Talk:Harmonic series (music)

message: Rather than saying 
"the difference between consecutive harmonics is constant.  But because our ears respond to sound logarithmically...,we perceive higher harmonics as 'closer together' than lower ones"....
would it not be more accurate to say that the distance between consecutive harmonics, as measured in semitones, decreases, but because of our ears respond to sound logarithmically, the frequency difference between consecutive harmonics is constant?
Kj4321 (talk) 17:19, 9 December 2008 (UTC)
replies: text: Here's another try: "The difference in number of cycles per second between the frequencies of consecutive members of any harmonic series remains constant, but because the human ear and brain compare frequencies logarithmically (by division rather than by subtraction), higher harmonics sound closer together intervalically." -- Another Stickler (talk) 19:04, 9 December 2008 (UTC)
replies: text: Thanks for the response.  Considering first the sound source, before it reaches the listener, each standing wave in a vibrating object with harmonic overtones is larger than the succeeding standing wave (1/2, 1/3, 1/4, 1/5...).  And this also conforms with tonal practice, with intervals of the harm series decreasing in size.  So is it just the distance between them which is constant?  Couldn't this 'constant frequency between harmonic numbers' be an "imprint" of the logarithmic human hearing system?   Kj4321 (talk) 03:02, 10 December 2008 (UTC)
replies: text: I don't know. -- Another Stickler (talk) 05:33, 10 December 2008 (UTC)
thread_title: Consonance & dissonance

message: On the use of the term "overtone", this page says: "Barbershop uses overtone colloquially in reference to the psychoacoustic phenomenon of close harmony." But the Barbershop music page and other info I've seen emphasis the use of Just intonation and the resulting resonant "ringing" sound. Unless I'm mistaken (quite possible), I don't think "overtone" just means "close harmony" in barbershop.. ? Pfly 07:47, 9 March 2007 (UTC)
replies: text: Concurred. There's no reason wide harmony can't ring too. I changed "close harmony" to "just intonation harmony". -- Another Stickler (talk) 22:19, 9 December 2008 (UTC)
replies: text: If memory serves, bluegrass singers call this phenomenon "the bird," referring to a mixing tone heard "above" the actual notes being sung. Can't recall where I read that, but will keep looking. __Just plain Bill (talk) 19:40, 10 December 2008 (UTC)
thread_title: Barbershop

message: This article is incorrect in:
===Terminology...===
Likewise, many musicians use the term overtones as a synonym for harmonics, though not all overtones are necessarily harmonic, some are inharmonic or non-harmonic. 
*No, they are harmonic, otherwise the overtone doesn't resonate. The 7/5 is a dissonant in the minor and major scale, but this doesn't mean the 7/5 isn't harmonic with the fundamental. The tone is enharmonic with all other tones of the interval: the perfect 5th (3/2), the perfect 4th (4/3), the major 3rd (5/4), the minor 3rd (6/5). So that's the reason why the 7/5 isn't in the major and minor scale, not because it's not harmonic with it the fundamental.
That is, an overtone may be any frequency that sounds along with the fundamental tone, regardless of its relationship to the fundamental frequency. The sound of a cymbal or tam-tam includes overtones that are not harmonics; that's why the gong's sound doesn't seem to have a very definite pitch compared to the same fundamental note played on a piano. 
*That's a different topic. A bell has a combined tone because of its non-perfect shape, this can (and in reality is always) be a combination of enharmonic notes.Houtlijm 07:37, 18 August 2007 (UTC)
replies: text: Please count again. In the current table Harmonic series (music)#Harmonics and tuning you can see that the order you claim is achieved by starting from the fundamental as 1. −Woodstone (talk) 15:05, 20 July 2008 (UTC)

text: Wow. I never noticed that pattern of multiples in the numbers. SharkD (talk) 19:18, 12 August 2008 (UTC)
thread_title: Overtones are always harmonic up to the 7th degree (the sixth overtone)

