Talk:Overtone

Definition

uncontested (basic definition rewritten) The following discussion has been closed. Please do not modify it.

I changed the basic definition of overtone as the orginal was very misleading. 12:47, 3 September 2007 (UTC) —Preceding unsigned comment added by Kevin aylward (talk • contribs) 11:48, 3 September 2007 (UTC)

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Formulas

A recursive arithmetic formula for the overtone series:

• a_n=a_n-1+x, for a₁=x

A recursive geometric formula for the octave "series":

• a_n=a_n-1*2, and/or
• a_n=a_n-1/2

I don't know if there is a way to do this in one formula.

Here's another formula for the overtone series for one-dimensional standing waves on a medium of length L and wave speed v:

• f_n=(nv/2L)

from Intervals, overtones, and axis PDF Hyacinth

http://www.chanceandchoice.com/ChanceandChoice/chapter2.html

Hyacinth, I think you mean harmonic series, not overtone series. Overtones can be inharmonic and overtones don't include the fundamental. Another Stickler (talk) 20:17, 12 February 2010 (UTC)

The simplest harmonic partial series formula I can think of is

fn = f * n

where f is some arbitrary positive real frequency, n is the ordinal or counting number of each harmonic partial from 1 up, and fn should be read as "f sub n" or "f of n", meaning the frequency of harmonic partial n. ~2026-34537-05 (talk) 17:53, 3 July 2026 (UTC)

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resolved ("overtone" not confined to acoustic waves) The following discussion has been closed. Please do not modify it.

The opening statement "Use of the term overtone is generally confined to acoustic waves, especially in applications related to music" is simply not correct. Overtones bands are extremely important in infrared spectroscopy. Overtone bands are used because they give nearly repetitive information at a variety of concentration ranges. They are therefore helpful when one cannot control either the pathlength or concentration of a particular sample or when one wishes to compare within one sample two species that have significantly different absorbtivities at their fundemental frequencies. unsigned

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resolved (use f, 1f, 2f, etc. instead of f1, f2, f3, etc.) The following discussion has been closed. Please do not modify it.

hmm... i don't think saying (f1, f2, f3) is any better. that looks to me like just any arbitrary frequencies. the idea is to show that the overtones are integer multiples of the fundamental frequency, like 100, 200, 300, 400 is f, 2*f, 3*f, 4*f, and going up by octaves yields 100, 200, 400, 800, 1600, or f, 2*f, 2*2*f, 2*2*2*f, etc. the formulas are right, though. maybe they could be put in the article?

Again, I think using f1, f2, f3, f4 isn't very clear. they could just represent frequency variables (f1 = 434 Hz, f2 = 12345432 Hz, etc.). 1f, 2f, 3f, 4f would at least be correct mathematically. - Omegatron

"Since the overtone series is an arithmetic"

So technically that should say harmonic series anyway... - Omegatron

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Octave Series

I thought "overtones" were an integer power of two multiplied by the fundamental frequency, so for example, the 'second overtone' of a fundamental frequency would be the fundamental frequency multiplied by two to the power of two, and the 'fourth overtone' would be the fundamental multiplied by two to the power of four, etc. Denelson83 01:37, 30 Nov 2004 (UTC)

so the overtone series would be 2f, 4f, 8f, 16f, 32f? that's the octave series. what field is this definition in? - Omegatron 15:44, Nov 30, 2004 (UTC)

Denelson83, there can't be any single series defined for all overtones for all resonating systems because overtones can be any frequency that a resonating system produces (that is higher than the fundamental). Depending on the system, that can include octaves of the fundamental (as in your proposed series), harmonics of the fundamental, or frequencies that are not multiples of the fundamental (inharmonic partials). It all depends on what the actual system produces. Another Stickler (talk) 20:11, 12 February 2010 (UTC)

resolved (not only "peculiar to barbershoppers") The following discussion has been closed. Please do not modify it.

== Barbershop sense ==

In barbershop music, the word overtone is often used in a different (though related) way. It refers to a psychoacoustic effect in which a listener hears an audible pitch that is higher than, and different from, the four pitches being sung by the quartet. This use of the word is peculiar to barbershoppers and not a standard dictionary usage. The barbershopper's "overtone" is created by the interactions of the overtones in each singer's note (and by sum and difference frequencies created by nonlinear interactions within the ear).

