Talk:Just intonation

Resonate?

The article begins saying "Just intonation is the tuning of a musical interval without beats. The result is an acoustically pure sound that resonates within the harmonic series." I have at least two problems with this statement: can a "musical interval" result in a "pure sound," and what is the meaning of "resonate" in the second sentence? Any explanation would be welcome. — Hucbald.SaintAmand (talk) 12:47, 23 April 2026 (UTC)

"Just" and "pure" are historically interchangeable descriptors for the sounding of an interval without acoustical beats. Those words reflect the sources.

The resonance aligns with the harmonic series. When it doesn't, you get the beats. I thought the Wikilink to the acoustical beat article sufficed to explain that, but I'll add a citation. Wendy Carlos goes into great detail about it in her article. Trumpetrep (talk) 14:32, 25 April 2026 (UTC)

I keep wondering how a musical interval could result in a pure sound. My problem is not so much about "pure" than about "sound." Doesn't an interval necessarily imply two sounds?

Just (or pure) intervals may beat, if the difference in frequency is less than about 30 Hz. That is to say that, e.g., a just major 3d would beat if the lower sound is below 120Hz. The definition given in our article refers to the New Harvard Dictionary, but Apel's definition in the Harvard Dictionary (1950, p. 384) refers only to simple frequency ratios. James Jeans, in Science of music (1937, p. 176), also defined just intonation in terms of frequency ratios. And one could easily find other sources which do not refer to the absence of beats.

As to sounds that "resonate" within the harmonic series, this reminds me of Rameau's idea that harmonic partials results from a résonance, misunderstanding Sauveur's experiment which evidenced harmonic partials with resonators. I am afraid that to say that "the resonance aligns with the harmonic series" sort of reproduces the same misunderstanding and I don't see any explanation of this "resonance" in the Beat (acoustics) article.

The harmonic series itself is defined as a series of integral multiples of a fundamental frequency. Do not these "integral multiples" bring back to the definition of just intonation as resulting from simple frequency ratios. — Hucbald.SaintAmand (talk) 17:05, 26 April 2026 (UTC)

The Harvard Dictionary definition which is cited in the article reads, "The beatless tuning of an interval, one that brings it into agreement with some analogous interval in the harmonic series. Such intervals are considered to be acoustically pure."

Perhaps this is a language issue. The interval itself has a sound. Yes, there are two notes, each with a sound of their own. But the pure sound of a justly tuned interval is the sound of the combination of those tones, just as a chord has a sound, a band has a sound, and an instrument has a sound. It's a malleable term.

I'll take a look at the beats article. Perhaps it needs to be clearer? The same combination principle applies here. Note A has its own set of overtones. Note B has a separate set. When they are sounded together as an interval, if their overtones resonate together (or agree per Harvard), the interval will sound pure or just. If they don't, they will beat.

I hope this is a clear answer to your concerns. Trumpetrep (talk) 19:43, 26 April 2026 (UTC)

@Trumpetrep,

1) I thought of the difference between "sound" and "note" (or, better, "tone" – a note, strictly speaking is a written tone or pitch). You are right on this point, but I feel nevertheless that it might lead to confusion.

2) The matter of beats is more important. The question may be whether "Just intonation" defines the tuning of possibly one single interval, or the attempt at tuning several just intervals in a scale. The New Harvard Dictionary first definition considers that all intervals between adjacent partials in the harmonic series form a "just intonation", but that includes intervals of a semitone or less, which indeed may be beatless, but which might not be "just" in the usual sense, while intervals that belong to just intonation, such as the major third of 5:4 frequency ratio, in some cases may beat. In other words, the meaning of "just intonation" may have changed between, say, Apel's Harvard Dictionary of 1969 (2d edition) and the New Harvard Dictionary of 1986 – or, to say it otherwise, definitons 1) and 2) in the New Harvard Dictionary may contradict each other. — Hucbald.SaintAmand (talk) 13:03, 27 April 2026 (UTC)

I should add that intervals that are "just" in the first definition of the New Harvard Dictionary may beat. As a matter of fact, any harmonic series of which the fundamental is lower than about 30 Hz has beating intervals between its own partials. Or we must conclude that Helmhotz was wrong... — Hucbald.SaintAmand (talk) 13:08, 27 April 2026 (UTC)

There are two definitions for just intonation. We discussed this during the overhaul. One of the first things I changed was to introduce to the article the primary definition of the term, which is reflected in the Harvard citation above. That history was entirely absent from the article.

The article now conveys that the pursuit of an entire tuning system that replicates just intonation of certain intervals has been a Holy Grail for some musicians. What comes first is the natural phenomenon: just intonation of intervals.

The attempt to replicate that natural phenomenon across an entire chromatic tuning system is how you end up in 7-limit land. It's like vulcanized rubber: the attempt to industrialize a miracle of nature.

Perhaps the article could use a few more words about the impulse to systematize just intonation? My focus was on simplicity, given the myriad issues with the article's sprawl. I think the text is clear, and neither the Harvard or the Grove entries for just intonation are particularly long. When an article starts trying to be all things to all people, it ends up being the kind of mess we just had to clean up. Trumpetrep (talk) 19:53, 27 April 2026 (UTC)

An additional problem is that Just intonation is described in our article as a "tuning," as it also is in the New Harvard Dictionary. One may sing in just intonation, without "tuning" anything. Apel described it as "A system of intonation and tuning," which makes more sense. That "just intonation" merely is "a system of intonation" seems ... obvious. — Hucbald.SaintAmand (talk) 17:15, 30 April 2026 (UTC)

I'm having trouble following you. The article gives two separate definitions for just intonation. It is the tuning of a musical interval. It is also a tuning system.

Those two definitions do not entirely overlap.

That is why it's important for the article to articulate both of them, which it now does.

It's hard to know if I'm addressing your concern, because it's unclear to me what they are. Trumpetrep (talk) 20:15, 30 April 2026 (UTC)

I see that the article Musical tuning also is unclear on this point. It says that "Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones." I agree with that: tuning is adjusting the pitch or the scale of an instrument. If you sing, say, in equal temperament, you don't need to "tune" your voice, but you attempt at matching the tuning of some instrument. Indeed, "tuning" is also used (once) in the article in the meaning of "matching" when it says "Tuning to a pitch with one's voice is called matching pitch."

This all to say that I feel that these articles are not strict enough in the meaning of the terms used. I admit that the common English usage may not be very strict, but I think that we should avoid that on WP. I don't think that "just intonation" ever is termed "just tuning," which indeed would mean something entirely different. Apel's description of just intonation as "a system of intonation and tuning" (I'd have written "intonation or tuning") is more cautious than the New Harvard Dictionary when it describes just intonation as either "the beatless tuning of an interval" or "any tuning that incorporates [...] acoustically pure types of intervals." In either cases, this concerns tuning instruments, not singing. — Hucbald.SaintAmand (talk) 10:03, 1 May 2026 (UTC)

Oof, that musical tuning article is a mess as well. The quote you pulled is the kind of unsourced riff that is all over our tuning articles. They read like someone ad-libbing a pre-concert lecture instead of writing a Wikipedia article.

I agree completely that our articles should be more precise. I do think there is a language gap at play here, though.

When you said, "One may sing in just intonation, without "tuning" anything", I didn't understand what you meant. Based on your latest post, it seems that your definition of tuning is limited to some kind of mechanical adjustment that the human voice is unable to make, such as adjusting a slide or a tuning peg.

Singers tune just like everyone else, though. Their mechanism is their voice. A classic example is Stimmung which can be translated as 'tuning'. It requires constant vigilance of pitch, and any rehearsal of it will include ample stretches where the singers focus on tuning their intervals.

I think I've understood you now, and I hope you'll clarify if I haven't. More importantly, I'm not sure how to address any concerns you have about the article. The language in it is a clear reflection of its trustworthy sources. If there are still points of confusion in its copy, let's focus on them. Trumpetrep (talk) 12:20, 1 May 2026 (UTC)

2026 Overhaul

User:lowercase sigmabot III leapfrogged the rather expansive discussion of this article's much-needed overhaul into the Talk archive. Please consult that discussion to see how this article's manifold problems were addressed. Trumpetrep (talk) 14:44, 25 April 2026 (UTC)

Re your edit that removed the information that Pythagorean tuning is a 3-limit, and the Ptolemy's is a 5-limit:

Why are you insisting that concepts introduced in this sections aren't classified by the limit class they belong to?

You claim the overhaul discussion contains some rationale on this referred to as "thus", but I can't see a real reason why removing a top level information (which is also prominent as a "Musical tunings" template's hierarchy) is an improvement. Zlamma (talk) 23:38, 25 April 2026 (UTC)

Additionally, re this edit that moved the limit classifications back to the end. I find your change detrimental. You say history is somehow important in this section, but history section is just above. This section is titled "scales" and IMHO should focus on laying out the taxonomy. The concept of limits give a very good high level classification which helps to digest what is to come. Whereas the current picks from history, with the current removal of information of limit classes, gives no structure to the section and makes it prone to further creep (this part of 5-limit article shows how there can be plenty variants of this just intonation class). Zlamma (talk) 00:27, 26 April 2026 (UTC)

Hi Zlamma, thanks for reading through the Talk page.

