53:12

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:
>
>> One issue I have grappled with is
>>> the mathematical versus the musical definition of harmonics. >>> The mathematical 'harmonic series' as I understand it > always
>>> represents harmonics as 1/n, whereas in music we often > talk
>>> about harmonics as whole number multiples, or what would > be
>>> called in math the 'arithmetic series'. What is your take on > this? >>> >>> Aaron >>
>> The reason for this is historical. We say whole number > multiples
>> today because everyone since Fourier talks about frequency >> measurements. In the old days, it was string length > measurements (or
>> still today, period or wavelength) where the numbers are > *inversely
>> proportional* to the frequency numbers. So the harmonic
> series in the
>> old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same >> harmonic series that today goes 1, 2, 3, 4 . . . > >
> Right; I am aware of this. There is yet a gap between > mathematicians and musicians here since the series 1 2 3 4 ... > is not called a 'harmonic series' in mathematics. It is instead > called an 'arithmetic series'. This was my point.
That's exactly what I explained above! Where did we lose each other?
> If we are > working with numbers all the time, shouldn't we adopt > terminology common to both mathematics _and music?
To some extent . . . but some people go too far trying to impose math terminology on musicians whose terminology is already fine (and most importantly, forms a common currency for musical communication) . . .
> For > example, we could call 1/1, 1/2, 1/3 ... a harmonic series and 1 2 > 3 4 ... an arithmetic series and be done with it ... almost. In music > we don't sum these series necessarily as is done in math, as > Gene pointd out. But at least we would be closer to common > agreement between these terms in both math and music. > > Aaron
In music the numbers have to *represent* something. 1/1, 1/2, 1/3, 1/4 . . . *is* the musical harmonic series if the numbers are representing periods, wavelengths, string lengths, air column lengths, etc. Numbers alone do not represent anything musical, though they are certainly a valid field of mathematical study.
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Message: 8752 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 16:55:51

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

> 1: 126/125 13.795 small septimal comma >Obviously 81/80 is simpler already. Sorry guys I'm behind.
Whoa, something is wrong with my algorithm. Manuel
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Message: 8753 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 16:04:15

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" <lists@k...> 
wrote:
> Does this look like something? > > 1 > 121/120
Thanks Kees, but obviously 81/80 also maps to 1 degree of 72 and it's got lower "expressibility" than 121/120.
> 35 > 7/5 4.7e-004 > 36 > 45/32 5.6e-003 > 64/45 5.6e-003 > 99/70 5.1e-005
This is clearly incorrect, 45/32 maps to 35 degrees of 72, not 36. It has to be the same as 7/5 since 225:224 vanishes in 72, as you know. I didn't even expect to hear from you, so thanks. I'm still hoping Gene and/or Manuel can give the solution, I didn't think it would be this hard for them given similar things they've posted before . . .
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Message: 8754 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 17:20:40

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

I forgot that the deviation was also weighted in.
So the result is now this, it became more uneven and also
improper but it's much better.