I remember in high school, one of my chorus teachers talked about an "overtone" in this sense. He demonstrated it by playing a note on the piano, then playing a loud chord that did not include that note, but it was still audible. We were not studying barbershop (in fact, I didn't even learn about barbershop until later on), so my question is, are we certain that this is generally called an "overtone" only in the context of barbershop? - furrykef (Talk at me) 15:28, 27 March 2006 (UTC)

No.

It would be interesting to find out more. I'll snip the phrase "peculiar to barbershoppers." Dpbsmith (talk) 15:32, 27 March 2006 (UTC)

resolved (discussion integrated into article) The following discussion has been closed. Please do not modify it.

fundamental sound physics question

If I understand correctly from the innuendo in the passege. Physically all Overtones are prodcued when a fundamental is played -Is that true? Are they less dominant because of their lesser energy in contrast to the fundamental which is played directly? If so, playing the same note on different instrument have a different "sound flavour" because of the non precise production of overtones of that specific instrument? if not, what causes it then ? --Procrastinating@^talk2me 22:44, 12 November 2006 (UTC)

All the overtones capable of being produced will be produced. THat is not to say that every sound source has the same overtone structure. A flute for instance produces very few overtones and sounds pure. An oboe a has a different overtone structure so sounds more 'reedy'. The relative amount of each overtone determines the quality of sound you get. Also some overtones are not harmonics on certain instruments. This also adds to the different character of different instruments--Light current 01:38, 13 November 2006 (UTC)

It is just not true. It is mostly true for struck instruments such as a guitar or bell. However, any overtone which has a node at the point being struck will not be excited. Some chinese bells take advantage of this to oscillate with two very different notes depending on where they are struck. A bowed or blown instrument will typically NOT oscillate at all of its possible overtones, as any bugle player knows. However, because these instruments are nonlinear, they DO sound with harmonics at integer multiples of whatever overtone is being excited, even if these harmonics are not overtones of the instrument (confusing, I know) David s graff 20:06, 8 February 2007 (UTC)

Now i'm even more confused.

• You say that all overtones (frequency integer multiplication) are produced, and then claims that their "amount" matters somehow. the amount is infinte if all are prododuced. So is it an instrument specific volume decaying format that makes the difference?

• • Overtones are not necessarily exact multiples of the fundamnetal frequency (although in most musical instuments like the piano the first 6 or seven are pretty near)

• • • Ok. So an overtone is a physical phenomenan occuring when a plucking a string to play a fundamnetal frequency , and the overtones configuration is the reason for the instrument specific "sound color" ? (and possibly dissonance)

• • • • Yes

• • • • Not entirely true, it depends on the amplitude of the different overtones as well as their frequency David s graff 20:06, 8 February 2007 (UTC)

• • • Do most western colture instrument produce overtones that are harmonic or almost? (integer multiplication of the base frequency)

• • • • Yes I think so (except certain untuned percussion, bells etc)

• • • • • Its not a western vs non-western thing. Its a one-dimensional vs higher dimensional thing. One-dimensional instruments are near-harmonic, but typical oscillators are not. See standing waves. So ALL drum heads and bells are inherently not harmonic.

• • • The Overtone article is VERY lacking. do they decrease in amplitude as the pitch gets higher? --Procrastinating@^talk2me 21:20, 13 November 2006 (UTC)

• • • • I agree the overtone article is not very good! Overtones generally decrease in amplitude at the number goes up but not always. The second overtone for instance may be larger than the first overtone. Dont ask what inst that would be!

• • • • • They do not necessarily decrease in amplitude, but we become less able to perceive high overtones at high frequencies, and our musical instruments are designed to only generate sound at a particular range of frequencies. Overtones are a general phenomena of all oscillators. A proton has an overtone of ~500GeV corresponding to about 10^26 Hz (pdb.lbl.gov) David s graff 20:06, 8 February 2007 (UTC)

• Please define "quality of sound".

• • I mean 'timbre'

• • • Which is ,If I understand currectly, the overall per instrument overtone configuration, yes?

