Talk:Five-limit tuning

Flat or sharp
The A# in the lower hand corner is mistaken (see more explanation at talk:just intonation). A#–D is not a third (5th harmonic) and Eb–A# is not a fifth (3rd harmonic). Rows should always read as a piece of the circle of fifths. Bb–D and Eb–Bb are correct. Just like the D's, these are not enharmonics, but just two tuning options. &minus;Woodstone (talk) 15:28, 6 July 2010 (UTC)

I replaced those lower-right A#'s with Bb's, and added the original "gray column" to my extension. Thanks for the heads-up!

However, I would suggest removing the "base ratios" as being too theoretical. I removed them in the extension because it looked better that way before the newly-edited tables above the extension. &minus; Glenn L (talk) 16:01, 6 July 2010 (UTC)


 * I agree to remove the base ratios. It looks much cleaner. Notes are universally considered to repeat per just octave (except of course for the Railsback curve on pianos). &minus;Woodstone (talk) 16:28, 6 July 2010 (UTC).


 * Ok, if you like, I'll delete the base ratio. --Paolo.dL (talk) 20:16, 6 July 2010 (UTC)


 * Done, thank you for the advice. I improved the text to make the computation method clearer (this involves two steps, one of which is computing the base ratios that I deleted from the table, so the information removed from the table must be given in the text). --Paolo.dL (talk) 09:03, 7 July 2010 (UTC)

Start with full enharmonics
I would propose to add rows in the first display to start with a full set of all flats and sharps. That way we show that the selection is a mask overlaid on the pattern to fit the key. Later copies can then be reduced to C only. We can also note that the choice if the much discussed Bb makes the difference for suitability for C major or A minor. &minus;Woodstone (talk) 16:38, 6 July 2010 (UTC)


 * I believe we should limit the complexity of this section, because it is already long enough and even if for you it's simple to understand, for ohers may not. For instance, I have not understood completely your picture yet. Is it D based? In my opinion, your suggestion can be implemented in the next section (Extension of the scale), where the larger table is given. --Paolo.dL (talk) 20:32, 6 July 2010 (UTC)

Intervals throughout the scale
The table that you asked me to transpose (in Talk:Just intonation) has the same direction as dozens of other tables in which the names of the intervals are found as row labels. I can't understand the reason why you prefer the transposition. The comparison table I edited in interval (music) has the same orientation.

As for the ratios, it would be very difficult for me to compute and visualize the ratios for each interval, and even more importantly, I don't know how to show fractions with excel. However, you should not use my table to read numbers or fractions. I really want the color code and the font enphasis (bold) to be more important than the numbers.

Have you noticed that all the just ratios are marked in bold? and that just ratios (different from those starting from C) come out even in the less often occurring intervals? Bold font shows this. --Paolo.dL (talk) 21:57, 6 July 2010 (UTC)


 * In most interval tables, the interval names are row headers and the values down a column. In your table the columns are for starting points, whereas the intervals are across each row. Don't you think that the flipped form below gives clearer sight on which intervals occur? I did this mostly in excel by using cell formatting as fractions. However it only works out well for fractions under 1 (an the P5 at F# is probably wrong, because it needs more than 3 digits). So I inverted them all, then made a copy of the "values", and manipulated them back. Colors would still be a welcome addition.


 * {| class="wikitable" style="text-align: center"