message: Likewise, many musicians use the term overtones as a synonym for harmonics, though not all overtones are necessarily harmonic, some are inharmonic or non-harmonic. 
*No, they are harmonic, otherwise the overtone doesn't resonate. The 7/5 is a dissonant in the minor and major scale, but this doesn't mean the 7/5 isn't harmonic with the fundamental. The tone is enharmonic with all other tones of the interval: the perfect 5th (3/2), the perfect 4th (4/3), the major 3rd (5/4), the minor 3rd (6/5). So that's the reason why the 7/5 isn't in the major and minor scale, not because it's not harmonic with it the fundamental.
That is, an overtone may be any frequency that sounds along with the fundamental tone, regardless of its relationship to the fundamental frequency. The sound of a cymbal or tam-tam includes overtones that are not harmonics; that's why the gong's sound doesn't seem to have a very definite pitch compared to the same fundamental note played on a piano. 
*That's a different topic. A bell has a combined tone because of its non-perfect shape, this can (and in reality is always) be a combination of enharmonic notes.Houtlijm 07:37, 18 August 2007 (UTC)
thread_title: Terminology...

message: From what I have experienced all overtones up to the 7th grade are clearly harmonic to the fundamental. If you go behind the 7th overtone, the string doesn't clearly resonate anymore, with the exception of the prime positions (8/1, 9/1, ->16/1). So the main dotted harmonic serie mentioned at moodswinger is the perfect consonant serie related to the fundamental tone. The 2nd smaller line serie  (8/1 -> 16/1) mentioned at the instrument are less consonant, because the string only clearly resonates at the prime overtone. (14/1 is clear, 14/3 or 14/5 for example cannot be heard clearly). 8/3 and 8/5 are maybe also consonant, but that's shaky. I'm still considering to put those tones in the scales, because they are important tones for the minor scale. They are the Plutos of the consonant harmonic serie.YuriLandman 07:58, 18 August 2007 (UTC)
thread_title: Harmonic serie on the moodswinger...

message: I'd like to point out two things: I think everything will be much clearer if the fundamental is numbered as O, the first harmonic as 1, so that the following order ensues: 
octaves 1,2,4,8,16 etc
fifths 3,6,12,24
thirds 5,10,20
sevenths 7,14,28 etc. This order seems logical and certainly practical. Above is mentioned "the 7th degree or sixth overtone which is utterly confusing. 
Secondly, nr 11 on the staff notation is given as a g-flat which is wrong as it is a raised f. Normally, special signs such as half flats and sharps come in handy here.Martinuddin (talk) 11:30, 20 July 2008 (UTC)
replies: text: Please count again. In the current table Harmonic series (music)#Harmonics and tuning you can see that the order you claim is achieved by starting from the fundamental as 1. −Woodstone (talk) 15:05, 20 July 2008 (UTC)

text: Wow. I never noticed that pattern of multiples in the numbers. SharkD (talk) 19:18, 12 August 2008 (UTC)
thread_title: Nomenclature...

message: A while back the table of harmonic series was changed to a new format. I think the old version was easier for novices to understand. The current table has a two dimensional array in place of a table column, which could perplex readers. SharkD (talk) 19:14, 12 August 2008 (UTC)
replies: text: I'm confused by this statement. A few lines above you expressed surprise that the table brings out the pattern so beautifully. In a single column table it is not visible that easily. −Woodstone (talk) 19:23, 12 August 2008 (UTC)
replies: text: It's still harder to grasp. It was necessary for User:Martinuddin to point the pattern out to me. I did not notice it on my own, so I don't think it is much more visible. How about adding the current table minus the "Variance" column as a second table? Assuming it is noteworthy, it could be used for the purposes of highlighting this pattern. SharkD (talk) 19:53, 12 August 2008 (UTC)
thread_title: Table