As to the structure of the Scales section, it is chronological like the rest of the article.

No article benefits by writing history inaccurately. The previous incarnation of this page had anachronistic statements like this, "During the second century AD, Claudius Ptolemy described a 5-limit diatonic scale in his influential text on music theory..."

Harry Partch's concept of a limit comes after thousands of years of history. Using it as a structural concept for a page about just intonation is like using the 426 Hemi as a frame for the history of transportation.

The Grove article on just intonation doesn't mention limits. A recurring feature of the improvement tags and the Talk page of this article was that readers didn't understand it. A big reason general readers and professional musicologists found it unintelligible was its myopic preference for limits instead of first principles.

The easiest thing to do when I started overhauling the article was to remove forked content. Let the limit articles speak for themselves. As to that, the Five-limit tuning is the same kind of shambles that this article was prior to the overhaul. It would be fantastic if you could find some quality sources for it and get some of that original research out of there!

Regarding the first edit you asked about, it removed two unsourced statements. In addition to not having citations, they blurred the chronological progress of the article. It moves gradually towards limits and the current state of the art. Trumpetrep (talk) 01:55, 26 April 2026 (UTC)

> As to the structure of the Scales section, it is chronological like the rest of the article. ... myopic preference for limits instead of first principles

While our recent edits found a way not to pour the baby with the bath, I will leave my thoughts here that find the preference for chronology in every section - even where history section exists - subjective, whereas, contrary to your tacit presupposition, understandability for modern reader is a first principle, arguably a very important one. Chronology may actually introduce chaos especially when expanding to cover cultures which didn't build on each other (possibly why pentatonic scale uses are not really chronological). Laying out modern understanding of a subject beforehand can help systematize a concept better than sticking strictly to the timeline (see, again, pentatonic scale's types, whose structure relies on modern classifications).

Regarding few other claims:

> No article benefits by writing history inaccurately.

Surely deciding to not write chronologically is not the same as "writing history inaccurately", and thus surely I have not introduced historical inaccuracies.

> anachronistic statements like this, "During the second century AD, Claudius Ptolemy described a 5-limit diatonic scale in his influential text on music theory..."

I'm not sure if I see this as a true anachronism, in that I'd be surprised to see readers take-away from such a statement that Ptolemy was necessarily aware of the limits classification, instead of thinking that the statement about historical fact is viewed through the lens of modern understanding, in the service of understandability by the modern reader. That this is a classification is evident due to the use of indefinite "a", and statements classifying a system are not statements of the system's literal content. As such I wonder if pointing this out isn't too nitpicky. In any case, I feel any confusion can be avoided simply by putting the classification in a clause that doesn't use Ptolemy as its subject, but more as a conclusion, like I actually did in the statements you removed ("thus *it* was a X-limit scale"). An extra care would be just informing about when the concept of limits was introduced, but really, because the concepts on Wikipedia are linked, the chronology can just be left to easily discover for the reader interested in it.

> A recurring feature of the improvement tags and the Talk page of this article was that readers didn't understand it ... A big reason ... was its ... preference for limits

Looking there, I feel it was mostly poor writing - written sloppily (not distinguishing with a consistent language the harmonic numbers present in interval ratios), but also possibly facts being random - a stuff of superfluous detail before a good overview has been distilled. Zlamma (talk) 11:19, 28 April 2026 (UTC)

Re your edit that removed my parentheticals, claiming that parentheticals mean that their content doesn't belong to an article, and that people should "stop adding them".

I found instructions in the Manual of Style that:

• outright encourage using parentheticals to explain technicalities inline instead of relying on linking: "Excessive wikilinking (linking within Wikipedia) can result from trying too hard to avoid putting explanations in parenthetical statements, like the one that appeared earlier in this sentence" (Wiki Manual Of style > Technical language)
• prescribes their use for content like: (see "some other wikipedia term") : '"See" and the like can be used to internally cross-reference other Wikipedia material. Do not italicize words like "see". Such a cross-reference should be parenthetical' (Wiki Manual Of style > Grammar and usage)
• condones using them along punctuation to clarify sentences: "Dashes can clarify a sentence's structure when commas, parentheses, or both are also being used" (Wiki Manual Of style > Punctuation)

And I found no statements discouraging them.

Can you point at a source of your claims? Zlamma (talk) 23:17, 27 April 2026 (UTC)

Thanks again for taking the time to discuss the article here. Parentheses certainly have their place on Wikipedia, as your citations support.

In this article, they have been misused. A big part of the overhaul was removing such conversational asides and tics (such as "Thus..."), because it was reading too much like someone's attempt at a music theory lecture rather than an encyclopedic article. The opening sentence was a classic example:

• "In music, just intonation or pure intonation is a tuning system in which the space between notes' frequencies (called intervals) is a whole number ratio."

Leads "should be written in a clear, accessible style with a neutral point of view." Clarity is concise. Barring dates and translations, if a Wikipedia article can't get through its first sentence without a parenthetical, something has gone wrong. Worse still, that parenthetical set the table for what was to come. It opened the door to far too many digressions that bewildered both readers and editors.

In the following instance, there are two problems:

• "When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is short of its destination (the initial note, but an octave higher)."

First, the destination is not an octave higher but seven per the cited source. Second, confining the definition of the destination to a parenthetical is unnecessary. It's fair to explain what the destination is, but there is no reason to cloister it behind parentheses.

In another article, that stylistic choice might not be an issue, but this article tends towards illegibility. Our job is to prevent it straying from the path. Trumpetrep (talk) 00:26, 28 April 2026 (UTC)

@Trumpetrep I just started editing Wikipedia again after a 10 or 15 year gap, but I doubt this stint will last long, because it still seems a losing battle, similar to fashioning a bust of Beethoven in taffy under the summer sun. I won't dis any fellow editor dedicating long hours to improving articles. In fact, I'll thank you, but I will say I've noticed you might have kind of a destructive bent. I nudge you to alter your course a little more toward the "improve not remove" philosophy. For example, "if a Wikipedia article can't get through its first sentence without a parenthetical, something has gone wrong." That reads to me like your personal preference, not a reflection of Wikipedia policy as cited by others. Parentheses are part of English sentence grammar. Their purpose is to delimit clarifying information. I then see no reason to disallow them in any English Wikipedia article, including in its opening paragraph. I hope that if you personally dislike parentheses so much that you are driven to edit a sentence to remove them, you'll find a grammatical way to rewrite that sentence, such as by using commas or dashes, or by breaking it into multiple sentences, that doesn't destroy any clarifying information. For example, "An article that uses MIDI to demonstrate musical passages is better off without them." That also reads as opinion not policy. If you prefer audio files over MIDI files, add them, but I hope you won't remove information previous volunteers spent hours on because of a personal dislike for a file format. There's nothing wrong even with having multiple formats in the same article. It might provide access to a wider audience using disparate devices and platforms. By the way, I also disagree with this sentence, "When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is short of its destination (the initial note, but an octave higher)," but perhaps for a different reason. The idea that some positive integer power of one prime (3) should have an expected "destination" of some positive integer power of another prime (2) is ridiculous. If that were possible, they wouldn't be prime! Jed ~2026-34537-05 (talk) 11:35, 3 July 2026 (UTC)

We use parentheses all the time in opening sentences for biographical dates or translations. The following usage is what I objected to:

In music, just intonation or pure intonation is a tuning system in which the space between notes' frequencies (called intervals) is a whole number ratio.

As a lead sentence in a Wikipedia article, that is well off the mark. It's discursive, and by the time the parentheses roll around they are too far from the word they are explaining to be helpful. A general reader might well read them as proof that frequencies are also called intervals.

And that was the opening sentence.

The article overflowed with such chatty asides which made for a difficult read. It left readers and editors confused. Years of discussion on this Talk page document this article's flaws. I addressed them. Yes, a lot of that work entailed destroying passages like this which cited no sources at all. There are sadly too many music theory articles on Wikipedia that are full of unsupported statements.

The chief improvement of the overhaul was tying every statement to a reliable source. If there's language you object to, it's because of the source. There are no shortage of reliable sources that can be cited in order to expand or improve the content of the article. What we can't do is just shoot from the hip. Trumpetrep (talk) 18:18, 4 July 2026 (UTC)

Re your edit that removed my redaction that helped to see that Ptolemy's scale is actually addressing the same problem as the solution before, do you really feel that the reader doesn't benefit from this information? Notably, you claimed earlier that chronology and historicity is important, while the "solution to add 5:4" has been hanging there completely without identifying which scale it even relates to, and its source is actually modern. My edit was actually helping to solve the problems your flag, by not making the "5:4" solution a separate concept and making space for its first chronological application. Justifying the removal by claiming that basic maths need to be sourced is IMO making it harder for this content to be accessible to the general reader (the intervals listed are actually opaque to them, and probably even most technical reader). Zlamma (talk) 01:08, 28 April 2026 (UTC)

Thanks again for talking through all of this. Your latest edit was bang on. It elegantly brought the past into the present, instead of the reverse.