! from || m2 || M2 || m3 || M3 || P4 || TT || P5 || m6 || M6 || m7 || M7
 * C ||16/15 || 9/8 || 6/5 || 5/4 || 4/3 || 45/32 || 3/2 || 8/5 || 5/3 || 9/5 || 15/8
 * Db ||135/128 || 9/8 || 75/64 || 5/4 || 675/512 || 45/32 || 3/2 || 25/16 || 27/16 || 225/128 || 15/8
 * D ||16/15 || 10/9 || 32/27 || 5/4 || 4/3 || 64/45 || 40/27 || 8/5 || 5/3 || 16/9 || 256/135
 * Eb ||25/24 || 10/9 || 75/64 || 5/4 || 4/3 || 25/18 || 3/2 || 25/16 || 5/3 || 16/9 || 15/8
 * E ||16/15 || 9/8 || 6/5 || 32/25 || 4/3 || 36/25 || 3/2 || 8/5 || 128/75 || 9/5 || 48/25
 * F ||135/128 || 9/8 || 6/5 || 5/4 || 27/20 || 45/32 || 3/2 || 8/5 || 27/16 || 9/5 || 15/8
 * F# ||16/15 || 256/225 || 32/27 || 32/25 || 4/3 || 64/45 || 757/499 || 8/5 || 128/75 || 16/9 || 256/135
 * G ||16/15 || 10/9 || 6/5 || 5/4 || 4/3 || 64/45 || 3/2 || 8/5 || 5/3 || 16/9 || 15/8
 * Ab ||25/24 || 9/8 || 75/64 || 5/4 || 4/3 || 45/32 || 3/2 || 25/16 || 5/3 || 225/128 || 15/8
 * A ||27/25 || 9/8 || 6/5 || 32/25 || 27/20 || 36/25 || 3/2 || 8/5 || 27/16 || 9/5 || 48/25
 * Bb ||25/24 || 10/9 || 32/27 || 5/4 || 4/3 || 25/18 || 40/27 || 25/16 || 5/3 || 16/9 || 50/27
 * B ||16/15 || 256/225 || 6/5 || 32/25 || 4/3 || 64/45 || 3/2 || 8/5 || 128/75 || 16/9 || 48/25
 * }
 * &minus;Woodstone (talk) 06:14, 7 July 2010 (UTC)
 * G ||16/15 || 10/9 || 6/5 || 5/4 || 4/3 || 64/45 || 3/2 || 8/5 || 5/3 || 16/9 || 15/8
 * Ab ||25/24 || 9/8 || 75/64 || 5/4 || 4/3 || 45/32 || 3/2 || 25/16 || 5/3 || 225/128 || 15/8
 * A ||27/25 || 9/8 || 6/5 || 32/25 || 27/20 || 36/25 || 3/2 || 8/5 || 27/16 || 9/5 || 48/25
 * Bb ||25/24 || 10/9 || 32/27 || 5/4 || 4/3 || 25/18 || 40/27 || 25/16 || 5/3 || 16/9 || 50/27
 * B ||16/15 || 256/225 || 6/5 || 32/25 || 4/3 || 64/45 || 3/2 || 8/5 || 128/75 || 16/9 || 48/25
 * }
 * &minus;Woodstone (talk) 06:14, 7 July 2010 (UTC)
 * Bb ||25/24 || 10/9 || 32/27 || 5/4 || 4/3 || 25/18 || 40/27 || 25/16 || 5/3 || 16/9 || 50/27
 * B ||16/15 || 256/225 || 6/5 || 32/25 || 4/3 || 64/45 || 3/2 || 8/5 || 128/75 || 16/9 || 48/25
 * }
 * &minus;Woodstone (talk) 06:14, 7 July 2010 (UTC)
 * &minus;Woodstone (talk) 06:14, 7 July 2010 (UTC)

Did you copy the ratios manually in each cell? My table is produced automatically from the 12 basic ratios (from C). For instance, to produce the asymmetric table which I just published in Five-limit tuning, I only had to change one value (9/5 instead of 16/9) in a single cell. How did you convert from Excel into Wikitable?

Colors in my table are not just a welcome addition. They are the code which conveys the most important information. Numbers or fractions, on the contrary, are an addition (and I am not sure they are welcome), conveying distracting information. My table would be very informative even without numbers, with a J representing "just intervals" (rather than bold font), and an empty cell for non-just intervals. Wouldn't that be nice? The black cells indicate very large deviations from J (larger than 22 cents). Isn't that all what we need?

The numbers can show the deviation from the "basic" intervals used to build the scale (those from C). That's why cents is better than ratios (remember that bold font shows which intervals are just). I might use deviations (difference from the corrisponding "basic" interval), rather than ratios. That would be probably more informative. But even if they might be useful, they are additional information, and I am not sure they are welcome. --Paolo.dL (talk) 09:45, 7 July 2010 (UTC)


 * No, I did not calculate or copy any fractions manually. I created the fractions in Excel and then completed the wikitable in Word with a few global substitutes. In the tuning discussed here all intervals are just. Only some are more just than others. You fall into the trap of thinking that deviation from equal tuning is somehow bad. Furthermore, since this is a 5-limit tuning, all prime factors of the enumerator and denominator are at most 5 (for all occurring intervals). So what is left to judge consonance is the size of the enumerator and the denominator. When these are small, the sound is more consonant then when they are large. That can be immediately seen from the fractions: much better than from the cents. For example it becomes directly visible where the m7 is 16/9 and or 9/5. Also the "wolf" intevals like 256/225 jump out. Yes color coding still helps and makes it nicer, but the information already visible. &minus;Woodstone (talk) 11:06, 7 July 2010 (UTC)