message: Dicklyon, I undid your replacement of the Harmonic numbering paragraph with something true, because 1) the something true is true but redundant with the paragraphs above it, and 2) the point of the original paragraph is to expose two harmonic numbering systems (not to be confused with overtone numbering or partial numbering). I put the  tag on it yesterday, hoping someone might defend it. We don't know for sure that it's not true, just that it's un-cited, and one of the wikipedia policies is to not destroy information, so let's leave it tagged for a while and see if someone champions it. If no one does, then eventually the Harmonic numbering subsection should be removed in whole. There's no need to replace it with anything. I personally suspect that the practice of numbering harmonics by excluding the fundamental is a terminological mistake by people who confused overtones with harmonics, and not worthy of encyclopedic note, but let's wait and see. -- Another Stickler (talk) 04:19, 10 December 2008 (UTC)
replies: text: OK, we can wait on that. Dicklyon (talk) 18:49, 10 December 2008 (UTC)
thread_title:  Harmonic numbering 

message: I'm a stickler myself on statements like "although the partials are not harmonics by the strict meaning of the word if they have any inharmonicity at all."  This is an inference from a strict interpretation, and unless it is sourced it is either an editor's opinion or original research.  I don't see how it adds anything to the statement before, which is that partials are commonly called harmonics even there's a bit of inharmonicity.  The reader can be left to deal with the implied bending of the definitions, or to do further research on how strictly the terms are used in various communities; if we want to tell him, we better find out, verifiably, before doing so. Dicklyon (talk) 01:18, 14 December 2008 (UTC)
replies: text: It's the "Terminology" section. That's why it's important to make extra sure the meanings are not accidentally blurred. I'll tone it down, but the point still must be made, harmonics are a special case of partials. -- Another Stickler (talk) 01:50, 14 December 2008 (UTC)
replies: text: But you linked a dictionary that says in music a harmonic is any overtone.  We need better sources if we want to make stricter statements. Dicklyon (talk) 02:19, 14 December 2008 (UTC)
thread_title:  Strict inference and sourcing 

message: D.L., why did you remove the harmonic citation ? The majority of dictionaries listed there have at least one sense of the word that exactly corresponds to the sense in the sentence I placed the citation after, the sense that harmonics are integer multiples of the fundamental, which is why we need to be sure to distinguish from partials, of which they are a subset. -- Another Stickler (talk) 02:08, 14 December 2008 (UTC)
replies: text: I removed as I said in the edit summary because the definition that it list in the music field does not agree with what is stated in the article.  Also, it was not a reference, but an in-line external link.  If you want to put it back as a citation, it would be good to quote both the physics and music definitions and maybe modify the text to suit.  Even better, seek a music book to cite, since dictionaries are actually pretty poor sources. GBS finds no sources that mention the notion of an "ideal partial", so that should probably be reworded. Dicklyon (talk) 02:15, 14 December 2008 (UTC)

text: Here's a book in which harmonics follow from the approximation of a sound as "essentially periodic".  This seems appropriate, and is quite distinct from what you seem to have in mind, which is the notion that partials can be found rather exactly, and then compared with an ideal to see if they are harmonics.  There's no theory I know of by which the latter makes much sense, though one can estimate inharmonicity of a partial in some cases. Dicklyon (talk) 02:27, 14 December 2008 (UTC)
replies: text: You didn't read all the dictionaries did you? You stopped at Random House Unabridged Dictionary that says in music harmonics are a synonym with overtones, which is clearly wrong. You have to read all of them, then you'll see that the majority do define it as we understand it, as an integer multiple of the fundamental. That's not saying actual instruments are capable of producing exact integer multiples. Partials can be harmonic or inharmonic. There's nothing wrong with an in-line link. If you want to improve it into a citation, do so, but don't remove the inline link until you do. Don't destroy information. -- Another Stickler (talk) 16:42, 14 December 2008 (UTC)
replies: text: The only dictionary entry I read was the one you cited. Dicklyon (talk) 06:07, 15 December 2008 (UTC)
replies: text: I thought so. Wait for it to load and use the scroll bar. There are six dictionaries if I remember right. No wonder you thought it was a bad citation. Try again, then put the citation back in. -- Another Stickler (talk) 07:08, 15 December 2008 (UTC)
thread_title:  harmonic 