There is probably already a source in the article that can support the statement you would like to make about 5:4.

What's important is not to misrepresent history. Further, this article does not need to cover everything. There already is an article about Ptolemy's intense diatonic scale. Have you looked at it? It's a total mess. Wright's book is only cited once in the lead sentence!

I would love to see some other editors pitching in to clean these messes up. I've been asking about this recently, and there hasn't been much interest. Trumpetrep (talk) 01:25, 28 April 2026 (UTC)

Re:

> Wright's book is only cited once in the lead sentence!

Wright's book doesn't even mention Ptolemy, so I fear the citation is just misplaced there. But that's just because 5-limit scales have many variants depending on where to place the just thirds (as observed here), and Wright was describing the entire class. I may try to find a source for the claim that Ptolemy's scale can be considered to be using the 5:4 third, but in the light of this being evident through basic mathematics over what's already well sourced, I can't think of anyone who would find it unhelpful if the simple fact was shown. Zlamma (talk) 01:59, 28 April 2026 (UTC)

To quote our old friend, "Even if you are sure something is true, it must have been published in a reliable source before you can add it." Trumpetrep (talk) 02:19, 28 April 2026 (UTC)

Thanks for finding the source (while I also found one).

I do feel that somehow joining the paragraphs about "5:4" and Ptolemic would still improve clarity so do consider achieving it again. Otherwise, I will refrain from making the change unless more editors opines here. Zlamma (talk) 11:01, 28 April 2026 (UTC)

Audio Files

Two cents on the revert dispute about whether audio files should have sources in the article. (CC: @Yangchengu)

1. Media files have source stated in the metadata, e.g. File:Primary triads in C just.mid , as such, I find it wasteful to require the information in the article.
2. If this is about "verifiability", then yes, if you inspect the metadata of the said files, many (all?) is "Own work". Nevertheless, I feel some files should be just considered multimedia part of the wikipedia's content. Just like graphical visualizations of concepts created by users. Some may have sources, but some may not but will provide encyclopedic information, just graphically. I was trying to look for any clear prescription about such cases in Wikipedia:Verifiability and its archive but I guess it needs a deeper dive, or just creating a talk subject. The closest I found was pertaining to images here, that "User-made images" and that "Technical data is uncopyrightable" which I feel kind of makes sense for generated sine waves, midi-recordings and such. @Trumpetrep do you really think there's no place for audio demonstrations created by users? Should Beat (acoustics) article's demonstrations be removed if not shown to be coming from some published work?

Zlamma (talk) 01:43, 28 April 2026 (UTC)

Yes, I think audio demonstrations should be verifiable just like everything on Wikipedia. I wish there were more audio (not MIDI) files on Wikipedia. In an article like this, audio files could be extremely helpful, but how can we verify that they are accurate? Without such an assurance, why would we include it in the article? Trumpetrep (talk) 02:17, 28 April 2026 (UTC)

@Trumpetrep

Specifically, what course of action (or lack thereof) and what medium, do you propose then for those intending to contribute to Wikipedia an audio demonstration of an audible phenomenon based on a audio file from a publication, that allows them to not infringe the copyrights of the publication, and at the same time allowing verifiability that the audio correctly contains the waves that demonstrate the same phenomenon?

Are you saying people should publish audio bitstream formats like WAV/FLAC and expect others to be verifying these (programmatically? or rely on aural senses?). Or do you expect the community to only use way lesser known file formats which only contain few basic sine-wave components and these to be verified.

BTW, Not sure why MIDI would be discouraged for musical concepts, as it is at least a widespread format allowing verifiability of notes. Zlamma (talk) 10:18, 28 April 2026 (UTC)

Hi Zlamma,

I agree with your view. User-created audio files (like MIDI demonstrations or sine wave examples) should be treated as primary sources – similar to how we treat user-made diagrams or visualizations. The metadata already states "Own work", and the content itself is a direct illustration of the concept being discussed.

Requiring an external published source for such self-evident audio demonstrations seems excessive. As long as the file accurately represents the described phenomenon (which can be verified by anyone listening or analyzing it), I think it has a valid place in the article, just like user-generated images.

Thanks for making this point clear. Yangchengu (talk) 09:14, 28 April 2026 (UTC)

An article that uses MIDI to demonstrate musical passages is better off without them. For theoretical examples like the ones in this article, MIDI can be useful.

The verification problem still exists. "Own work" satisfies nothing. It just tells readers who made a file. How are they to know it is accurate?

Saying the file is "self-evident" runs right into the core tenet of verifiability, "Even if you are sure something is true, it must have been published in a reliable source before you can add it".

User-made diagrams or visualizations require sources just like everything else on Wikipedia. Why would we exempt audio from this policy? Trumpetrep (talk) 11:41, 28 April 2026 (UTC)

Re:

> An article that uses MIDI to demonstrate musical passages is better off without them

Can you elaborate? Do you really think a general reader does not benefit from having the score playable? The example you gave is actually using the <score> format which guarantees that the audio files match the sheet, so I am especially intrigued why the general reader is "worse off" by being able to listen to a midi rendition of the notes?

Other than this, just making sure, I hope you can answer my specific question on how to ensure verifiability and no copyright infringement of audio bitstream formats (best to do it in appropriate thread). Zlamma (talk) 17:13, 28 April 2026 (UTC)

Yes, the article is improved by access to real audio of a Boccherini performance, rather than a MIDI realization.

As to verification of user-generated audio, I don't know the answer to this problem, which is why I have asked about it a few times at various articles.

I was just perusing the inarticulate mess that is the Pythagorean comma article, and this very issue came up on the talk page. It spread over two separate conversations:

Talk:Pythagorean_comma#Audio_sample_correct?

Talk:Pythagorean_comma#MIDI_errors_(2)

What emerges is a kind of peer review process for audio samples which I doubt anyone would recommend given its violation of Wikipedia's ethos. Trumpetrep (talk) 18:31, 28 April 2026 (UTC)

Thanks for your thoughts on this. I see where you're coming from, but I think there may be a misunderstanding.

You've been arguing that user-generated audio files need to come from a published reliable source to be verifiable. But Wikipedia already allows user-made diagrams and visualizations to illustrate concepts, as long as they accurately represent what they're supposed to show. A MIDI file demonstrating a just intonation triad is essentially the same kind of thing — anyone can listen and verify whether it's correct.

WP:V states that information must be verifiable, not that it must already be "published" in the traditional sense. For simple audio demonstrations like sine waves or MIDI examples, the file itself is the demonstration, and verification is straightforward: anyone with basic music theory knowledge can check it against the description.

The Boccherini example feels like a different situation. A MIDI performance of a complex classical piece has limitations — I agree. But that's not what we're talking about here. For basic intervals and chords in just intonation, MIDI works perfectly fine and is genuinely helpful for readers.

Could you clarify how you would handle this scenario: If a user wants to create an audio demonstration of a basic acoustic phenomenon (like a beat frequency or a pure interval), what would be an acceptable way to do that without violating copyright or relying on existing publications that may not exist for such elementary concepts? Yangchengu (talk) 09:27, 29 April 2026 (UTC)

The Image use policy still requires a source for user-made diagrams, "In such cases, it is required to include verification of the source(s) of the original data when uploading such images."

To answer your specific question, it seems to me that a user shouldn't want to "create an audio demonstration of a basic acoustic phenomenon", because it violates Wikpedia's basic requirements that facts be verifiable. The demonstration should be tied to a source.

If there isn't a publication to support the demonstration, then why would we demonstrate it? This isn't wikiHow. Trumpetrep (talk) 14:33, 29 April 2026 (UTC)

Oh dear, none of my hundreds of mathematical images have such verification. I'm not even sure what “the original data” would be. —Antonissimo (talk) 00:38, 30 April 2026 (UTC)

Those are lovely, User:Antonissimo.

The policy is clear enough, and it tracks with the rest of Wikipedia's verifiability requirement. If there's no source, it doesn't belong in an article.

The biggest issue with this article when I overhauled it was that it lacked sources for statements. Editors were just spitballing. When I reset the article to a baseline of verified statements, the only thing that couldn't be sourced were the audio samples. Trumpetrep (talk) 01:01, 30 April 2026 (UTC)

Re:

> "The Image use policy still requires a source for user-made diagrams, "In such cases, it is required to include verification of the source(s) of the original data when uploading such images."

That just refers to diagrams visualizing data, and requires the source for the data, not for the visualization. What the policy is also clear about is that:

"user-made images may be wholly original. In such cases, the image should be primarily serving an educational purpose" (source) Zlamma (talk) 20:19, 2 May 2026 (UTC)

Re:

> a user shouldn't want to "create an audio demonstration of a basic acoustic phenomenon", because it violates Wikpedia's basic requirements that facts be verifiable. The demonstration should be tied to a source.

I feel this is something that would be worth discussing with the wikipedia community, because ultimately the first principle is for Wikipedia to be more valuable, and better at explaining concepts to the user. Not all best explanations are symbolic.