I am absolutely not falling into that trap. I never mentioned equal intervals as a reference. The reference intervals are those published in the comparison table I carefully edited some weeks ago, as you know. And please see also the reference ratios that I published today right in this page. You, however, clearly (but also momentarily, I am sure) fall into another trap, when you write "In the tuning discussed here all intervals are just". That's absolutely false, as you can see in my table (and as you already and certainly know), the main purpose of which, BTW, is showing that they are not all just! Again, you can see it in my table by looking at the emphasis (those which are not bold are not just), not at the numbers! Q.E.D.

--Paolo.dL (talk) 13:03, 7 July 2010 (UTC)


 * Hi Paolo, it's probably a matter of definition. My definition of just is that the notes making up the interval are both harmonics of a common note. In other words, there is a harmonic series that contains them both. That is clearly true for all pairs of notes here, since their ratio can be expressed as a fraction of two integers. It is clearly not true (apart from octavation) for any pair of ET notes. My statement is still that justness (wolfness) of an interval is not determined by the deviation from ET, but by the hight of the location in thier common harmonic series. &minus;Woodstone (talk) 15:05, 7 July 2010 (UTC)

Just or juster than
Yes, it is a matter of definition. Thank you for explaining. According to the definition in Just intonation just refers to ratios of small numbers (not ratios of powers of small mumbers). Do you suggest to change that definition? According to your definition, the only non-just intervals would be those produced by radicals (irrational numbers), such as those in TET or quarter-comma. Can we safely state, for instance, that 45/32 is considered by everybody strictly just? And in this comparison table, according to your definition all the intervals shown for Pythagorean tuning would be just, because they are produced with powers of 3 and 2, wouldn't they? I agree that they are "juster" than a wolf fifth, but not that they are strictly just. --Paolo.dL (talk) 16:42, 7 July 2010 (UTC)


 * As the definition talks about "small" numbers, there is no "strict" justness. It's all a matter of degree. What is a "small" number? I would prefer to call the intervals with higher numbers in the ratio less just. Just tuning makes all intervals "just" to a certain extent. Pythagorean goes indeed to very high numbers. I have used to word "pure" interval in the context of chords to avoid this definition problem. Pure would then stand for the lowest used ratio numbers only. A fifth of 3/2 would be "pure", all others would not be. &minus;Woodstone (talk) 16:56, 7 July 2010 (UTC)

Would you say that a man is short if the definition of short is "having a small length"? Of course, a basketball player can be shorter than another, but not short. So, 3/2, 5/4, and similar ratios are "strictly just", not only juster than others. There's no way to decide univocally who is a short man and who is not, it is difficult to decide exactly the threshold, but this does not mean that it is impossible or not allowed. I would say that these ratios, being the "justest", must be also strictly just, as well as a typical NBA player is strictly tall. Thus, the others are not. (the strict justness of 45/32 might be controversial.) This based only on a (very reasonable) opinion, not on a convention, but this does not mean that the concept of strictly just is not implied in the definition! (there's no convention to decide whether a man is strictly short or not, but everybody is entitled to reasonably judge the difference between short and tall, and there's some agreement among experts).

By the way, I like the word "pure" in this context, and I used it sometimes as well. But "strictly just" is ok as well, to express exactly the same concept. --Paolo.dL (talk) 17:27, 7 July 2010 (UTC)


 * PMFJI, but while the comparative and superlative words for just can be expressed as juster and justest, wouldn't more just and most just look and sound preferable to you, as it does to me? If you think I'm off the wall, let me know and I'll never discuss this again. &minus; Glenn L (talk) 19:48, 7 July 2010 (UTC)


 * I am not sure, I am not english mother tongue. --Paolo.dL (talk) 21:38, 7 July 2010 (UTC)