• • • • Yes

• How can an overtones be non harmonic? it goes against the very defention of "Absolute harmony"

• • Beacause some ot the physical systems used to produce sounds are not perfect. (like a piano string for instance. Its seventh overtone (or partial) is slightly flat) Also some of the overtones from a cymbal are not harmonically related

• • • • Actually most of the physical systems which produce sound are not close to perfect. And the typical overtone in a piano string is actually slightly sharp of the seventh harmonic. But the seventh harmonic of any instrument is slightly flat in most tuning systems since 8/7 falls between a whole step at 9/8 and a just minor third at 6/5. Many instrument designers try to minimize the seventh harmonic by e.g. putting the hole in a guitar about 1/7 along the string. David s graff 20:06, 8 February 2007 (UTC)

• • • Another defenition is needed. A "perfect sound" prodcuing system will play only the fundamnetal frequency , and/or it's overtones will be of absolute harmony ?

• • • • Yes (if you like perfection!)--Light current 21:25, 13 November 2006 (UTC)

• • • • • So overtones can not be avoided, they can just to be with absolute harmony to the base frequency. We call this (no where to be found in wikipedia!) a "perfect sound". near perfect instrument are flutes, and tuning forks, right?

• • • • • • Some instruments are lacking in overtones like the flute. Also a tuning fork will have no overtones to speak of. I wouldnt say they are 'perfect' instruments because they have no overtones and therefore sometimes sound very boring. They are however 'perfect' in the sense that, because they produce no overtones, the overtones cannot be out of tune with the fundamental. It really is difficult to say which is the most harmonious instrument. I would have to think and research that one! Anything I think that produces overtones is going to 'foul up' eventually as the overtones go higher! 8-)--Light current 22:54, 13 November 2006 (UTC)
          • A tuning fork does create overtones, that is why one must wait before listening to it after striking. There's something missing about this defenition, is there a wiki article I can read/you can write -to clear this up? --Procrastinating@^talk2me 13:11, 14 November 2006 (UTC)

• • • A tuning fork does have overtones, but they have a much higher frequency than the fundamental and also decay very quickly compared to the fundamental. Any system driven to a high amplitude will become non-linear and hence also generate harmoincs, but a tuning fork is pretty close to linear, like the classical mass-on-a-spring from physics, so once the amplitude drops, it also doesn't generate many harmonics. David s graff 20:06, 8 February 2007 (UTC)

• These are hard questions, How sure are you..? :)

• • @Reasonably! 8-)

• • • you rock! Thanks. --Procrastinating@^talk2me 15:59, 13 November 2006 (UTC)

UR welcome --Light current 16:37, 13 November 2006 (UTC)

I've been looking for real answers regarding harmony and it's preception for a very long time now, this is startgin to get clearer now..Yet thw wiki artciles about music are very very lacking. hopfulyl this discussion will help clear them up..:) Procrastinating@^talk2me 21:22, 13 November 2006 (UTC)

Great, thanks for all the answers(I got it!). Now all that is needed is for someone to update this info..: --Procrastinating@^talk2me 21:34, 13 November 2006 (UTC)

How about U? I could look over any edits you make if you like. Its always better when 2 people are working in 'harmony' 8-)--Light current 21:42, 13 November 2006 (UTC)

Sure, I will. please answer my last question whic you have missed.Procrastinating@^talk2me 21:49, 13 November 2006 (UTC)

Ok. I have rewrote the article, and integrated our discussion. --Procrastinating@^talk2me 21:09, 20 November 2006 (UTC)

Yup. Looks good to me 8-)--Light current 22:25, 20 November 2006 (UTC)

Just did a bit more tidying. See what U think!--Light current 23:03, 20 November 2006 (UTC)

This article lacks references and order. Corrections needed namewise ("overtone is wrong - correct scientific name is "partial") to reflect scientific practice (this is an encyclopedia), mentioning IN PASSING the usage in common practice (musical or other). --David Be (talk) 08:39, 10 November 2008 (UTC)

Notation

[] Notation und MIDI-Sound --88.73.222.2 08:20, 4 December 2006 (UTC)

That link points to a nice illustration of a Harmonic series (music) compared with 12-tone equal temperament, but it's not usable in this article to illustrate overtones because overtones are not necessarily harmonic, and the illustration includes the fundamental, which by definition is not an overtone. Another Stickler (talk) 17:51, 12 February 2010 (UTC)

Guitar string

"This means that halving the physical string length, does not halve the actual string vibration length, and hence, the overtones will not be exact multiples of a fundamental frequency. The effect is so pronounced that well set up guitars will angle the bridge such that the thinner strings will progressively have a length up to few millimeters shorter than the thicker strings. Not doing so would result in inharmonious chords made up of two or more strings. Similar considerations apply to tube instruments."