Also take note that the difficulty of verifiability of an audio file is not an unsurmontable problem and may very well be a temporary technical limitation. Not only verifying a synthetic WAV file may become possible (especially with the advent of AI code generation), but also it's also an option to introduce a data format that leaves the waves out in the open (here one file format I found, but simply JavaScript code could be another option), and generates the sound programmatically. Given that I feel that doing today the best possible (so, e.g. relying on users catching buggy files) and in the process scaffolding the best encyclopedia to be, while solving technical problems later, is an approach that the community would actually benefit, rather than suffer from. Zlamma (talk) 20:45, 2 May 2026 (UTC)

Re:

> Yes, the article is improved by access to real audio of a Boccherini performance, rather than a MIDI realization.

You earlier said:

> an article that uses MIDI to demonstrate musical passages is better off without them

which is a different claim. Zlamma (talk) 20:26, 2 May 2026 (UTC)

Thanks again for the constructive response, Zlamma. I have asked the music theory project about this issue. Any other suggestions about where to take it?

You are right about the feasibility of verifying audio files. The ad hoc peer review that happened at the Pythagorean comma article is a window into how it could work. The question remains, is such a peer review process compatible with Wikipedia's guidelines?

My main concern is simply creating the best product we can for readers. In a lot of these articles, there are sloppy, unexamined assertions and audio examples that detract from the subject, rather than illuminate it.

Regarding MIDI and audio, I've described both sides of the coin. Trumpetrep (talk) 01:52, 3 May 2026 (UTC)

Thanks.

I also decided to action my take on it - to me, this is a foundational problem that applies throughout Wikipedia, related to lack of rules' distinction between verifiability of base claims and verifiability of correctness of works demonstrating the claims, so here I asked to consider even hinting at the distinction in the Verifiability rules. Zlamma (talk) 12:30, 21 May 2026 (UTC)

Revert of fixes to the Scales > Pythagorean comma problem

@Trumpetrep regarding your revert containing only unnecessary disparaging characterization of my edits without addressing the problem that they were fixing:

Using own words ('DIY language', in your disparaging vocabulary) is not forbidden on Wikipedia, and even required. In fact, the current wording of the paragraph are also just editor's words. I see this because I have actually gone to the cited source and compared. Secondly, because the words were actually those that create 'gibberish' (using your vocabulary), which I did point out in my edits. That's because creating 'twelve chromatic tones' requires only 11 just fifths (the base note is already counted as the first of the twelve) and so the 'destination' to these is just the last twelveth chromatic tone which is not yet supposed to be octave-equivalent of the base note. The destination that the cited source refers to exists only in the context of attempting to solve the problem of what they call 'scale formation', which currently doesn't even feature in the paragraph. The paragraph's only goal set is to create twelve chromatic tones. Straightening this confusion while trying to retain as much form of the original as possible took me a while, but now what you have done is reverted to the original 'gibberish'. I now invite you to 'riff' (again, from your vocabulary) in such a way that is suitable to you, but removes the inaccuracy by defining the 'destination' correctly. Zlamma (talk) 19:22, 4 July 2026 (UTC)

Hi Zlamma, I'm sorry my language felt disparaging. I genuinely did not understand how we went from this:

"When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is wide of its destination, which should be seven octaves higher than the initial note. This gap is the Pythagorean comma." (39 words)

To this:

"When justly tuned fifths are used to generate the twelve tones of the chromatic scale, and then instead of octaves, an attempt is made to use another fifth to add a thirteenth note as an octave equivalent to the initial (for a full octave scale to be playable), that fifth (the twelfth in sequence, because initial note counts as first of the thirteen) ends up near the desired initial note's octave equivalent (seven octaves higher at this stage), but is actually wide of it. This gap is the Pythagorean comma." (90 words)

I couldn't make heads or tails of the passage when I first read it, and I think it would look like gibberish to a general reader. That is why I used the word. I apologize for hurting your feelings.

I understand what your concern is. I think it's a little pedantic. In general usage, scales are understood to run to the octave. In my view, the original summation of Campbell & Greated's language is clear enough. The destination of the scale in question is the original note. For other editors, this is the original language of the cited passage. The summarized passages are in bold:

It might appear at first sight that the circle of fifths gives the complete answer to the problem of scale formation, but unfortunately this is not the case. From the harmonic series we know that the frequency ratio between any two adjacent notes in the circle should be 3/2 and there are 12 intervals, thus one complete circle will increase the frequency by (3/2) 12 = 129.75. It will be seen, however, that the final note should be exactly 7 octaves higher than the starting note, i.e. the frequency ratio corresponding to one complete circle should be 27 = 128. The discrepancy is known as the Pythagorean comma. (Campbell, Murray and Clive Greated. The Musician's Guide to Acoustics. Oxford University Press, 1994. 172.)

You are right that this is a confusing point, and it will improve the article to make it as clear as possible for the reader. Perhaps another source articulates it better, or can help us expand the language without creating our own. That is what we have to avoid, those improvisations that explain what a text means. There are plenty of sources. We shouldn't have to explain anything ourselves. Trumpetrep (talk) 21:33, 4 July 2026 (UTC)

You have not hurt my feelings. You have just wasted my time and while yourself not taking any time to understand the issue at hand. I don't think the remark is pedantic, since the section that you quote has circle-of-fifths in its context, and as a 'destination', while the paragraph in the article doesn't, and that created gibberish for anyone who actually wants to understand the text (as evidenced by yesterday's confusion of the user above, which made me notice the illogicality of this paragraph). The paragraph doesn't even mention that the goal is to create a scale in the sense of reaching an octave with solely the fifths (it shouldn't, because this goal is unattainable, so it should merely talk of 'attempting' it, so as not to confuse a general reader that this is a sane thing to do). But regarding assuming that a scale is meant, I tried to see it your way that scales 'scales are understood to run to the octave' before you even came with the revert, but I couldn't find the consistency, which you talk of, in practiced writing. For the most parts, most scales invoke in their definition just the base note and all the notes before the next octave, e.g. Diatonic scale talks of 7 notes, not 8, pentatonic also centers its definition on the five. This kind of makes sense because a melody can be played resolving on the base tone and not needing the octave. Furthermore, being forewarned by your earlier activity about needing to back any information by a citation, and, for some reason, reverting work that doesn't maintain chronology (your personal preference IMO), I could not find a source to back the claim when the Pythagorean tuned instruments had a goal of incorporating both the fifths-derived tones and a higher octave in the same instrument/melody, to encounter the problem. If you want to introduce a premise that this problem was an natural obstacle to performing music in the said scale, I take it you have a citation for this claim, since you speak here mostly from a position of someone who introduced here only sources-backed information. Zlamma (talk) 22:16, 4 July 2026 (UTC)

I'm having a little trouble following what you are saying. In my view, the passage summarizes the source. I think the reader will have no difficulty following the narrative of the article which has taken them from the natural phenomenon of just intonation, through a brief history, and into the section on scale formation.

The first paragraph explains how the just fifth can be used to easily generate a 5-note scale. The second paragraph explains how that technique comes up short for a 12-note scale. As I read those paragraphs together, the progression is quite clear: the system that works for a pentatonic scale does not for a chromatic one. I'm sorry that I don't comprehend your point about the goal. So, I'm doing my best to respond as I understand what you said.

The passage also leads us into the Pythagorean comma. Both Pythagorean tuning and Pythagorean comma have articles of their own where these subjects can be dealt with at far greater length. In an article about just intonation for the general reader, there doesn't need to be much more material than what is already here. If those Pythagorean articles didn't exist, it might make more sense to expand this section, but it seems best to give a simple explanation and point readers who want to know more in the right direction.

The simple issue is that the 12th note is in just intonation with the 11th note, but not with the 1st note. I could be wrong, but I feel like we are in agreement that the summary of Campbell & Greated does not spell this problem out so succinctly. Our job is to find a source that does. In my view, the passage is clear enough, but I agree that it could be clearer. Trumpetrep (talk) 23:02, 4 July 2026 (UTC)

Re The simple issue is that the 12th note is in just intonation with the 11th note, but not with the 1st note I understand this fact very well (notably, with the 1st note, the 11th is a fourth away from it, so its the just-fourth that is violated, though the wolf just fifth violation also features, with the appropriate overtone of the 1st note), but before I did, I was the general reader, and I was mislead by this paragraph. The paragraph uses the example from the book, and the style of narration, which only talks of a dissonance of the 13th note, with some other note higher by a number octaves (7, because it avoids at all costs using down-octaves in the process for transposing the notes to form the actual 12 chromatic tomes, which I am not sure helps). This 13th note is not part of the process defined as 'When justly tuned fifths are stacked to generate all twelve chromatic tones' and the paragraph doesn't even make it clear that all this happens at the 13th note.

Re Both Pythagorean tuning and Pythagorean comma have articles of their own where these subjects can be dealt with at far greater length. In an article about just intonation for the general reader, there doesn't need to be much more material than what is already here., I totally agree that existing articles describe the whole matter better (because they don't avoid expressing goals of reaching octave intervals) and I don't see much need in this article having a new go at it, which I expressed in summaries my edits (1, 2). I would much rather point to those articles, but if the current detail is to stay here in JI, it should stop confusing.

