Talk:Pythagorean tuning

message: I edited the text about collapsing the series of perfect fifths into a one octave scale (by shifting low or high notes up or down by one or more octaves).  My intent was to provide the musical motivation for this part of the method.  In the as-is text, it was stated that this compression was done by convention, but no rationale was given.
I hope this is an improvement but comments and corrections would be welcome
Mark.camp (talk) 04:35, 28 December 2010 (UTC)
thread_title: Added to the text about collapsing the series into a compact scale

message: This article has been deemed too confusing by (apparently) a U.S. junior high school student (see Wikipedia:Reference_desk/Miscellaneous#PYTHAGORAS). Make of that what you will, but the discussion contains content and links that might help to make this article more accessible. Sandstein 17:47, 3 June 2006 (UTC)
thread_title:  Confusing? 

message: In the "Method" section, why were the letternames chosen? D and A are so nowhere near 200Hz and 300Hz that it's idiotic. The frequiencies given are an awful lot closer to G (196Hz) and D (294Hz). It's a handy illustration of the ratios being discussed, and such approximation of the frequencies (within 2%) is not unreasonable, but you can't pull note names out of thin air. You have to make an effort to sound like you know what you're talking about. 68.124.137.79 03:08, 12 May 2007 (UTC)
thread_title:  Notes and frequencies 

message: The article there is very detailed and contains many images and tables. Anyone here knows French and can translate the entire article to English? This will be very helpful! I'm willing to help with translating the text in the images (just tell me what to write). Barak Sh (talk) 03:32, 27 May 2008 (UTC)
thread_title:  French Wikipedia 

message: The table in the article says that in order to simplify the ratios involved, intervals are tuned down from unison as well as above. Is this the preferred method, or is only tuning upward also common? SharkD (talk) 02:37, 12 August 2008 (UTC)
replies: text: It does not make any difference to the resulting frequencies. Only the numbers shown are smaller. The ratio of two consecutive notes in the circle of fifths is always the same 3:2. It is just where you choose to start with 1. If you put one at the start, you get all factors 3 in the enumerator. If you put 1 in the middle they are in the denominator in one half of the circle. −Woodstone (talk) 07:47, 12 August 2008 (UTC)
replies: text: There is a difference. For instance, after re-modulating the interval to fit within the same octave according to the rules outlined in the article, the eighth fifth after unison has a ratio of 6561:4096 (815.64 cents, a minor sixth) when tuning upward. When tuning downward the ratio is 128:81 (792.18 cents). The difference between the two is the Pythagorean comma--the same as the difference between Eb and D# in the table within the article. The next fifth is 19683:16384 when tuning upward and 32:27 when tuning downward--also a difference equal to the Pythagorean comma. And so on. I haven't checked to see where and when wolf intervals occur. SharkD (talk) 09:11, 12 August 2008 (UTC)
replies: text: The only difference is created by the choice where the "break" in the circle is located. The pythagorean comma is why the "circle" is actually not a circle but a spiral. To bend it into a circle, a piece the length of the comma needs to be cut. This does not influence what happens to tuning up or down. If the circle starts at 1 going up, the last one is 312/219=531441/524288=~1.014. If you start in the middle with 1, the highest is 36/29=729/512=~1.424 and the lowest is 210/36=1024/729=~1.405. The difference between the extremes is still the same (729/512)/(1024/729)=(729*729)/(512*1024)=531441/524288 and verified by 1.424/1.405=~1.014. −Woodstone (talk) 12:36, 12 August 2008 (UTC)
replies: text: Still, the frequencies themselves are affected. For instance, the table and image in the article "break" at different points, leading to discrepancies. SharkD (talk) 16:25, 12 August 2008 (UTC)
thread_title: Tuning up or down?