The angle of the bridge isn't really for correcting overtone-problems, but for correcting intonation problems (relating to the fundamental more than the overtones that is) resulting from difference in mass etc. of the strings. —Preceding unsigned comment added by 83.253.57.162 (talk) 18:26, 23 February 2008 (UTC)

It is true that guitar strings are slightly inharmonic, especially depending on how dirty and old they are, but it is also true that bridge intonation adjustment is not related (much) to this inharmonicity. Instead, bridge intonation compensates for stretch in the fretting of notes primarily, which differs based on string mass and material.Backfromquadrangle (talk) 05:36, 12 October 2010 (UTC)

resolved (overtone versus partial) The following discussion has been closed. Please do not modify it.

Excellent - now go over to the 'harmonics' section and rewrite that ...

This is a fine example of good, clear writing.

I have just come from the 'harmonics' article and its a mess. Could the guy who wrote this article please go over there and clean it up? The harmonics article seems to have been put together by a room full of monkeys.

Thanks!

--Dpolwarth (talk) 07:41, 3 August 2008 (UTC)

Actually, "Overtone" and "Overtones" NEEDS TO HAVE ITS NAME CHANGED TO "PARTIAL" (with redirects created), because IT IS AN INCORRECT and SUPERFLUOUS name for "partial". I don't know how to do it (name change), and unfortunately lack time and references, but have done some editing (albeit, and alas, here and there)to reflect the scientific standing. Common practice notwithstanding, this IS an encyclopedia, and should show correct usage of terms keeping into account (but NOT furthering) common practice when erroneous. --David Be (talk) 08:37, 10 November 2008 (UTC)

If the terminology is wrong as you say, you'll need to produce a source to verify that. Dicklyon (talk) 01:32, 11 November 2008 (UTC)

David Be is incorrect. Overtone is not synonymous with partial. Reference: . It is a poor term though. See my new topic below. -- Another Stickler (talk) 18:25, 8 December 2008 (UTC)

resolved ("overtone" not recommended) The following discussion has been closed. Please do not modify it.

overtone is deprecated

I was taught that overtone and overtones are poor terms to use because overtones refers to all the sines making up a complex tone except the fundamental. That's why there's "over" in "overtones". (See for multiple dictionaries defining it so.) This makes it confusing when trying to relate it to partials and harmonics, because the first overtone is the second partial. If there are people who think overtones is synonymous with partials or harmonics, it's because of inaccuracy in teaching/learning. Even if the term were ever to be redefined as a synonym, it would become redundant, and still deprecated. Though obsolete, it still has its place; you need to know it to understand what old books are saying. -- Another Stickler (talk) 18:17, 8 December 2008 (UTC)

Overtone singing

"Overtone singing (wrongly known also as throat singing)..." Tuvan's and Mongol's Overtone singing IS called Throat Singing, it is NOT only the Inuit Katajjaq... (Written by someone from Finland and who is a practitioner of TUVAN THROAT SINGING.) —Preceding unsigned comment added by 81.175.200.134 (talk) 21:00, 22 May 2009 (UTC)

Circular drums

The article claims that the first overtone of a circular drum is 2.4 times the fundamental frequency. But from the Vibrations of a circular drum article, it seems that it should be a₁₁/a₀₁, where a_mn is the n-th positive zero of the Bessel function J_m. This is 1.5933..., so I think "2.4" should be changed to "about 1.6". --Zundark (talk) 12:15, 18 June 2010 (UTC)

I've changed it to "about 1.6", and added a reference (which says 1.593). --Zundark (talk) 12:18, 19 June 2010 (UTC)

"Overtone" was deprecated in the 19th century!

I dare say that any competent and intelligent student of musical acoustics at least understands that "overtone" was deprecated over a century ago. Please see the note in the article about Ellis' translation of Helmholtz (which I have read, some time ago). I'm EXTREMELY* disappointed that Wikipedia seems to still give this term so much credence. *I use all caps only very rarely!

Such a respected author as Thomas Rossing, co-author of a fine book about acoustics, as I understand it (I don't own a copy) uses the term "partials", instead of "overtones", pointing out that there are two kinds: harmonic, and inharmonic. The almost-harmonic partials of plucked or struck strings do not detract from a sense that the notes produced have a definite pitch, as hammered dulcimers, pianos, and harpsichords -- as well as guitars, lutes, banjos, and many others plainly demonstrate.