Re The first paragraph explains how the just fifth can be used to easily generate a 5-note scale. ... As I read those paragraphs together, the progression is quite clear: the system that works for a pentatonic scale does not for a chromatic one. The system would not work for the pentatonic scale if the same exact contrived exercise was conducted as for the 12-tone, i.e. the higher octave was attempted to be produced using only the fifths. There's no reason why the 'exercise in attempting to reach a higher octave' is applicable for chromatic scale, and not for pentatonic, since both scales can equally perform music with or without that higher-octave tone - such music will still be described as of that scale. Zlamma (talk) 00:00, 5 July 2026 (UTC)

Ah, I didn't realize you were a general reader as well. I feel like I understand a bit more this confusion about the "goal".

The tonic note isn't really the goal of the scale. It's just a note like any other. Classifying it as a goal is kind of misleading. The issue here is that if we start on C and go up seven octaves in a 2:1 ratio we end up in a slightly different place than if we go up in 3:2 fifths. The difference is the Pythagorean comma. That's the issue described in the source, and reflected in the summarized text.

To put it another way, the goal isn't to "reach a higher octave". The goal is to make notes available in a consistent tuning system. When you stack 3:2 fifths from C and return to the pitch 7 octaves later, it's in a location that's different enough to be audible. A circle of 3:2 fifths is actually a spiral, and it yields two different tunings of the same note.

The Pythagorean comma article is painful to look at. It's almost entirely original research, and then it has illegible images like this. I don't see the expressing of goals, you describe. All I see is an article that violates a lot of Wikipedia's policies. Trumpetrep (talk) 01:37, 5 July 2026 (UTC)

No, you misunderstand me. I no longer lack the grasp of music theory, so save explaining it to me. That's why I said I "was" a general reader. But using my memory of past lack of understanding, I'm currently analyzing the logical correctness of the paragraph, needed for a person who is reading it without the conceptual mental model (that often makes people not notice slight inaccuracies) and using the text to build such a model. And in my analysis, I am finding the text incorrect indeed. Please read below with only that in mind:

The 'goal' I refer to is just the logical goal stated in the paragraph, which says 'When justly tuned fifths are stacked to generate all twelve chromatic tones', making the 'goal' just generating the twelve tones. It then refers to 'the final note in the series' which clearly is just the twelfth tone (achieved with only 11 fifths), and that one is (roughly) a fifth short from the higher octave equivalent of the initial (e.g. F when C was the initial). But then follows a claims that some 'destination' of this very note 'should be seven octaves higher than the initial note'. Clearly it shouldn't! It may be off by 23.5 from being a fifth from initial note's octave equivalent, but not from being an octave equivalent! This is misleading. The book sets up a different (correct) 'destination' which indeed is an octave equivalent of the initial note, by setting its 'goal' as closing the circle of fifth *after* generating the 12 tones, which makes the note that needs to reach this goal the 13th one (which requires 12 fifths).

If you argue that the 12th tone also has its 'destination' 23.5 off from where the exercise puts it, just by the virtue of not sounding just with the initial tone (and an error is just in 'which should be seven octaves higher than the initial note'), then that is not a valid requirement for chromatic scales. Chromatic tones are just a division where each is a semitone distance apart, and are not even defined absolutely - different tunings, and in some tunings even different 'centre' of the stack, will put them differently (as seen Chromatic scale), so when the paragraphs talks of 'destination' for the 12th tone, it is no other than being 'roughly' a semitone apart from its neighbors. Chromatic scale has no requirement for the 7th semitone to be a perfect fifth/fourth from the tonics (it's never that in 12-tet for example because it's 700 and not the just 701.955, and it's that only for JI for some tonics). It's necessary to posit the need for an octave equivalent tone and get one hugely dissonant with it, that isn't any of the already decided 12 tones, to show that the set of notes is no longer chromatic (a 5-limit tuning will stop being chromatic at this point only if it doesn't use the real 2:1 octaves for the base of the scale in each new octave, but attempts to use fifths). Zlamma (talk) 09:44, 5 July 2026 (UTC)

The word "goal" is still causing confusion. The observation from Sethares sums it up quite pithily, "It is theoretically impossible to return to the unison". I think the Grove definition also sums it up well: 12 just fifths ≠ 7 perfect octaves.

That paragraph is not about the size of the semitone. It's continuing the discussion of how a Pythagorean system of tuning by fifths generates problems.

The article begins by explaining the ratios of fifths and octaves. Should we reiterate the ratios in the scales section? The simple fact that fifths and octaves proceed in different ratios seems clear to me at this point in the article. As Jed said, this isn't complicated math. Trumpetrep (talk) 13:29, 5 July 2026 (UTC)

Re: The word "goal" is still causing confusion .. Please, by goal I mean, (in an abstract sense) "the thing that the first sentence of the paragraph tells is the purpose of 'stacking the fifths'". In the non-abstract sense, with the current wording it happens to be " generating all twelve chromatic tones". Please propose a better word for it. I'm lost for ideas. Or just think of it as "the thing that the first sentence of the paragraph tells is the purpose of 'stacking the fifths'".

Re: That paragraph is not about the size of the semitone I never made any such claim, so this is a detraction. When I was talking about size of semitone, I was not referring to 'what the article is about', but what chromatic scale definition is about, which the paragraph chose to only refer to. I said all that in the context an internal of the Talk page - a hypothetical argument that there's a destination for the 11th tone. If this is beyond grasp, let's just leave it. I'm reducing the discussion to the minimum below.

Re: The observation from Sethares sums.... Don't expect the reader to know what context has been laid in the source. I am trying to fix readability of the article, for which contents of cited sources are irrelevant. The article must be readable without having to read sources.

Re: The article begins by... An overwhelming majority of Wikipedia sections don't require to read any preceding sections, and that's for a good reason: many users use ToC to read only the section of interest (and they won't even read the lead when they are already familiar with the concept). Many users arrive straight to a section from a link in another article. Moreover, a section is expected to set its own context - see "Put a brief overview statement at the top of each section.". Furthermore, if there's a reliance on some content from above section, if the section doesn't make it explicit, the user may be bound not to guess it.

Re: Should we reiterate the ratios in the scales section?

I don't know what you mean but some 'reiteration' doesn't sound too good to me. What should be there is something that lacks the contradiction present there currently.

Now, trying to understand what your mindset is that prevents you from seeing the misinformation, I don't think I can take any more detraction, before you answer the following questions:

Please consider just the following excerpt of the current text:

"When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is...'.

and answer:

1. Which note this excerpt is calling 'the final in the series'?

If your answer isn't an equivalent of

"one of the twelve notes of the chromatic scale just generated, specifically the twelfth note at the end of stack of fifths that creates the 12-tone chromatic scale, e.g. if the initial note was C, it would be F",

then please further answer:

2. Which specific sentence(s) in the article have caused you to conclude this?

3. Why did the immediate precedence of "justly tuned fifths are stacked to generate all twelve chromatic tones" not cause you to think that the context is being redefined in the very paragraph, and "the series" is referring precisely to a note in the 12 (and only 12) chromatic tones just generated, and the the final note isn't the 12th tone at the end of the stack? Zlamma (talk) 20:42, 5 July 2026 (UTC)

Again, I find it a little hard to follow what you are saying, but I think the gist of it is clear enough. You are confused about the language of stacking fifths, and if you are confused, it's fair to assume other readers might be as well.

I have restated the ratios of the intervals at the outset of the section. I also reworded the statement about 12 fifths vs. 7 octaves. I hope that's clearer. Trumpetrep (talk) 21:43, 5 July 2026 (UTC)

Again, I find it a little hard to follow what you are saying - please just answer my 1. 2. 3. questions at the end of my message.

And now, with the new version, please answer another one:

4.Why is 'a stack of 12 justly tuned fifths' mentioned by the way 'tuning all twelve chromatic notes' if only 11 is sufficient to tune all twelve chromatic notes? Zlamma (talk) 22:02, 5 July 2026 (UTC)

Again, I don't understand what you are asking. There are 3 reliable sources cited in this passage that all explain this fairly clearly. The language reflects what they say. It seems like your quibble is with the demonstration, which dates back centuries. Trumpetrep (talk) 22:20, 5 July 2026 (UTC)

Re: your quibble is with the demonstration, which dates back centuries

No, my contention is how the article incorrectly narrates over the subject. It presents broken reasoning chain. Incorrect narration of the article has nothing to do with the sources, and the fact that sources are written correctly cannot rescue you from incorrectly written article. Again, readers are not expected to need to read sources.

(The broken reasoning chain is that it claims that there's readily a problem when merely trying to 'tune 12 chromatic tones', and that problem shows at the 12 fifths. But there is really no problem with 'tuning 12 chromatic tones' because one only needs 11 fifths to tune a version of 12 chromatic tones just fine.)

But please, just try to fulfill one request for me in this long evading communication.

I don't understand what you are asking - please don't read into the intents behind the question. Just read the question. If you truly don't understand it, can you advise which word or clause you don't understand? Zlamma (talk) 22:43, 5 July 2026 (UTC)

Again, the reasoning chain is the sources'. Trumpetrep (talk) 22:49, 5 July 2026 (UTC)

What I'm saying is that the article fails to represent the sources chains of reasoning. You can't answer to that 'the sources chain of reasoning is sound'. The article still has the problem.