message: Looking for a source, I found this one that makes it clear that inharmonicity is a property of an individual partial, not of a tone.  There may be other definitions, so need to look more before changing this in the article. Dicklyon (talk) 02:13, 14 December 2008 (UTC)
replies: text: That's my understanding too. Inharmonicity refers to individual partials, not tones. -- Another Stickler (talk) 16:56, 14 December 2008 (UTC)
thread_title:  Inharmonicity 

message: Excuse me if I'm not filling this form with correct standards, but I'm new to this. My understanding of the harmonic series is that the pattern 1/2, 1/3, and 1/4 makes a statement regarding fourths and fifths relationship with octaves. That being said I didn't understand how a fifth that was only 2 cents off could translate into a fourth that was 29 cents off. So I did the math and personally found the equal temperament fourth to in fact be 2 cents sharp. Am I simply misunderstanding the table?
24.21.9.176 (talk) 22:03, 23 May 2009 (UTC)
replies: text: fast response, I'll remember to check back more consistently in the future! I've always understood the fourth and fifth as summing to an octave. I've read in numerous places that the ratio 3:2 is equivalent to the perfect fifth. Since the 1/3 that makes the 3:2 splits an octave, I was explained that you could conclude that 4:3 was a fourth since it and a perfect fifth make the octave. If you plug that into the equation you showed you get 4.98, or 2 cents flat of 5 ET semitones. The chart doesn't list that as one of the ratios however...I guess I'll reread the article, but in case you answer swiftly again where's the source for those specific intervals?24.21.9.176 (talk) 21:40, 30 May 2009 (UTC)

text: I think this points at a general problem with this section.  It seems to imply that, for example, the occurrence of the fourth in Western music has something to do with its relationship to the 21st harmonic, and that it is only the use of equal temperament that makes this inexact (The Western chromatic scale has been modified into twelve equal semitones, which is slightly out of tune with many of the harmonics).  In fact the fourth corresponds to 4:3 in just intonation, and has nothing to do with the 21st harmonic.  The same is true of most of the other intervals. 68.239.116.212 (talk) 02:27, 1 December 2009 (UTC)
thread_title:  Table's fouth inaccurate 

message: For example, a perfect fifth, say 200 and 300 Hz (cycles per second), produces a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz); that is, an octave below the lower (actual sounding) note. This 100 Hz first order combination tone then interacts with both notes of the interval to produce second order combination tones of 200 (300-100) and 100 (200-100) Hz and, of course, all further nth order combination tones are all the same, being formed from various subtraction of 100, 200, and 300.
I'm not sure what this is intended to mean.  Is it supposed to describe some psychoacoustic effect?  Certainly no such tones are actually produced (unless the sound is passed through some sort of nonlinearity). 68.239.116.212 (talk) 02:33, 1 December 2009 (UTC)
replies: text: I have no idea what that paragraph is trying to say, either.  But there really are physical combination tones produced by nonlinearities in the normally functioning cochlea, as well as psychophysical effects even with binaural presentation of two tones.  See combination tone.  Tartini noticed that you could use these to tune a fifth to an appropriate temperament, since you can hear beats between the f2-f1 and the 2f1-f2 combination tones.  For example, with 200 Hz and 301 Hz, the f2-f1 is 101 Hz and the 2f1-f2 is 99 Hz, so you hear a 2 Hz beat.  The beat is slow enough that you can rely on it to get a particular tuning of the fifthen when tuning a piano or organ (see this page for example. Dicklyon (talk) 07:10, 1 December 2009 (UTC)
thread_title:  "Combination tones" 

message: Why, what, where, and how does this article need additional citations for verification? Hyacinth (talk) 03:29, 18 October 2010 (UTC)
replies: text: Tag removed. Hyacinth (talk) 17:17, 15 July 2011 (UTC)
thread_title: Additional citations/ref improve