There are certain choices our society stubbornly holds on to, even though there are quite-good reasons to change. (Consider the ridiculous arrangement of the letters on nearly all keyboards, for instance.) If Helmholtz is not good enough of a reference, what is? (Rossing, et al?)

Moreover, the notice at the top of the article that says that no references are cited ignores Ellis. Seems to me that it should be taken down. Nikevich (talk) 14:36, 1 July 2010 (UTC)

Added comments

I tried to rewrite, substituting "partial" for "overtone", essentially where it didn't violently clash with how musicians use the term. I also added some text. However, it seems that, when nearly done, I created an editing conflict with myself (all too doggoned easy to do, hang it!) and lost about four hours' work.

Some more references, inexact (I'm running out of energy; sorry!):

• Juan G. Roederer, Intro. to the Physics and Psychophysics of Hearing (approx. title, possibly correct)
• Arthur [ ] Benade, Horns, Strings, and Harmony (I think that is the title of his- book on musical acoustics)
• Ganot's Physics (Google books; 19th-century text) -- Very popular in its time; of interest partly for illustrations and descriptions of scientific apparatus)
• Rossing, Moore, and Wheeler, The Science of Sound -- Excellent work on acoustics

One of these (Roederer?) points out that, as only fairly-recently learned, perceived timbre is influenced a great deal by initial transients that start a musical note; these die out within a fraction of a second.

As well, please, let's not help perpetuate the "(n-1)" nonsense of calling, say, five times the fundamental the fourth overtone (or harmonic). While it seems odd to designate the first harmonic as the fundamental, that's not much of a nuisance, imho.

No mention of Chladni figures? Good gosh.Nikevich (talk) 16:29, 1 July 2010 (UTC)

I linked the Chladni figures article, which maybe didn't exist 16 years ago, or you probably would have linked it. ~2026-34537-05 (talk) 19:00, 3 July 2026 (UTC)

resonating strings

Instruments such as cellos show spontaneous vibration of adjacent open strings when certain notes are being played and music written in certain keys seems to exploit this property for a richer sound (i.e. the prelude to Bach cello suite 1 in G major). Instruments like sitars use resonating strings. Are these also examples of overtones?

I added to the See also section an internal link to the Sympathetic resonance article. ~2026-34537-05 (talk)

"...sharpness or flatness of... overtones [makes] waveforms not perfectly periodic"?

In Musical usage term, it states "The sharpness or flatness of their overtones is one of the elements that contributes to their unique sound. This also has the effect of making their waveforms not perfectly periodic."

I think that if the fundamental, and the overtones, are constant over time, then the resultant waveform will be perfectly periodic, but the period will simply be longer than if the overtones were exact multiples.

In a physical instrument the waveform is very unlikely to be perfectly periodic, but for other reasons: the amplitudes, and to a certain extent frequencies, of the fundamental and various overtones vary over time. FrankSier (talk) 12:34, 8 February 2013 (UTC)

Harmonics, overtones, partials

I can't make sense of the lead's attempt to distinguish these terms: "Using the model of Fourier analysis, the fundamental and the overtones together are called partials. Harmonics, or more precisely, harmonic partials, are partials whose frequencies are integer multiples of the fundamental (including the fundamental which is 1 times itself)." Anyone interested in helping me make this easier to understand? ~Kvng (talk) 15:11, 16 April 2015 (UTC)

===============================================

There seems to be an inconsistency between the text on overtones and the table under "Musical Usage Term". The text states:

"Because "overtone" makes the upper partials seem like such a distinct phenomena, it leads to the mathematical problem where the first overtone is the second partial." 

whereas the table lists the second overtone as being the same as the second partial. I believe that the text is correct.

Another issue is that the article takes a particular usage of "harmonic" that is not universal; i.e.

"A harmonic frequency is an integer multiple of the fundamental frequency." 

A lot of technical work concerns the harmonics of resonators, which are not integer multiples. I think this is the reason for the term "true harmonic", in which case the sentence should read "A true harmonic frequency is an integer multiple of the fundamental frequency".

I note that the whole area of harmonics, overtones and partials is already full of semantic inconsistencies (possibly due to the terms having been redeveloped by multiple independent sources).