 * I don't think it's correct to say one just interval is more just than another just interval. The set of just intervals is determined by how you define "just", and each interval would either be in the set or not in the set, 100% or 0%, depending on the definition. Within the set, there's no further comparison of justness. You might be able to say one just interval is less complex than another when compared in a certain way (such as by multiplying the numerator by the denominator within each interval) or suggest that one seems more consonant than another, but those kind of comparisons probably belong in a different article such as Tuning systems or Consonance and dissonance. 108.60.216.202 (talk) 02:39, 7 June 2015 (UTC)

Rearranged occurence table
Interval table sorted by the circle of fifths in the same manner as for the Quarter Comma Meantone. The choice was made to take three rows of 4 notes, C-G centered. Bringing out the construction pattern by heavier lines. Note that the blue "corners" are moved from their rightful places left and right next to the table to fit into the missing portions. &minus;Woodstone (talk) 03:48, 18 July 2010 (UTC)

D-based construction table
I am copying here a comment posted elsewhere by Woodstone supporting a D-A centered stack of fifths in 5-limit tuning. I agree fullhearthedly. Am I authorized to edit the article?

&minus; Paolo.dL (talk) 13:15, 27 July 2010 (UTC)

[...omissis...] "If you have a piece in C major, the notes you will use mostly are the white keys (F C G D A E B). So you would like the intervals between those to be as just as possible. That implies a non-symmetric choice for the mapping. So actually the symmetric choice around D (surprise!) might be best (not the symmetric one around C!). If you want to play in A minor, you still want the white keys, but preferably also the F# and G#, needed for the melodic scale (F C G D A E B F# C# G#), so for that key you would go for a symmetric choice around or A.or E Since C major and A minor are related keys, there is good chance that you would need both in a piece. For the combination a choice with (D A) in the (left/right) center seems justified. These remarks are actually more valid for 5-limit than for QCM, but nevertheless give some insight here as well." [...omissis...]

Posted on Talk:Quarter-comma meantone by Woodstone (talk) 19:20, 15 July 2010 (UTC)


 * Quasi-wolf fifth. Notice, also, that the D-based 5-limit has a "quasi-wolf" fifth in the same familiar position as the wolf fifth in Pythagorean tuning and Quarter-comma meantone (G♯-E♭). This "quasi-wolf" is, in my opinion, not large enough to be called a true wolf, yet it is audibly dissonant, as it deviates from just intonation almost as much as the wolf fifth in Pythagorean tuning. Namely, it is sharper than the just size of 702 cents by about 19.5 cents (almost one syntonic comma), while the wolf fifth in Pythagorean tuning is flatter by about 23.5 cents (a little more than one synthonic comma). Before equal temperament tuning, musicians were used to avoid playing this interval.


 * If we use the C-based 5-limit, however, this quasi-wolf fifth occurs in a different and possibly less familiar position (i.e. F♯-D♭ rather than the more familiar G♯-E♭). This is one of the reasons why we used a D-based construction table in Quarter comma meantone.


 * &minus; Paolo.dL (talk) 15:02, 27 July 2010 (UTC)

Unexpected conclusion: C-based is preferable
Here is a quite interesting reference with a very good point against the D-based construction table in 5-limit tuning: newmusicbox. The D-based table creates a wolf fifth in C-G (ratio 40/27 = about 680 cents), making it impossible to play a C-major chord! You can see it in Woodstone's table right above. So, I guess the C-based construction table currently shown in this article is the best choice and will remain as it is.

By the way, notice that the ratio 40/27 for wolf fifths such as C-G (in D-based 5-limit) or D-A (in C-based 5-limit) deviates from the corresponding pure ratio (3/2) by exacly one syntonic comma (i.e. by 81/80, or about 21.5 cents). This is why it is not automatically marked as a wolf in my tables in which colors and font emphasis are automatically produced using Microsoft Excel's conditional formatting, and where wolf intervals are operationally defined as intervals made up of 3, 4, 5, 7, 8, or 9 semitones (i.e. thirds, sixths, fifths, fourths and enharmonic equivalents) which deviate from their just intonation by a factor greater than the syntonic comma (otherwise too many wolf intervals would appear in my table; namely, two P5's, two P4's, six m3's, six M6's, four M3's, and four m6's, i.e a total of 24 wolf intervals).