message: Why and where does this section need additional citations for verification? What references does it need and how should they be added? Hyacinth (talk) 00:22, 30 January 2012 (UTC)
thread_title: Timbre of musical instruments

message: the A-natural above C is a Major Sixth not a Minor Sixth. I don't know how to edit this table but it seems inaccurate.   — Preceding unsigned comment added by Shastymax (talk • contribs) 02:30, 23 July 2016 (UTC)
replies: text: The 13th harmonic is close to a neutral sixth, half way between minor and major sixths, but closer to minor (41 cents off).  Is the note name wrong in the table? No, it says A flat.  Dicklyon (talk) 02:49, 23 July 2016 (UTC)
replies: text: Yes, in the table is a wrong note name. The mathematically reliable name must be A-natural with a flattering by tridecimal comma.  Shastymax is right, because the talking is about the tridecimal version of Major Sixth  and the table requires correction. --93.76.26.31 (talk) 19:26, 3 August 2016 (UTC)
thread_title: Table again

message: Simultaneous sounding of all overtones of harmonic series forms the so-called klang 
Riemann by Shedlock 1876, p. 143:
«Since it has been known that the sounds of our musical instruments are not simple tones, but compounded of a series of simple tones which can be distinguished by a most attentive listener (but commonly are not thus distinguished), the term S., in scientific works, has been replaced by the more general, comprehensive one, C, whilst sound is applied to the simple sounds as part of the C. The height of the C[lang] is determined by the pitch of the lowest, and, as a rule, the strongest of its compound tones»
Partch 1974, p. 70:
«Harmonic Content: practically synonymous with quality of tone; the term used to indicate that characteristic of a musical tone which is determined by the distribution and comparative energy of its partials, or harmonics; the klang»
usually audible as one pitch
IEV 1994, 801-29-01:
«pitch	
that attribute of auditory sensation in terms of which sounds may be ordered on a scale extending from low to high
Note 1 — The pitch of a complex wave depends primarily upon the frequency content of the stimulus, but it also depends upon the sound pressure and the waveform.
Note 2 — The pitch of a sound may be described by the frequency of that pure tone having a specified sound pressure level that is judged by subjects to produce the same pitch»
http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-29-01
at the level of fundamental ie the lowest, first partial.
IEV 1994, 801-30-02:
«partial
sinusoidal component of a complex sound wave»
http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-30-02
The timbre of klang depends on the loudness distribution on the set of all partials and can move sensation of klang pitch on the level of second or third or some other partial in selected cases of close to zero loudness of some partial subsets.
Vertical harmonic series or klang as a kind of complex sound,
IEV 1994, 801-21-06:
«complex sound
sound that is not a simple oscillation»
http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-21-06