  0:          1/1               0.000 unison, perfect prime
  1:         81/80             21.506 syntonic comma, Didymus comma
  2:         45/44             38.906 1/5-tone
  3:         33/32             53.273 undecimal comma, 33rd harmonic
  4:         25/24             70.672 classic chromatic semitone, minor chroma
  5:         21/20             84.467 minor semitone
  6:         35/33            101.867
  7:         15/14            119.443 major diatonic semitone
  8:         27/25            133.238 large limma, BP small semitone
  9:         12/11            150.637 3/4-tone, undecimal neutral second
 10:         11/10            165.004 4/5-tone, Ptolemy's second
 11:         10/9             182.404 minor whole tone
 12:          9/8             203.910 major whole tone
 13:         25/22            221.309
 14:          8/7             231.174 septimal whole tone
 15:         81/70            252.680 Al-Hwarizmi's lute middle finger
 16:          7/6             266.871 septimal minor third
 17:         33/28            284.447 undecimal minor third
 18:         25/21            301.847 BP second, quasi-tempered minor third
 19:          6/5             315.641 minor third
 20:         40/33            333.041
 21:         11/9             347.408 undecimal neutral third
 22:        100/81            364.807 grave major third
 23:          5/4             386.314 major third
 24:         44/35            396.178
 25:         14/11            417.508 undecimal diminished fourth or major third
 26:          9/7             435.084 septimal major third, BP third
 27:         35/27            449.275 9/4-tone, septimal semi-diminished fourth
 28:         21/16            470.781 narrow fourth
 29:         33/25            480.646 2 pentatones
 30:          4/3             498.045 perfect fourth
 31:         27/20            519.551 acute fourth
 32:         15/11            536.951 undecimal augmented fourth
 33:         11/8             551.318 undecimal semi-augmented fourth
 34:         25/18            568.717 classic augmented fourth
 35:          7/5             582.512 septimal or Huygens' tritone, BP fourth
 36:         99/70            600.088 2nd quasi-equal tritone
 37:         10/7             617.488 Euler's tritone
 38:         36/25            631.283 classic diminished fifth
 39:         16/11            648.682 undecimal semi-diminished fifth
 40:         22/15            663.049 undecimal diminished fifth
 41:         40/27            680.449 grave fifth
 42:          3/2             701.955 perfect fifth
 43:         50/33            719.354 3 pentatones
 44:         32/21            729.219 wide fifth
 45:         54/35            750.725 septimal semi-augmented fifth
 46:         14/9             764.916 septimal minor sixth
 47:         11/7             782.492 undecimal augmented fifth
 48:         35/22            803.822
 49:          8/5             813.686 minor sixth
 50:         81/50            835.193 acute minor sixth
 51:         18/11            852.592 undecimal neutral sixth
 52:         33/20            866.959
 53:          5/3             884.359 major sixth, BP sixth
 54:         42/25            898.153 quasi-tempered major sixth
 55:         56/33            915.553
 56:         12/7             933.129 septimal major sixth
 57:        140/81            947.320
 58:          7/4             968.826 harmonic seventh
 59:         44/25            978.691
 60:         16/9             996.090 Pythagorean minor seventh
 61:          9/5            1017.596 just minor seventh, BP seventh
 62:         20/11           1034.996 large minor seventh
 63:         11/6            1049.363 21/4-tone, undecimal neutral seventh
 64:         50/27           1066.762 grave major seventh
 65:         15/8            1088.269 classic major seventh
 66:         66/35           1098.133
 67:         21/11           1119.463
 68:         48/25           1129.328 classic diminished octave
 69:         64/33           1146.727 33rd subharmonic
 70:         88/45           1161.094
 71:        160/81           1178.494 octave - syntonic comma
 72:          2/1            1200.000 octave

Manuel


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Message: 8755 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 17:15:33