You can only persuade that this is not the case by answering question 4., so showing how the article (ideally which sentence) demonstrates that problem at the 12th fifth occurs while trying to 'tune 12 chromatic tones' which require only 11 fifths. Zlamma (talk) 23:15, 5 July 2026 (UTC)

This is your own particular issue, Zlamma. I don't know how to help you other than to reprint what our cited sources say:

• "A series of 12 Pythagorean 5ths provides a fully chromatic scale..." ("Pythagorean intonation", Grove)
• "The ‘Pythagorean comma’ (or ‘ditonic comma’) is 23.46 cents, being the difference between twelve 5ths and seven octaves." ("Comma", Grove)
• "It might appear at first sight that the circle of fifths gives the complete answer to the problem of scale formation, but unfortunately this is not the case. From the harmonic series we know that the frequency ratio between any two adjacent notes in the circle should be 3/2 and there are 12 intervals, thus one complete circle will increase the frequency by (3/2) 12 = 129.75. It will be seen, however, that the final note should be exactly 7 octaves higher than the starting note, i.e. the frequency ratio corresponding to one complete circle should be 27 = 128. The discrepancy is known as the Pythagorean comma." (Campbell & Greated)
• "Much to Pythagoras’ chagrin, however, there is a problem. When extending the scale to a complete tuning system (continuing to multiply successive terms by perfect 3/2 fifths), it is impossible to ever return to the unison. After 12 steps, for instance, the ratio is..." (Sethares)

Trumpetrep (talk) 23:28, 5 July 2026 (UTC)

The 3 last points have correct reasoning chain that the article is missing. Neither of them characterizes the 'problem' as occurring when generating just the 12 chromatic tones:

(Campbell & Greated) shows correctly that the problem happens with trying to complete the circle of fifths.

(Sethares) talks of 'completing the tuning system' (however nebulous this is, it's not "generating 12 tones")

("Comma", Grove) - that passage just explains what the comma is, but not where it causes a problem, so it's not quite relevant (it doesn't back the part of the paragraph that claims there's the problem, but just the part that claims what value the Pythagorean comma has).

Now, most importantly:

("Pythagorean intonation", Grove)'s "A series of 12 Pythagorean 5ths provides a fully chromatic scale..." is just an off-by-one error (pretty easy to commit, thus common).

Compare it with:

• Margo Schulter - Pythagorean Tuning and Medieval Polyphony "To derive a complete chromatic scale of the kind common on keyboards by around 1300, we take a series of 11 perfect fifths"
• "This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale" Wikipedia (Granted this is Wikipedia and I'm not sure if this has a source, which I would suggest to ignore for now, but just use that text to verify which claim is correct. 12 or 11?)

Zlamma (talk) 00:24, 6 July 2026 (UTC)

The notion that the Grove dictionary is mistaken and medieval.org should be cited instead is indicative of a larger problem in identifying reliable sources. Trumpetrep (talk) 00:49, 6 July 2026 (UTC)

Setting the source authority aside, you yourself have no doubt? Is your stance based on just knowing a source, or understanding? If the latter, can you demonstrate how generating the sequence from Medieval, the "Eb Bb F C G D A E B F# C# G#", when starting from Eb, requires "12 Pythagorean 5ths", not eleven? Zlamma (talk) 01:08, 6 July 2026 (UTC)

Due to your concerns with the language, the passage was updated to make it clear that the problem in question is the Pythagorean comma. I'm grateful that you pointed out your confusion over the language.

Needless to say, at 11 just fifths, there is no Pythagorean comma. It arises at the 12th, which is why the language reflects what is in the five cited, reliable sources.Trumpetrep (talk) 01:15, 6 July 2026 (UTC)

It's you who is looping this discussion. Please answer questions and don't write statements unrelated to the question that instead going back to subjects from higher topics. At this level in the thread, you were giving credibility to "A series of 12 Pythagorean 5ths provides a fully chromatic scale...", but not to the claims that say 11th is enough. There's no pythagorean comma in any of these claims. There are "Pythagorean 5ths" and they come from the source you prefer. Can you demonstrate where the 12 fifths are in:

"Eb Bb F C G D A E B F# C# G#", if Eb is the base note? Zlamma (talk) 01:37, 6 July 2026 (UTC)

The passage in question, and the sources that are cited to support the statements in it, is about the Pythagorean comma. Trumpetrep (talk) 01:49, 6 July 2026 (UTC)

The singular clauses I only cited, the "A series of 12 Pythagorean 5ths provides a fully chromatic scale...", and the clause that claims the number is 11, both make self-contained claims that does not involve Pythagorean comma.

This response is just a continuation to not answer a self-contained question about your understanding of the maths that make you give credibility to the '12' and discredit the '11'. You write unrelatedly, while accusing of over-verbosity and prolonging the discussion.

In any case, at this point, I feel the more constructive thing to do is answer on the fresh proposal below. Zlamma (talk) 07:42, 6 July 2026 (UTC)

A fresh idea to resolve:
(building on my earlier "I tried to see it your way that scales 'scales are understood to run to the octave' before you even came with the revert, but I couldn't find the consistency, which you talk of, in practiced writing.")

Assuming that at least some writings use 'scale' in reference to the sequence running to the octave, so having 13 notes in case of chromatic, 8 for heptatonic, 6 for pentatonic, etc. (I would not admit this assumption for usages of "X tones of scale", where X is "the numeral minus one"), what is needed is just disambiguation.

To that effect, would you accept the starting of the paragraph that was (exactly, or your close-paraphrase of, if you deem unfit) one of:

• A problem exists when this technique is used to tune a chromatic scale spanning from the base note fully to the top of the octave (13 notes, separated by 12 fifths).
• A problem exists when this technique is used to tune a chromatic scale spanning fully to the octave.
• A problem exists when this technique is used to tune a chromatic scale.

instead of the current "Using this technique to tune all twelve chromatic notes creates a problem"?

Zlamma (talk) 07:27, 6 July 2026 (UTC)

None of the cited sources use this proposed language. Why should we? It's cumbersome and unidiomatic. The language that is on the page reflects the reliable sources that are cited. Trumpetrep (talk) 12:09, 6 July 2026 (UTC)

My main proposed (and considered necessary by me) change is to replace the "to tune all twelve chromatic notes" with "to tune a chromatic scale".

If you claim no sources use the version after the change, then you're wrong. See the sources you yourself brought here. ("Pythagorean intonation", Grove) talks of the scale, not the 12 notes.

If you claim that no sources use any of the sentences I gave, then that's also true of the current "Using this technique to tune all twelve chromatic notes creates a problem". Zlamma (talk) 12:25, 6 July 2026 (UTC)

@Trumpetrep I can see that you edited, going with the third option, i.e. just replace the confusing 'twelve chromatic notes' with the 'chromatic scale'.

I'm glad this progress is made, but you may be surprised to learn that I am still hoping for more. You might have thought I gave each 3 options because they're all acceptable by me, but actually, I only added the last one, the one you went for, to even understand whether there's any part of my changes on which we see eye to eye. So far I was at a loss, especially because your edit came as a sweeping revert of my work, instead of improvement of it, and I also couldn't make out from the revert's summary or the discussion what was incorrect with it (in fact until my above response made you reconsider, you seemed to have not agreed with any aspect of my fix, so perhaps my impression was accurate).

And before I explain why it's not enough, I want to throw in another alternative:

• Using this technique to tune the octave repeating chromatic scale creates a problem.

(I prefer it less than the first two, because it doesn't help to explain the number 12 that appears later, but it has a pro of being succinct and not requiring to swap the sentence like the others are doing)

Justification

(Forgive the verbosity. Because I don't know which part of my thinking is not mutual, and you seem to not see the problems when I merely point at them, I'm left no other option but present my entire mindset. I'm also wary of editorial errors I make while trying to manage such a long text in a short time, leading to artifacts of past versions scattered in the sentences - I hope that this time, you will allow me to improve sentences that are unclear by pointing at them, before discarding the entire writing.)

The reason I see the need for further disambiguation of 'chromatic scale', is because I see all around the literature and Wikipedia itself, that the term 'scale' can be used to refer two different things. And which thing it is, mostly depends on the context:

• As I'm sure you know, in the context of playing music, performing, the scale is probably always thought of as the playable sequence (AKA progression), which is always (probably) played with both the unison and the octave included (whichever direction). This can be seen in:
  • a scale is "any consecutive series of notes that form a progression between one note and its octave" - Wikipedia - Scale (music)
  • The ascending and descending chromatic scale is shown below (followed by image with 13 pitches) - Benward and Saker - Music in Theory and Practice, quoted on quoted on Wikipedia - Chromatic scale
• In the context of music theory however, the term scale is very often used defined as only the set of unique pitch classes. This, naturally, doesn't include the octave, as it's the same pitch class as the unison. Examples:
  • "set of 12 notes known as the chromatic scale" Encyclopædia Britannica - What Are the 12 Notes of the Chromatic Scale?
  • 'chromatic scale, then, is a collection of all the available pitches' - Piston W. - Harmony, quoted on Wikipedia - Chromatic scale
  • Emphatically, it is explicitly defined this way in Wikipedia's Chromatic Scale article, which will be the natural place to visit for a person reading the disputed paragraph in case they don't know the concept: 'a set of twelve pitches within an octave'. Interestingly, in the same article, in its section below, a definition is given again but it's different: "The chromatic scale is a musical scale with twelve pitches", which is more towards the 'progression' interpretation if one follows the musical scale link. Interestingly the lede used to have the same definition until this edit.