message: Does this article need some serious looking at or am I missreading it?
Under the "Method" heading it states "The table below (starting at E flat) illustrates this". That has me confused. The table doesn't start on E flat.
It goes on to say "showing the note name, the ratio above D...." Why would it show the ratio above D if it started on E flat? The table shows the ratio above C, not D or E flat.
The article continues with "Pythagorean tuning uses the above 12 notes from E flat to D sharp shown above" thus compounding the confusion.  —Preceding unsigned comment added by 212.32.105.249 (talk) 01:57, 2 January 2009 (UTC)
replies: text: You are right. It arose because people have changed the table without adapting the surrounding text. I may have a look to straighten it out. −Woodstone (talk) 12:29, 2 January 2009 (UTC)

text: Had a go at it. Should be in better shape now. Clearly a Pythagorean tuning is key-specific. In a piece in C, there will usually be harmonies including both flats an sharps. The center of D is chosen for the table to show equally many flats and sharps. I have not been able to find literature on where it is usual to place the cut in the circle. −Woodstone (talk) 14:35, 2 January 2009 (UTC)
replies: text: That was a bit better but there were still a few notes wrong. You'd still got a few "Gs" in there that should have been "Ds". I've put it right (I think)  —Preceding unsigned comment added by 212.32.84.76 (talk) 15:49, 2 January 2009 (UTC)
thread_title: Horribly confusing

message: I am very much interested in this article, but I simply cannot understand it. It does not seem to explain itself in plain English, using a number of technical terms that I do not understand. The article begins with three confusingly-named music files that all sound the same (what does “diatonic scale in 12 et” mean?).
The article then begins to exaplin that Pythagorean tuning is “based on a stack of perfect fifths, each tuned in the ratio 3:2”. Sorry, what does this mean? I didn’t come to this article because I’m a professor of music and already know these terms, I came here to try and learn.
The article goes on like this using unfamiliar terms with no explanation of the basics, assuming the reader knows absolutely everything about music apart from the article in question.
Please can somebody re-write this whole article to make it make some sense. I am actually a musician and I know a lot more about music than most people, yet even I find this article baffling.
Grand Dizzy (talk) 20:41, 21 February 2009 (UTC)
thread_title: This article makes no sense

message: Below the tables it says, To get around this problem, Pythagorean tuning uses the above 12 notes from A flat to G sharp shown above [sic], and then places above the G sharp another A flat, starting the sequence again. Shouldn't this be "from A flat to C sharp"? I.e., instead of the last step in the table, return to the first line?  —Preceding unsigned comment added by 95.222.120.117 (talk) 21:30, 21 May 2009 (UTC)
thread_title:  Looping around 

message: We are discussing the sign of the Pythagorean diminished second in Talk:Pythagorean comma. If we apply the definition of diminished second given on Wikipedia, it should be negative (see also Pythagorean interval):
:
Name             	Short	Ratio        	Cents  	ETCents
diminished second	d2   	524288/531441	-23.460	0
Pythagorean comma	     	531441/524288	23.460 	0
Please share you opinion with us on Talk:Pythagorean comma. Do not answer here, because it would duplicate the discussion. This messagge will point the editors of this article to the other talk page.
− Paolo.dL (talk) 22:10, 15 August 2010 (UTC)
thread_title:  Sign of the Pythagorean diminished second 

message: In Talk:Pythagorean comma, we are discussing the sign of ε for Pythagorean tuning. ε is defined at the end of this section, as the positive difference between the size of the fifth and 700: 
Definition of ε in Pythagorean tuning
By definition, in Pythagorean tuning 11 perfect fifths (P5 in the table) have a size of approximately 701.955 cents (700+ε cents, where ε ≈ 1.955 cents).
Please share you opinion with us on Talk:Pythagorean comma. Do not answer here, because it would duplicate the discussion. This messagge will point the editors of this article to the other talk page.
− Paolo.dL (talk) 22:10, 15 August 2010 (UTC)
thread_title:  Sign of &epsilon; 

message: The Pythagorean scale is made possible by the mathematical coincidence, discovered by the Pythagoreans, that 3^{12} = 531, 441 is approximately equal to 2^{19} = 524,288. In fact, their ratio is 1.01364..., within 1.37 %. That the third part of the octave is close to the pure major third is due to another, independent coincidence, that 2^7 = 128 is approximately 5^3 = 125. In this case, 128/125 = 1.024, for an error of 2.4%.Manfred Schroeder. Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise, W.H. Freeman, 1991. Reprinted by Dover, 2009.
I removed this text. It might be interesting, but there's no explanation about the use, in Pythagorean tuning, of the mentioned numbers:
* 3^{12}
* 2^{19}
* 2^{7}
* 5^{3}
Also, this note does not deserve to be the first section of the article. Paolo.dL (talk) 20:08, 27 February 2012 (UTC)
thread_title: The Underlying Mathematics