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> > I forgot that the deviation was also weighted in. > So the result is now this, it became more uneven and also > improper but it's much better.
Thanks so very much Manuel. Your other methods may prove useful as well, I appreciate the time you've taken to code all of them. I hope George, if it's not too late, will consider using these ratios for his 72-equal keyboard diagram -- the unevenness is probably not important for him since he told me he intended the ratios to show how *intervals* look on the keyboard, not as a representation of the *pitches* (those of you who have been reading my posts for years know what i mean -- intervals I notate as a:b, while pitches I notate a/b) . . . George, you already have some 3-digit numbers, so the below shouldn't be a problem, should it? If this isn't acceptable, maybe a 17-limit version of same? Or feel free to ignore me. -Paul
> > 0: 1/1 0.000 unison, perfect prime > 1: 81/80 21.506 syntonic comma, Didymus comma > 2: 45/44 38.906 1/5-tone > 3: 33/32 53.273 undecimal comma, 33rd harmonic > 4: 25/24 70.672 classic chromatic semitone, minor chroma > 5: 21/20 84.467 minor semitone > 6: 35/33 101.867 > 7: 15/14 119.443 major diatonic semitone > 8: 27/25 133.238 large limma, BP small semitone > 9: 12/11 150.637 3/4-tone, undecimal neutral second > 10: 11/10 165.004 4/5-tone, Ptolemy's second > 11: 10/9 182.404 minor whole tone > 12: 9/8 203.910 major whole tone > 13: 25/22 221.309 > 14: 8/7 231.174 septimal whole tone > 15: 81/70 252.680 Al-Hwarizmi's lute middle finger > 16: 7/6 266.871 septimal minor third > 17: 33/28 284.447 undecimal minor third > 18: 25/21 301.847 BP second, quasi-tempered minor third > 19: 6/5 315.641 minor third > 20: 40/33 333.041 > 21: 11/9 347.408 undecimal neutral third > 22: 100/81 364.807 grave major third > 23: 5/4 386.314 major third > 24: 44/35 396.178 > 25: 14/11 417.508 undecimal diminished fourth
or major third
> 26: 9/7 435.084 septimal major third, BP third > 27: 35/27 449.275 9/4-tone, septimal semi- diminished fourth > 28: 21/16 470.781 narrow fourth > 29: 33/25 480.646 2 pentatones > 30: 4/3 498.045 perfect fourth > 31: 27/20 519.551 acute fourth > 32: 15/11 536.951 undecimal augmented fourth > 33: 11/8 551.318 undecimal semi-augmented fourth > 34: 25/18 568.717 classic augmented fourth > 35: 7/5 582.512 septimal or Huygens' tritone, BP fourth > 36: 99/70 600.088 2nd quasi-equal tritone > 37: 10/7 617.488 Euler's tritone > 38: 36/25 631.283 classic diminished fifth > 39: 16/11 648.682 undecimal semi-diminished fifth > 40: 22/15 663.049 undecimal diminished fifth > 41: 40/27 680.449 grave fifth > 42: 3/2 701.955 perfect fifth > 43: 50/33 719.354 3 pentatones > 44: 32/21 729.219 wide fifth > 45: 54/35 750.725 septimal semi-augmented fifth > 46: 14/9 764.916 septimal minor sixth > 47: 11/7 782.492 undecimal augmented fifth > 48: 35/22 803.822 > 49: 8/5 813.686 minor sixth > 50: 81/50 835.193 acute minor sixth > 51: 18/11 852.592 undecimal neutral sixth > 52: 33/20 866.959 > 53: 5/3 884.359 major sixth, BP sixth > 54: 42/25 898.153 quasi-tempered major sixth > 55: 56/33 915.553 > 56: 12/7 933.129 septimal major sixth > 57: 140/81 947.320 > 58: 7/4 968.826 harmonic seventh > 59: 44/25 978.691 > 60: 16/9 996.090 Pythagorean minor seventh > 61: 9/5 1017.596 just minor seventh, BP seventh > 62: 20/11 1034.996 large minor seventh > 63: 11/6 1049.363 21/4-tone, undecimal neutral seventh > 64: 50/27 1066.762 grave major seventh > 65: 15/8 1088.269 classic major seventh > 66: 66/35 1098.133 > 67: 21/11 1119.463 > 68: 48/25 1129.328 classic diminished octave > 69: 64/33 1146.727 33rd subharmonic > 70: 88/45 1161.094 > 71: 160/81 1178.494 octave - syntonic comma > 72: 2/1 1200.000 octave > > Manuel
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Message: 8757 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 02:29:31

Subject: Re: enumerating pitch class sets algebraicallyy

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote:

>> Can you give any examples of pre-serial atonal music? >
> You cant do any better than Webern op 5-16. You can > download mp3s of 0p 5 and 6 (both landmark works) from here: > > Anton Webern * [with cont.] (Wayb.) > > Dante
i've made mp3's of some samples from my MIDI-files, of course with my usual exaggerated rubato (for which i will never apologize) : Schoenberg's 1st Quartet and Webern's "String Quartet (1905)" are the two earliest examples (AFAIK) of atonality in music. Webern - climax from "String Quartet (1905)"   └ * [with cont.] (Wayb.) Schoenberg - opening of 1st Quartet   └ * [with cont.] (Wayb.) and since Dante mentioned an mp3 of op. 6, here's mine: Webern - 1st piece from _6 OrchesterstŘcke, op. 6_   └ * [with cont.] (Wayb.) and this is perhaps the most famous example of non-serial atonal orchestral music: Schoenberg - opening of 1st piece from _5 OrchesterstŘcke, op. 16_   └ * [with cont.] (Wayb.) there's lots more in my "A Century of New Music in Vienna", between 1905 and 1920. A century of new music in Vienna, (c) 1999-200... * [with cont.] (Wayb.) -monz
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Message: 8759 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 02:30:45

Subject: Re: reply to Carl

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:
>
>> And while I'm on it, serial tonal music?
the best example (AFAIK) of serial tonal music is Ben Johnston's _6th Quartet_. i've posted a lot on this piece on the main tuning list. check the archives. -monz
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Message: 8760 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 18:05:47

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" <lists@k...> 
wrote:

> Hi Paul, > > 81/80 maps to 1.29 steps and was rejected in the first pass of the > algorithm, where I look for the simplest ratio of successive values > in the series.
I'm unclear on what that means.
> 45/32 was accepted in the second pass. I agree this causes an > unevenness in acceptance. Still, these are best ratios for the > complexity.
45/32 should *only* map to 35 steps of 72, never to 36 steps of 72, if you are constructing your periodicity block correctly.
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Message: 8761 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 02:32:21

Subject: Re: enumerating pitch class sets algebraicallyy

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote:
>>> Can you give any examples of pre-serial atonal music? >>
>> You cant do any better than Webern op 5-16. You can download mp3s
> of 0p 5
>> and 6 (both landmark works) from here: >> >> Anton Webern * [with cont.] (Wayb.) >
> Do you know of a good source for midi versions of Webern?
i have made several, but i believe that they're all still under copyright and thus i'm not supposed to share them. (but i could be wrong about the pieces from 1905-1909.) -monz
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Message: 8762 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 18:04:08

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:
> >>>> the harmonic
>>> series in the
>>>> old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same >>>> harmonic series that today goes 1, 2, 3, 4 . . . >>> >>>
>>> Right; I am aware of this. There is yet a gap between >>> mathematicians and musicians here since the series 1 2 3 4 > ...
>>> is not called a 'harmonic series' in mathematics. It is instead >>> called an 'arithmetic series'. This was my point. >>
>> That's exactly what I explained above! Where did we lose each > other? > >
> No, I haven't lost you, but my point must not be clear. So, I will try > to be clearer this time. > > If we call 1,2,3,4 ... a harmonic series, then a mathematician will > say we are wrong; we are using the wrong term, because in > mathematics this is not called a harmonic series, it is called an > arithmetic series.
That would be an awfully pedantic and solipsistic mathematician, unless "we" patently failed to make it clear we were talking about *frequencies* above (assuming, of course, that we were!) . . .
> So, I suggest that if we are using terminology which is common > to both math and music, such as 'the harmonic series', we > should try to have as much agreement as possible, and on this > very basic point I feel that we could benefit from adopting the > common terms of a harmonic or arithmetic series from the > terminology of mathematics.
They don't apply. Mathematics deals with pure number. In music numbers can signify many things, but have no meaning on their own. If we decide to refer to "1/1 1/2 1/3 1/4" as the "harmonic sequence" regardless of whether the numbers represent frequencies or periods or wavelengths, we are stripping ourselves of the ability to make a crucial distinction, and refer to an important musical referent. You can replace "music" with "physics" in everything I've written on this topic and it remains true. No physicist would accept what you are proposing. Are you aware of the history of the term "arithmetic series", "arithmetic division", etc., in music?
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Message: 8763 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 02:37:48

Subject: Re: Digest Number 864

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote: >
>> I mean, Schoenberg actually believed it when he said he >> had "discovered a method of composing that would ensure >> the supremacy of German music for the next 100 years". >
> After which, of course, he moved to the Big Orange.
ah, but this is what i find really interesting about that famous comment by Schoenberg: i'm convinced that his early "free atonality" style came about as a result of his decision not to go with microtonality. and now, here we are a century later, on what i think is truly the threshold of the "microtonal era". again, i refer to my "A Century of New Music in Vienna", the years 1908 to 1914, especially 1908-10. A century of new music in Vienna, (c) 1999-200... * [with cont.] (Wayb.) -monz
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Message: 8765 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 03:31:25

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> > What's the 11-limit TM-reduced basis of 72-tET again? See Yahoo groups: /tuning-math/message/7392 * [with cont.]
for these. Its <225/224, 243/242, 385/384, 4000/3993>
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Message: 8766 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 19:24:27

Subject: Re: Digest Number 862

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> If we call 1,2,3,4 ... a harmonic series, then a mathematician will >> say we are wrong; we are using the wrong term, because in >> mathematics this is not called a harmonic series, it is called an >> arithmetic series. >
> That would be an awfully pedantic and solipsistic mathematician, > unless "we" patently failed to make it clear we were talking about > *frequencies* above (assuming, of course, that we were!) . . .
A pedantic, solipsistic mathematician would say it is wrong because it isn't a series at all, though usage is sometimes sloppy on this point. A number theorist usually calls a sequence of the type a, a+b, a+2b, a+3b, ... where a and b are integers an arithmetic progression. Arithmetic Series -- from MathWorld * [with cont.] Arithmetic Sequence -- from MathWorld * [with cont.] Harmonic Series -- from MathWorld * [with cont.]
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Message: 8767 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 03:43:35