&minus; Paolo.dL (talk) 19:59, 28 July 2010 (UTC)


 * It's very enlightening, and a great improvement to the coloring scheme that in the "sweet zone" (minor, major, perfect) only the single comma 81/80 appears to occur, and only the two commas 128/125 and 2048/2025 in the augmented and diminished region. No arbitrary limits anymore. In the text you might want to change the words "most observed" into "reference" (as in the picture). &minus;Woodstone (talk) 06:23, 29 July 2010 (UTC)

Alternative C-based asymmetric construction tables
Good catch about the centering. And it was staring us in the face all the time! Of course if you play in C, you need the famous 3 chords F, C, G, which obviously need to be pure, so adding the fifth of G this forces a choice of rows F-C-G-D and up a third A-E-B. This block is the core of just tuning. Question remains how to allocate the accidentals. For the related melodic A minor, one would like to have F# and G#, so after including the intervening C#, we get:

Two more notes to choose. Only 3 ways: either Eb-Bb or Bb-D# or D#-A#. All even less centered. Let's think about it. &minus;Woodstone (talk) 04:10, 29 July 2010 (UTC)

Other possibilities still exist. One may go for a pentatonic core:

All these examples include diminished or augmented intervals, whereas the current choice has only minor and major ones. Lots of food for thought. &minus;Woodstone (talk) 05:18, 29 July 2010 (UTC)


 * I finished my job here, and it took too much of my time already! :-) Thank you for your great suggestions. If you want to add a section about alternative construction tables, feel free to do it, but I won't help you. By the way, we still have an open discussion about your diagrams. Paolo.dL (talk) 11:56, 29 July 2010 (UTC)

Dreadful!
This article is simply dreadful. Very obviously "personal essay" throughout. Wouldn't it be better to simply publish a paper (or book)? In the spirit of inclusionism, though, before nominating the article for deletion, I'll contemplate trying to strip it down to something appropriate to Wikipedia.Frank Zamjatin (talk) 21:20, 27 June 2011 (UTC)


 * I might agree that the style of this article perhaps not fully encyclopedic, but the content is quite enlightening and correct. Deletion is a rather heavy-handed approach. I see no reason for deletion. If you seriously propose delion you should refer to the applicable formal grounds of WP, not some self invented description. You might have a go at improvement yourself. I will remove the notice. &minus;Woodstone (talk) 12:07, 28 June 2011 (UTC)


 * I agree with Woodstone. Moreover, generic judgements (such as "dreadful" or "personal essay") about the whole article are of little use. Examples or comments on specific sentences or paragraphs can be discussed more effectively. Consider that most parts of the article have been already discussed in detail on Talk:Just intonation or on this page. Paolo.dL (talk) 13:47, 28 June 2011 (UTC)

Sorry you guys but Wikipedia is not the place to publish original research. Let's take a look: "Five-limit tuning, or 5-limit tuning is one of the most commonly used methods to obtain a scale in which most of the intervals between notes are justly tuned." Reference? This isn't "common knowledge", "QED"... or found in any music dictionary.

"The notes of the diatonic or chromatic scale are obtained by multiplying a given initial note (called base note) by powers of 2, 3, or 5, or a combination of them." Reference? Hopefully there isn't a printed reference for this, as it is bizarre. How do multiply a "note" by anything? Do you mean to say that freqencies are multiplied by frequency ratios?

And, do you not realize that "diatonic or chromatic scale" is here an alarm for anyone who actually knows what they're talking about? A "5-limit" tuning may have little to do with diatonic or chromatic scales. What of "5-limit" diamonds, Fokker blocks, ancient Greek tetrachords?Frank Zamjatin (talk) 20:08, 29 June 2011 (UTC)


 * You are right, but how does that justify the deletion of this article? See if you like my recent edits. Please, add a new section about 5-limit diamonds, Fokker blocks and ancient Greek tetrachords. Paolo.dL (talk) 09:12, 30 June 2011 (UTC)

Very nice- the first section now has a neutral, general stance that can be referenced and is appropriate to Wikipedia, thanks! Now it's possible to go through the article, put in references, etc. Frank Zamjatin (talk) 10:25, 30 June 2011 (UTC)


 * I didn't see a neutral, general stance in the opening, but this is 2015, so maybe it was added and then edited out again. Anyway, I rewrote the intro 5 minutes ago to be more inclusive. Zamjatin, if you ever read this, Paolo's request for you to add the 5-limit diamonds, Fokker blocks and ancient Greek tetrachords you mention looks like it's still open. I don't see those things mentioned yet, but my intro sets it up for you now, even mentioning Partch's odd-limit. It's always better to edit than to complain, folks. 38.86.48.38 (talk) 22:11, 6 June 2015 (UTC)

Objection to edits
Hi Woodstone. I object to some of the modifications you recently made. Here's why:

1) You moved the word powers, which is OK, but you also stripped out the link to the Exponentiation article, which an early editor put in, which adds info for the reader who needs to learn more about what powers are, which I preserved, and which you should also preserve. You lost information.