makes active some group of nerve fibers separated by groups of inactive fibres and the active fibers correspond to partials. The properties of of nerve fibers to conduct separately an appropriate for each partial sensation develop, probably before the birth of organism under the influence of tireless sound of mother's heart. The interaction of nerve fibers in the perception partials of several complex sounds have a significant impact on the overall assessment of pleasantness/unpleasantness of the received sensations. The interaction of nerve fibers in the perception partials of several complex sounds have a significant impact on the overall assessment of pleasantness /  unpleasantness of the received sensations. In case perception of several klangs no doubt a dependence of feeling the consonance / dissonance from the expressed by rational numbers frequency relations of fundamentals.
Datta, Sengupta, Dey and Nag 2006, p. 21:
«Іt is suggested that the perception of complex tones begins at a later stage of development of the foetus. The fluid in which a foetus is immersed transmits the throbbing of the mother's heart, activating continuously the auditory processes. The concept of partials is developed at that stage. It is an experimental fact that when musical notes, which are in simple, integral ratios, are sounded together they sound pleasant. This does not always hold true for pure tones [15]. One theory is that when some upper partials of the two tones match consonance results. In fact, Helmholtz [16] in 1862 explained the dimension of consonance in terms of the coincidence and proximity of the overtones and difference tones. These tones arise when simultaneously sounded complex tones excite real non-linear physical resonators, e.g. the human ear. To the extent that an interval's most powerful secondary tones exactly coincide, it is consonant or sweet sounding. To the extent that any of its secondary tones are separated in frequency by a small enough difference to "beat" at a rate, which he puts at around 33 c/s, it is dissonant, or harsh. Same effects are observed even when these two sounds are not simultaneous but in succession. A complex wave would give rise to a series of active groups of nerve fibres separated by groups of inactive fibres. The active fibres correspond to partials [12], Recent studies revealed prints of harmonic structures in the auditory cortex A1 [17]. The other theory of preference of pair of tones relies on the similarity of the temporal patterns of neural discharge for tones having simple frequency ratios [18]»
Important role in perception of klang playing ability of its partials collectively reproduce the sensation of pitch of missing fundamental, or some other partial from such most notable, ie low frequency partial in a given klang.
Schouten 1940, pp. 356-61:
«It is a fact that a pitch equal to that of the fundamental tone is ascribed even to those sounds in which the fundamental tone is not present <...> one or more components may be perceived which do not correspond with any individual sinusoidal oscillation, but which are a collective manifestation of some of those oscillations which are not or scarcely individually perceptible. These components (residues) have an impure, sharp tone-quality <...> the harmonics highest in frequency are perceived as a subjective component almost lowest in pitch»
Therefore, the sensation of klang pitch in general is a sensation of exact unison between pitch of taken separately fundamental and collectively expressed pitch of its residue, as the set of all partials of given klang without the fundamental, ie the lowest from partials.
\version "2.16.2"
\header {
   tagline = ##f
   title = "Some partials of almost harmonic series in A"
   subtitle = "Notation in sense of 24EDO temperament. Click to hear MIDI"
   subsubtitle = "Schouten's residue is located in accolade of the upper voices. In the bass — fundamental"