What may add further confusion to the learner, is that there seems to be no stipulation in the literature that, in addition to the base note, its second version (typically octave above it) has to even feature in musical playing within a scale. So, a person educating themselves about the music practice should think that there may very well be music entirely limited to one instance of each of the scale's pitch classes. This is easily concluded when one learns that there's music with just one or two pitches. Surely then, an instrument that's limited to any scale but excludes the octave will still play (or at least accompany) any music in that scale. I imagine such instruments must exist and have existed: tin-whistle have just 6 holes, so in the hands of the players who can't blow overtones will play only 7 tones, some simplified glockenspiels may be similar (found this children 5-note-only pentatonic one). These examples are not the ones that people have to tune, but I'd imagine that instruments which require tuning must also exist in this category. E.g. stringed ones, like the venerable lyres (very much on the mind of someone while reading about the early history of tuning musical instruments), must exist in versions with only as many strings as the number of unique pitch-classes in the scale known by the folk. Given that, I don't think we can reasonably expect that all readers arriving at this paragraph hold an image in their head that tuning the top-of-the-octave note is necessary to tune to any scale.

So, concluding, somebody who built their knowledge on literature which defined the chromatic scale in the second way, will wonder why is there any need to use 12 fifths. In this article's use case, it's especially likely that this will be the case: it's going to be someone who went to the chromatic scale page to [learn/refresh their memory] about the chromatic scale, while trying to understand the tuning problem, and learns that it's "a set of twelve pitches". A simple disambiguation that we mean the full scale that includes the top not will make sure they have everything they need to understand the maths of what follows.

Your contention based on the fact that sources don't provide this distinction is IMO missing an important fact. The reason why the dichotomy of interpreting the term 'scale' is rarely acknowledged is not because it's not a useful thing to do. It's just for the said sources it is redundant - articles and books can assume the comfort of a sequential reader, who can be assumed to have read earlier chapters or paragraphs, so it can be assumed that they learned about related concept from the same source, and apply the same interpretation to the term. In some of the sources you cite, the Pythagorean comma is said to be the problem not merely when tuning the 12 notes, but when attempting to 'complete' a 'the circle of fifths' or, 'the tuning system'. Another reason may be that books and articles often confine themselves to only one of 'theory' or 'practice', so never encounter the need to disambiguate. Wikipedia is different however - its general readers may have very little familiarity with music (which says nothing about their lack of familiarity with other concepts like maths), and haven't necessarily arrived from a text with the same context - for a Wikipedia reader, it's very easy to drop into a music theory article from a musical practice article and vice-versa. What may seem like pedantic to a person already familiar with the music world, is actually a sign of the general reader - someone who is only in the process of learning about the lore of the discipline, so interprets everything using the meanings of the terms from outside of that discipline (e.g. treating 'set' as purely the mathematic unordered set, and never the ordered set). An encyclopedia is more effective if it avoids "Ignotum per ignotius" for the largest population of readers, by requiring prior-knowledge of only the minimum of the most basic terms. When you wrote "general usage, scales are understood to run to the octave", that is some claim about some use being "general" in the musical world. That doesn't appeal to the "general reader", who should not need to know which of a number of interpretations is "the general", but quite the opposite - it expects the reader to not be the layperson. Zlamma (talk) 16:05, 9 July 2026 (UTC)

Congratulations on making a mountain out of a molehill.

It does seem that there is a fundamental langauge barrier here for you, Zlamma. The article reflects the cited sources, which are written in common English. Trumpetrep (talk) 18:18, 9 July 2026 (UTC)

Re: The article reflects the cited sources. You repeatedly fall back on "the sources say it" but the article's "Using this technique to tune a chromatic scale creates a problem" is not a sentence from any of the source, and each source uses a different narration for explanation - some easier to grasp and some harder. What you're demonstrating is just that you think you have some automatic right to restore the sentences you came up with, and you don't need to listen to anyone's feedback. Indeed such is your conduct with most my edits here, while the amount of effort needed for the very few changes I managed to get through you is unlike anything I ever saw on Wikipedia.

Re: Congratulations on making a mountain out of a molehill. I don't know how you have no qualm to respond with sarcasm and condescension to a person's work that provides you with explanation for their stance. If ignoring the points they raised was not enough.

Re: It does seem that there is a fundamental langauge barrier here for you, Zlamma. One fundamental barrier is evident, in that you never manage to digest any text that is longer than a few sentences - something that a LLM can do just fine, finding no language problems and being able to pick up the points made (try it for yourself on my last response and you may discover that barrier of comprehension sits on the reader side). Zlamma (talk) 00:46, 10 July 2026 (UTC)

You wrote 1,490 words to respond to a 30-word comment. I don't think it's unreasonable to view that as out of proportion.

Your feedback has been helpful, and I have repeatedly thanked you for it. The language in the article reflects all of the cited sources. That's what Wikipedia requires of us. If it's not in a reliable source, we can't put it in the article. Trumpetrep (talk) 01:10, 10 July 2026 (UTC)

If you don't intend to address any new points I made, there's no need to repeat the same argument.

Neither there's any point in diverting the thread to the number of words written in an explanation to an obstinate interlocutor. The number does depend on how attentive the interlocutor was to the earlier simpler messages. In any case, the number doesn't make the author's claims wrong, so cut the off-topic irrelevancies already. Zlamma (talk) 01:26, 10 July 2026 (UTC)

If you look at my first reply, my twofold concerns have been consistent throughout this discussion: respecting your perspective and Wikipedia's policies about sourcing. It's clear we are going to disagree about language. That's not going to prevent us from working together to improve the article, which I believe we have. Trumpetrep (talk) 02:04, 10 July 2026 (UTC)

And if you look at the edit summary, and a number other replies like the 'congratulations' two messages up, the claim of consistent respect is shown to be ludicrous.

The consistency in responding without addressing counterarguments as if you haven't even read them, but just repeating the same claim without any more substance, which I admit is present (along with consistency of going off-topic), even still in this thread which continues to ignore my entire message, is exactly the opposite of what helps. Without this 'consistency', it would take much less messages to finally get rid of the confusing 'to generate twelve chromatic tones' pointed as a serious issue in my first message, or remove a single word providing no value. Zlamma (talk) 11:05, 10 July 2026 (UTC)

It is 'theoretically' impossible often means it is 'practically possible'. Remove it

@Trumpetrep Preventing turning the reverts of my edits into an edit war:

Recap:

1. . Me: edit link
   Change: It is theoretically impossible to return to the unison.
   Summary: Remove incorrect (or very unusual) use of the word 'theoretically', which also doesn't exist in the cited source. 'Theoretically' usually means *only* in theory (and not in practice, not actually). See https://www.merriam-webster.com/dictionary/theoretically and follow second meaning https://www.merriam-webster.com/dictionary/theoretical
2. . @Trumpetrep: revert link
   Change: It is theoretically impossible to return to the unison.
   Summary: Sethares specifically uses the word "theoretical" in this passage on page 54 when explaining how the system works. "...it was constructed using only the theoretically ideal ratios..."
3. . Me: edit link
   Change: It is theoretically impossible to return to the unison.
   Summary: Please. Source saying 'theoretically ideal ratios of 3:2 ... Using ratios of 3/2 it is impossible to...' does not mean the source is also saying 'Using ratios of 3:2 it is theoretically impossible to'. Facts 3:2 can be proven mathematically, so that part isn't theoretical
4. . @Trumpetrep: revert link
   Change: It is theoretically impossible to return to the unison.
   Summary: The theory is what makes this impossible. It seems like there might be a language issue here? The source is crystal clear.

My response to last edit without a revert here. Please answer:

1. . From your 'The theory is what makes this impossible.' I take it that you interpret your words of 'theoretically impossible' only as 'impossible as proven by the theory', right? If so, what do you think of the danger of interpreting it according to the first and second entry from 'theoretically' at dictionary.com, ("in principle; according to reasonable assumptions or rules, as opposed to what actually happens" or "in a way that is purely speculative; hypothetically."), which would make the reader think of it as 'impossible hypothetically, but actually possible'?
2. . Wikipedia MoS mentions that sentences should contain no unnecessary words.. Why can't the sentence be without the word 'theoretically'? What new information does that word give to the reader that the user doesn't have?

Zlamma (talk) 23:42, 5 July 2026 (UTC)

Is there a language barrier here, Zlamma? I have noticed that it is often difficult to understand what you have written. Likewise, you also have a very specific view of English usage which seems uncommon. For fear of giving offence, I would not ask under normal circumstances. Given the sprawling issues here over language, and our communication difficulties, it seems worth asking.