2) Harry Partch did not name his limit "odd limit". He just called it "limit". That's why I specifically said "meaning odd limit", not "named odd limit". Also, you removed the fact that it does not mean prime limit, which is an important distinction that should be explicitly stated for contrast against odd limit. Also, when I started the paragraph with "Note:", that set the odd-limit definition apart from the surrounding text. Your removing that made it seem like part of the surrounding text, which could confuse readers unfamiliar with the topic into thinking that some of the surrounding text refers to odd limit, which none of it does. IMO, your revision resulted in less information for the reader, not more.

3) You destroyed my step by step explanation of how we get 5/4 using factors of the three primes in 5-limit for people who don't already know how to do that and which explained the math formulas that immediately followed it for people who are familiar with music but not math. Loss of information.

4) You added 81/64 as a major third, which only confuses the idea. Remember, this is the 5-limit article. 81/64 is the 3-limit major third. The whole reason to go to 5-limit is to be able to use 5/4, and the lead explanations make the point that 5-limit affords you the basic building blocks of 2/1, 3/2, and 5/4. Yes you can get 81/64 in 5 limit, but the whole point of 5-limit is to be able to get 5/4 and its powers, such as 25/16, so you don't have to use 81/64, 6561/4096, etc.

5) You changed the word "intervalic" to "intervallic". Both spellings are valid. http://dictionary.reference.com/browse/intervallic I believe the Wikipedia general convention for multiple valid spellings is to keep the spelling that is entered by the person who first typed it, otherwise the corrections would ping pong forever. For example, color and colour. What if every editor in the USA changed it to "color" every time they saw it, and every editor in the UK changed it to "colour" every time they saw it? The edits would never cease. 108.60.216.202 (talk) 18:46, 9 June 2015 (UTC)


 * Response to obejctions above.
 * 1) That was a case of overlinking. Normal words need not be linked. Anyone who wants to find out what it means can search for it.
 * 2) The focus was on the person proposing it instead of the term. I reworded to state the facts first, with an attribution trailing. If some clarity was lost feel free to rephrase. Using the word "Note" this way is discouraged by WP:MOS (see Editorializing).
 * 3) The explanation was much too long for the introduction section. Just facts are enough.
 * 4) Any 3-limit interval is also a 5-limit interval. Some attention for the non-uniqueness of the choice for tuning is an essential element in understanding limit-tuning. I just picked one example.
 * 5) This rule only applies to equally common spellings. In this case double l is dozens of times more frequent (see google ngram)
 * &minus;Woodstone (talk) 08:20, 10 June 2015 (UTC)

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Actual Music
Maybe the moderators of this article would like to refer to www.ji5.nl for actual sound files demonstrating the alledged suitability of five limit tuning for organ music. Music by J.S. Bach, L. Boellmann, M. Reger and others. Kind regards, Erik Zuurbier, 22 October 2018


 * Wikipedia does not have moderators. You can add this resource yourself. &minus;Woodstone (talk) 05:06, 27 October 2018 (UTC)


 * Just tuning is to a large extent used when playing violin. There are several reasons for this.
 * When tuning a violin (in perfect fifths) you use the absence of beats where "the harmonies meet" between the second harmony of the high pitch string, with the third harmony of the lower string, as a very accurate measure for getting the four strings in tune. Once the violin is accurately tuned, the resonance ringing of the looses strings guides the violinist where to put his fingers. This resonance ringing only appears on just intervals (where the harmony meets). The lower the prime number, the stronger the effekt. For higher ratios the ringing gets outside the audible range. Oleviolin (talk) 14:55, 20 May 2024 (UTC)

Reference to / conflict with 7-limit tuning
There is a fine article at https://en.wikipedia.org/wiki/7-limit_tuning

which is worth linking here.

This page also features a 7-limit table, but confusingly, it uses different fractions than the above mentioned page. I don't know which one is (more) correct, or what the difference means.

217.104.69.33 (talk) 23:49, 9 February 2019 (UTC)commonpike