}
melody = 
{
\clef bass \key a \major
\time 4/1
\set Staff.midiInstrument = "flute"
{
\override NoteHead.font-size = #2 \override NoteHead.style = #'harmonic
a,1_\markup{\column{\italic{A} Loudly audible pitch}}
  r1^\markup{\column{Well audible pitch \italic{A}}}
    a,1_\markup{\column{\italic{A} Loudly audible pitch}}
      r1^\markup{\column{Well audible pitch \italic{A}}}
a,1_\markup{\column{\italic{A} Quietly audible pitch}}
  r1^\markup{\column{Not audible pitch \italic{A}}}
    a,1_\markup{\column{\italic{A} Quietly audible pitch}}
      r1^\markup{\column{Not audible pitch \italic{A}}}
a,1_\markup{\column{\italic{A} Loudly audible pitch}}
  r1^\markup{\column{Well audible pitch \italic{A}}}
    a,1_\markup{\column{\italic{A} Loudly audible pitch}}
      r1^\markup{\column{Well audible pitch \italic{A}}}
a,1_\markup{\column{\italic{A} Quietly audible pitch}}
  r1^\markup{\column{Not audible pitch \italic{A}}}
    a,1_\markup{\column{\italic{A} Quietly audible pitch}}
      r1^\markup{\column{Not audible pitch \italic{A}}}
        \bar "|."
}}
text = \lyricmode { }
\score {
<<
\new PianoStaff
<<
\new Staff
{
\clef "treble^15" \key a \major
\set Staff.midiInstrument = "piccolo"
{
\override NoteHead.font-size = #2 \override NoteHead.style = #'harmonic
<a''''
 c''''
 b'''
 ais!
 e>1^\markup{\column{
\line{
Example that supports the statement of Schouten about ability of \italic{residue}  
}\line{
to collectively express the pitch of fundamental even if the last is no
}}}
       ~<a''''
         c''''
         b'''
         ais'''
         e>1~<a'
                 c''''
                 b'''
                 ais'''
                 e>1~<a' 
                         c''''
                         b'''
                         ais'''
                         e>1
r1
  r1
    r1
      r1
<a''''
 c''''
 b'''
 ais!
 e>1~<a'
         c''''
         b'''
         ais'''
         e>1~<a'
                 c''''
                 b'''
                 ais'''
                 e>1~<a' 
                         c''''
                         b'''
                         ais'''
                         e>1
r1
  r1
    r1
      r1
}}
\new Staff
{
\clef "treble^15" \key a \major
\set Staff.midiInstrument = "piccolo"
{
\override NoteHead.font-size = #2 \override NoteHead.style = #'harmonic
<aeh' 
 dih>1~<aeh' 
           dih>1~<aeh'
                     dih>1~<aeh'
                               dih>1
r1
  r1
    r1
      r1
<aeh' 
 dih>1~<aeh' 
           dih>1~<aeh'
                     dih>1~<aeh'
                               dih>1
r1
  r1
    r1
      r1
}}
\new Staff
{
\clef treble \key a \major
\set Staff.midiInstrument = "piccolo"
{
\override NoteHead.font-size = #2 \override NoteHead.style = #'harmonic
1~1~1~1
r1
  r1
    r1
      r1
1~1~1~1
r1
  r1
    r1
      r1
}}
>>
\new Voice = "mel" \melody
\new Lyrics \lyricsto mel \text
>>
\layout {
indent = #0
line-width = #180
}
\midi { }
}
Overtone as partial tone, in opposed to partial, is not the single pure tone,
IEV 1994, 801-21-05:
«pure sound
pure tone
sinusoidal acoustic oscillation»
http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-21-05
but a complex sound, which formed by specific subset of partials of any klang, ie klang in klang, as stated in a detailed description
Тюлин 1966, pp. 41-3:
«each overtone <...> has its own natural skala in the number of overtones [of klang with pitch] of fundamental tone()»
with musical example:
\version "2.16.2"
\header
{
 tagline = ##f
 piece = "Тюлин 1966: 42"