Subject: Re: Enumerating pitch class sets algebraically

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i think the main reason harmonic analysis would be > characterised as an "art" is precisely *because* of > the ambiguity available to a composer like Brahms, > whether his intended tuning is 12edo or a meantone > (the only two likely possibilities for Brahms IMO).
Speaking as a veteran retuner, meantone is fine for Francois Couperin, Meade Lux Lewis or Buddy Holly, but don't try it with Brahms. It won't work. Speaking of which, I've recently finished a grail version of his Piano Concerto #2, which does work, but I've corrupted the piano into a jazz piano in this version. It makes me think it wasn't just Beethoven who was one step away from total boogie.
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Message: 8770 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 21:37:29

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" > <perlich@a...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" > <cfaah@e...> wrote:
> But does "we" in quotes mean that you are insulted > when I say "we"? No. > "You" will have to explain "yourself" to a mathematician when > "you" say that a harmonic series goes 1,2,3,4 ... because > according to the mathematician, the harmonic series does not > go 1,2,3,4 ... is goes 1/1, 1/2, 1/3 ...
Indeed -- except that, as Gene just pointed out, the latter is not called the "harmonic series" by mathematicians until you sum up the terms.
> I thought we have been discussing mathematicians, not > physicists, but I would guess that physicists would prefer to > speak for themselves.
OK, I guess so far only one has.
> I would wager that all of the imaginary > physicists are not on "your" side.
Imaginary? Why imaginary?
> It's quite likely that there are > those who would agree with me and there are those who would > agree with "you".
The "harmonic series" is a frequent referent in physics texts and journals that discuss acoustics or even nonlinear dynamical systems. Physicists refer to a set of *periods* or *wavelengths* as 'harmonic' if their proportions belong to the set {1/1, 1/2, 1/3, 1/4 . . .} Physicists refer to a set of *frequencies* as 'harmonic' if their proportions refer to the set {1, 2, 3, 4 . . .} I challenge you to find a counterexample in any physics text or journal article. Let me now begin to collect supporting examples, which I hope not to have to spend too much time doing: How harmonic are harmonics? * [with cont.] (Wayb.) đ¤ ▒ * [with cont.] (Wayb.) Harmonic Series * [with cont.] (Wayb.) http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n1002.htm http://www.physics.odu.edu/~hyde/Teaching/5 http://www.physics.northwestern.edu/classes/2001Spring/135- 1/Projects/3/sound.html&Physics_harmonic_series.htm Physics 4830 Course Notes * [with cont.] (Wayb.) Page not found * [with cont.] (Wayb.) 404 Not Found * [with cont.] Search for http://www.physics.northwestern.edu/classes/2001Spring/135- in Wayback Machine 1/Projects/3/sound.html
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Message: 8773 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 18:36:10

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

No problem Kees -- my question did concern periodicity blocks, but I 
shouldn't have assumed that you'd have read every post. No apology 
necessary from your end.

--- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" <lists@k...> 
wrote:
> I totally agree Paul. I just threw an algorithm together and gave the > raw results. I didn't even consider periodicity blocks. I just tried > to find relatively best rationals for the steps. That, of course, > doesn't have to result in consistent mapping. > Sorry if I only caused confusion. > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" > <lists@k...> >> wrote: >> >>> Hi Paul, >>>
>>> 81/80 maps to 1.29 steps and was rejected in the first pass of > the
>>> algorithm, where I look for the simplest ratio of successive > values
>>> in the series. >>
>> I'm unclear on what that means. >>
>>> 45/32 was accepted in the second pass. I agree this causes an >>> unevenness in acceptance. Still, these are best ratios for the >>> complexity. >>
>> 45/32 should *only* map to 35 steps of 72, never to 36 steps of 72, >> if you are constructing your periodicity block correctly.
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Message: 8774 - Contents - Hide Contents

Date: Wed, 10 Dec 2003 23:35:53

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> imaginary because you invoked an army of like-minded > physicists in order to rebut my point of view, and this seemed > rather fantastical and unnecessary to me.
Fantastical and imaginary as in untrue?
> By the way, Paul, it wasn't necessary to go to the trouble.
If it seemed untrue/imaginary/fantastical before, but now seems true, then certainly *something* changed your mind . . .
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