The usage of the word "theoretically" here is in line with the cited source and common usage. I don't feel it is unnecessary. Sethares explains the theoretical method for the tuning, and explains why it results in a theoretical impossibility.

Your interpretation of the sentence is unidiomatic, which is why I began by asking about a language barrier. Please forgive me if the question is hurtful. My only intent is to help us communicate, which has been a struggle. Trumpetrep (talk) 23:57, 5 July 2026 (UTC)

I'm not sure if there is a language barrier as such a diagnosis is a subjective matter, and I'll refrain from speculating whether it's with the writer or the reader.

Can you instead just answer questions the 1. and 2. above? If not, what's stopping you? Zlamma (talk) 00:27, 6 July 2026 (UTC)

Subjectively, your English usage seems to be a persistent problem, which is why I asked if it is a foreign language. You frequently use words in ways that do not make sense, which is why I found myself unable to understand many of your questions.

The definition of "theoretically" you cited has four meanings. The one that applies to the usage in this article is #3, "according to, or in a way that relates to, a scientific theory." Definition #4 is closely related, "in a way that relates to the set of principles or methods belonging to a particular subject, art, or activity."

I don't think most English readers would misunderstand this usage. This is an article about music theory, and this particular section is about the theoretical work of Pythagoras. In that context, the word "theoretically" is perfectly clear. Yet, here we are discussing it, which makes me wonder if there is a language barrier that is creating difficulties for you. Trumpetrep (talk) 00:42, 6 July 2026 (UTC)

Re question 1., so the fact that there's an ambiguity, thus a possibility of misinterpreting to mean 'hypothetically but not actually', is to you not a reason to avoid it?

Can you also answer question 2? Zlamma (talk) 00:55, 6 July 2026 (UTC)

I already did answer question 2. The way discussions with you loop in circles because of misunderstandings such as this is similar to speaking with someone who does not fully comprehend the language of the discussion. Is English a foreign language for you?

The fact that you think there is an ambiguity in the usage is part of what prompted me to wonder about your familiarity with English. The usage is perfectly clear in the context of a music theory article, as previously stated. Trumpetrep (talk) 01:04, 6 July 2026 (UTC)

The question 2. is not about ambiguity. It is:

2. Why can't the sentence be without the word 'theoretically'? What new information does that word give to the reader that the user doesn't have?

You haven't answered it. Zlamma (talk) 01:11, 6 July 2026 (UTC)

In my first reply, I did answer your question. Trumpetrep (talk) 01:17, 6 July 2026 (UTC)

I missed that you thought of this as a response, as it wasn't answering "Why can't the sentence be without the word 'theoretically'?", but actually to "Why did you add the word?", which I didn't ask. My request was for a necessary condition. You answered a request for a sufficient condition, which I didn't make.

I don't think there's anything missing if the word is removed, and given it is ambiguous, it's better when it's not there. I will ask for a Wikipedia:Third opinion to help resolve this, when I find time. Zlamma (talk) 01:25, 6 July 2026 (UTC)

It's not ambiguous, as stated.

Remember when UpTheOctave! asked if you could be less verbose? This is the kind of thing she was talking about. What you refer to as "question 2" is actually a paragraph consisting of one sentence, and two complete questions. All three of those sentences were labeled as #2. What exactly am I meant to respond to? All three sentences? Both questions?

Given the amount of time these conversations have taken up, I've been replying succinctly. I said as much as necessary to respond to "question 2" in order to make my view clear. Trumpetrep (talk) 01:47, 6 July 2026 (UTC)

On the topic. Yes, these are your opinions, and I hold opposite ones. The source literally says "It is impossible to ever return to the unison" and you find it necessary to say "It is theoretically impossible to return to the unison." for the above reasons, while I gave mine.

On the off-topic remarks, If if didn't occur to you, I too could share guesses at personal tendencies being the cause for writing much of unneeded dead weight to messages in this and other threads, or I could give links to where other editors made guesses at your other personal tendencies. The reason I don't do it here or anywhere else (except hinting at their existence just now), is because it does nothing to help us learn what the objective basis of the particular differences of opinion is, to be able to resolve it. I instead try to just ask questions about the basis for conviction and earnestly answer questions in the way their author stated them, and I avoid meandering, answering misrepresentations of the questions, or going off-topic.

Back to the point. A Wikipedia:Third opinion will help us determine if 'theoretically' is needed in reference to 'impossible'.
UPDATE1: it seems that one came already. The word 'theoretically' seems not useful to that user. Zlamma (talk) 06:02, 6 July 2026 (UTC)

UPDATE2: For reference, it seems your undoing of that user's edit has also conceded my point. If this was intentional, the case is closed. Zlamma (talk) 12:08, 8 July 2026 (UTC)

Re: Your interpretation of the sentence is unidiomatic. Care about the general reader, not the editor. See MOS:IDIOM: "In general, if a literal reading of a phrase makes no sense given the context, the sentence needs rewording. Some idioms are common only in certain parts of the world, and many readers are not native speakers of English; articles should not presume familiarity with particular phrases."

There's many times more English learners as a second language than native speakers. Given that, actually, even with editors, avoiding idioms is just a good way to not be misunderstood. Zlamma (talk) 11:24, 6 July 2026 (UTC)

What is an example of "theoretically impossible but practically possible"? Examples of the inverse are common. —Antonissimo (talk) 19:03, 11 July 2026 (UTC)

The whole of this article is about the struggle between theory and practice. It is practically quite possible to return to the unison via fifths. The passage in question is about the theory of Pythagorean tuning, wherein it is not, hence the inclusion of the word used in the cited source. Trumpetrep (talk) 22:05, 11 July 2026 (UTC)

Rather than "practically possible to match" I prefer "possible to practically match" (not in that exact construction). It is obviously possible to get within any finite tolerance, including the resolution of human pitch discrimination. —Antonissimo (talk) 08:14, 17 July 2026 (UTC)

Re: Trumpetrep's "...hence the inclusion of the word used in the cited source"

Please read and don't ignore points made in the revert summaries and talk pages.

As I raised already (1, 2), your claim of the word being used in the source is entirely moot.

Compare:

Book: '… theoretically ideal ratios of 3:2 … [paragraph break] … Using ratios of 3/2 it is impossible to...'

You:                                                             Using ratios of 3:2 it is theoretically impossible to…'

The word is used in the source, but on a different adjective ('ideal') from a different sentence in a different paragraph, to achieve a different meaning ('theoretically ideal' vs your 'theoretically impossible'). Zlamma (talk) 06:50, 18 July 2026 (UTC)

Latest revisions

User:Epixix, I reverted your latest revisions because they ignored the conversation we had in March on these very issues. They set the article back by muddying the waters and confusing the lead again. The text reflects cited sources.

Simply saying "I prefer this instead of 'the tuning of a musical interval without beats'" is insufficient on Wikipedia. The definition in the lead of the article is the classic definition of the term found in multiple authoritative references which are cited in the article. We've got to use reliable sources for all statements in the article. We can't just wing it based on our preferences. Trumpetrep (talk) 22:31, 13 July 2026 (UTC)

I understand your concern that I was basing my edits on personal preferences. I had another thought which I didn't state for the current definition on the page:

"Just intonation is the tuning of a musical interval without beats"

My original concern was that "musical interval" was ambiguous, whether it was referring to the simple "difference in pitch" or the theoretical intervals that arise in common practice Western theory (A-G with sharps and flats, major, minor, perfect, diminished, augmented, etc.). After reviewing the Harvard Dictionary entry which is linked in the Definition section, I can conclude they indeed do intend the former:

"(1) The beatless tuning of an interval, one that brings it into agreement with some analogous interval in the harmonic series...When an interval can be expressed by adjacent partials in the series it has the form (x+1):x."

In retrospect, this makes sense, because a definition relying on the Western system would be completely disregarding of non-Western cultures.

With this, the definition I had put:

"In music, just intonation comprises intervals and chords that sound without beats"

is logically equivalent, with the benefit of not requiring the user to check the sources to disambiguate. Epixix (talk) 22:57, 13 July 2026 (UTC)

Thanks for discussing the changes. Your version of the lead sentence is not logically equivalent. It gets rid of "tuning" and adds "chords". It also inserts the comically unnecessary "In music...".

We don't write, "In warfare, a tank is an armoured fighting vehicle..." We just start the article. Leads should be clear and concise. This lead reflects the cited source. Yours is a variation, not a logical equivalent. Trumpetrep (talk) 23:15, 13 July 2026 (UTC)

On the lede wording, I prefer @Trumpetrep's original version.

On the remaining reverted content and the conduct, 2 cents:

@Trumpetrep please start following the Partial reverts guideline. @Epixix has followed good editing practices and split their edits, to enable separate reverts, while you made a wholesale revert, and have not justified, as of yet, why other change, like the switch to lilypond were reverted (as Epixix pointed out).

A number of my edits (1 2) which were also granular and separately justified in their summaries, met the same heavy-handed, sparsely justified fate. This generates further burden on the other editor to have to add to the discussion's volume. Please respect the time spent on contributions by taking effort to at least justify the revert of each change. Zlamma (talk) 15:35, 14 July 2026 (UTC)