I believe that this makes it advisable to use only those few terms that are well-defined (note?1) for developmental text, and describe the possible multiple meanings of ambiguously defined terms under their separate headings?

Note?1: i.e. true harmonic, overtone and (possibly) partial (though I personally remain uncomfortable with the use of overtone number because it so distorts the numeric relation between overtones of the fundamental and overtones of its harmonics). PhysicistQuery (talk) 21:34, 4 September 2020 (UTC)

relation between harmonic (1st harmonic, 2nd harmonic, ...) and overtone (1st overtone, 2nd overtone, ...)

what is the relation between harmonic and overtone? in Wikipedia page of harmonic, it say: 2nd harmonic correspond to 1st overtone, but in Wikipedia page of overtone, it say: 2nd harmonic correspond to 2nd overtone. which is right? Xnsxsnx (talk) 06:54, 5 June 2025 (UTC)

partial overtone sentence ambiguity

This sentence from the "Overtones in music composition" section really bugs me. It mixes partial and overtone in exactly the kind of way that causes ambiguity. I'd fix it if I knew which interpretation they meant.

"The primacy of the triad in Western harmony comes from the first four partials of the overtone series."

If they meant harmonic partials 1 2 3 4, that's a modern equivalent to Pythagoras's "tetraktys" in the 500s BC, which cannot produce a triad containing 5, so it's wrong.

If they meant harmonic overtones counted 1 2 3 4, equivalent to harmonic partials 2 3 4 5, it can produce a triad containing 5, but why leave out the fundamental?

With 1 added, that's equivalent to Zarlino's expansion of the tetraktys to his "senario" 1 2 3 4 5 6 in the 1500s, which Rene Descartes wrote in the 1600s could produce all consonant intervals: octaves, fifths, fourths, thirds, and sixths (though I'm not sure how he got a minor sixth without 8).

Rameau in the 1700s expanded the senario to his "corpse sonore" 1 2 3 4 5 6 7 8, which allowed him to theorize about the construction of seventh chords, not just intervals and triads.

Any ideas? ~2026-34537-05 (talk) 19:54, 3 July 2026 (UTC)

Help fixing reference "Fineberg, Joshua (2000). Guide to the Basic Concepts and Techniques of Spectral Music"

All the links under the reference are currently broken or lead to an abstract instead of the full text. I found a full-text URL and tried to add it, but when I edit the References section, the editor shows only the section title with no other content. If anybody else knows how to add it, please do. Here's the URL.

https://cmp.ischool.illinois.edu/courses/tipei/M408E/Notes/spectral.pdf

~2026-34537-05 (talk) 06:27, 4 July 2026 (UTC)

Citation doesn't support text

"Because the overtone series rises infinitely from the fundamental with no periodicity, in Western music the equal temperament scale was designed to create synchronicity between different octaves.[2][22]"

That's reads as ridiculous to me. The very definition of harmonic partials is that they're all periodic with the fundamental. Also, every interval in 12-tone equal temperament except the octave is irrational therefore aperiodic. So I checked the references, and, as expected, they don't support the claim.

What's the proper Wikipedia way to deal with that? Adding a citation needed tag is meaningless, because it already has two citations. Is there an "I call B.S." tag?

~2026-34537-05 (talk) 07:02, 4 July 2026 (UTC)

There are a couple of ways to do it. You could use Template:Failed verification. What saves the most time is to simply edit the passage to accurately reflect the cited source. Wikipedia encourages editors to be bold for precisely this reason. Otherwise, errors like these remain uncorrected for years.

Too many of our music theory articles are problematic. You've found a rare instance where there are actually two cited sources for a statement. The more common issue is that there are no sources to be found. Trumpetrep (talk) 01:29, 10 July 2026 (UTC)

Thanks for the template link. I'll use that for now. ~2026-38734-14 (talk) 05:40, 25 July 2026 (UTC)

Superharmonic

There's a notice at the top:

"Superharmonic" redirects here. For functions in mathematics, see Superharmonic function.

I searched, and there are only two instances of superharmonic found in this article, the very instances in that notice. It might have been a mistake to redirect it here instead of to Superharmonic function.

Can whoever redirected it here, or anyone else, justify that? ~2026-38734-14 (talk) 16:53, 11 July 2026 (UTC)