message: The dedication to Pythagoras is very popular in modern handbooks (and probably since the "Latin middle ages"), but I changed the beginning, because there is no evidence (a general problem since Walter Burkert's Pythagoras study by the 1960s), that it was used to an earlier time than by Eratosthenes, if we would like believe Ptolemy and Boethius who copied Ptolemy's divisions and his ascriptions in his music treatise. Please check the quotation before note 1 and 2, because readers cannot be sure, who was really quoted here. I "ascribed" the quotation now to Leo Gunther, but quite frankly, what is said here, is so common, that it should be better said by someone's own words (what should a reader imagine to be a "major chord"?). Hence, I bet it was taken from the handbook of Benward and Saker. Platonykiss (talk) 14:04, 3 September 2013 (UTC)
thread_title:  Pythagoras and unnecessary, and uncorrectly formatted quotations 

message: I propose to remove the last two columns of the table found under Pythagorean tuning#Pythagorean intervals in order to make it simpler, and replace them with the Greek and Latin names of the pitch ratios. --kupirijo (talk) 03:14, 21 September 2019 (UTC)
replies: text: What is called Pythagorean tuning in current music theory is a stack of pure second and third harmonics (and their inverses). So any mention of using a fifth harmonic is out of place. Indeed the last two columns of the table may be better removed. However the long worded segment above with assumptions on what the real Pythagoras would have tuned like is out of place as well. −Woodstone (talk) 06:31, 8 May 2021 (UTC)
thread_title:  Table of intervals 

message: The first sentence is "Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2" at the time I'm making this comment. I am not DISPUTING this sentence. I say it is confusing. I don't say it's wrong. I don't KNOW. But if you tune, say, a piano, so that every interval that is a "fifth" between two notes is THE SAME (exactly 3/2), then what you're doing is making the octaves out of what to achieve fifths that are exact. For instance if I start with A being 440, and use the symbols "~" to mark notes in octaves higher than the A that will be approximately 880, I will tune as follows (in some cases approximately rounded):
A:440.0
E:660.0
B:990.0
F#:1,485.0
C#:2,227.5
G#:3,341.25
Eb:5,011.875
Bb:7,517.8125
F:11,276.71875
C:16,915.078125
G:25,372.6171875
D:38,058.92578125
A:57,088.388671875
And I don't say that's wrong, but that last "A" I tuned isn't the same as the "A" I'd have tuned if I'd tuned all the "A"s on the piano first and made each "A" exactly twice (or half) of some earlier "A". Then I'd be tuning an "A" to 56,320.0 (not 57,088.388671875). So, again, I'm not saying that's wrong. But isn't it worth saying in that first sentence that the choice to tune every note to make it EXACTLY 3/2 of the note a fifth below it causes the octaves to be wider than a 2:1 tuning? Again, I'm totally saying that for all I know the Pythagorean tuning means "fifths precisely at 3:2, octaves wider than 2:1", but I would make it clear in the first sentence that the octaves are being sacrificed so that the fifths can be precise. But then later on the article has this sentence (right now, as I'm typing this) "The Pythagorean scale is any scale which can be constructed from only pure perfect fifths (3:2) and octaves (2:1)". But isn't that impossible? Isn't it impossible to be "loyal to" BOTH of the ratios 2:1 and 3:2? For so it would seem from my attempt at following a 3:2 tuning as listed above. It seems to me that to tune things so that every possible pair of levers on a keyboard such that their strings ring in a 3:2 ratio would require that a semitone interval be set as "the seventh root of 3/2" or to say it another say "(3/2) to the (1/7) power" and that you just can't do that if you have any pair of levers space 12 apart tuned in the ratio 2:1. A later paragraph seems to me to show an Ab and a G# that are two different notes. Okay, there was such a thing as splitting levers on keyboards once. You could split the ebony between G and A into a G# and an Ab. But would it then still be true that EVERY LEVER on the keyboard still has some other lever that rings in a ratio of 2:3 or 3:2 with that first lever? I'm just not getting how this works. Please explain.108.234.62.240 (talk) 05:02, 9 November 2023 (UTC)
thread_title: You lost me at the first sentence