}
{ 
\new PianoStaff 
<<
 \new Staff \with { \remove "Time_signature_engraver" } \relative c''
 {
 \clef treble \key c \major \time 10/1
 <<
 {
 <c b! a f d>1
  <e c>1
 } \\
 {
 <bes g e c>1_\markup { \bold 8 }^\markup { \bold 16 }^\markup { \bold ↑ }
  <d bes g e c>1_\markup { \bold 8 }^\markup { \bold 20 }^\markup { \bold ↑ }
   1_\markup { \bold 9 }^\markup { \bold 30 }^\markup { \bold ↑ }
    <c bes g e c g c,>1_\markup { \bold 8 }^\markup { \bold 32 }^\markup { \bold ↑ }
     <e d b gis e b e,>1_\markup { \bold 10 }^\markup { \bold 40 }^\markup { \bold ↑ }
      <f d b g d g,>1_\markup { \bold 12 }^\markup { \bold 42 }^\markup { \bold ↑ }
       <f d bes f bes,>1_\markup { \bold 14 }^\markup { \bold 42 }^\markup { \bold ↑ }
        <g e c g c,>1_\markup { \bold 16 }^\markup { \bold 48 }^\markup { \bold ↑ }
         <a fis d a d,>1_\markup { \bold 18 }^\markup { \bold 54 }^\markup { \bold ↑ }
          1_\markup { \bold 20 }^\markup { \bold 60 }^\markup { \bold ↑ }
 }
 >>
 }
 \new Staff \with { \remove "Time_signature_engraver" } \relative c'
 {
 \clef bass \key c \major \time 10/1
 <bes g e c g c, c,>1_\markup { \bold 1 }^\markup { \bold 7 }
  <g c, c,>1_\markup { \bold 2 }^\markup { \bold 6 }
   <g g,>1_\markup { \bold 3 }^\markup { \bold 6 }
    c,1_\markup { \bold 4 }
     e1_\markup { \bold 5 }
      g1_\markup { \bold 6 }
       bes!1_\markup { \bold 7 }
        c1_\markup { \bold 8 }
         d1_\markup { \bold 9 }
          e1_\markup { \bold 10 }
 \bar "|"
 }
>>
}
The score clearly show an essence that with involvement of alphabetic notation of Helmholtz looks more clear perhaps as math expression:
\left\{\begin{matrix}
:\\
~ ~~c^3{:}[16]\\
~ ~~b^2{:}[15]\\
bes^2{:}[14]\\
~ ~~a^2{:}[13]\\
~ ~~g^2{:}[12]\\
~ ~~f^2{:}[11]\\
~ ~~e^2{:}[10]\\
~ ~~d^2{:}[9]\\
~ ~~c^2{:}[8]\\
bes^1{:}[7]\\
~ ~~g^1{:}[6]\\
~ ~~e^1{:}[5]\\
~ ~~c^1{:}[4]\\
~~ ~~g{:}[3]\\
~~ ~~c{:}[2]\\
~~ ~~C{:}[1]\\
\end{matrix}\right\}\supseteq
\left\{\begin{matrix}
:\\
~ ~~c^3{:}[16]\\
~ ~~b^2{:}[15]\\
bes^2{:}[14]\\
~ ~~a^2{:}[13]\\
~ ~~g^2{:}[12]\\
~ ~~f^2{:}[11]\\
~ ~~e^2{:}[10]\\
~ ~~d^2{:}[9]\\
~ ~~c^2{:}[8]\\
bes^1{:}[7]\\
~ ~~g^1{:}[6]\\
~ ~~e^1{:}[5]\\
~ ~~c^1{:}[4]\\
~~ ~~g{:}[3]\\
~~ ~~c{:}[2]\\
~~ ~~C{:}[1]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~e^3{:}[20]\\
~ ~~d^3{:}[18]\\
~ ~~c^3{:}[16]\\
bes^2{:}[14]\\
~ ~~g^2{:}[12]\\
~ ~~e^2{:}[10]\\
~ ~~c^2{:}[8]\\
~ ~~g^1{:}[6]\\
~ ~~c^1{:}[4]\\
~~ ~~c{:}[2]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~b^3{:}[30]\\
~ ~~a^3{:}[27]\\
~ ~~g^3{:}[24]\\
~ ~~f^3{:}[21]\\
~ ~~d^3{:}[18]\\
~ ~~b^2{:}[15]\\
~ ~~g^2{:}[12]\\
~ ~~d^2{:}[9]\\
~ ~~g^1{:}[6]\\
~~ ~~g{:}[3]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~c^4{:}[32]\\
bes^3{:}[28]\\
~ ~~g^3{:}[24]\\
~ ~~e^3{:}[20]\\
~ ~~c^3{:}[16]\\
~ ~~g^2{:}[12]\\
~ ~~c^2{:}[8]\\
~ ~~c^1{:}[4]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~e^4{:}[40]\\
~ ~~d^4{:}[35]\\
~ ~~b^3{:}[30]\\
gis^3{:}[25]\\
~ ~~e^3{:}[20]\\
~ ~~b^2{:}[15]\\
~ ~~e^2{:}[10]\\
~ ~~e^1{:}[5]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~f^4{:}[42]\\
~ ~~d^4{:}[36]\\
~ ~~b^3{:}[30]\\
~ ~~g^3{:}[24]\\
~ ~~d^3{:}[18]\\
~ ~~g^2{:}[12]\\
~ ~~g^1{:}[6]\\
\end{matrix}\right\}\cup
\cup
\left\{\begin{matrix}
:\\
~ ~~f^4{:}[42]\\
~ ~~d^4{:}[35]\\
bes^3{:}[28]\\
~ ~~f^3{:}[21]\\
bes^2{:}[14]\\
bes^1{:}[7]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~g^4{:}[48]\\
~ ~~e^4{:}[40]\\
~ ~~c^4{:}[32]\\
~ ~~g^3{:}[24]\\
~ ~~c^3{:}[16]\\
~ ~~c^2{:}[8]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~a^4{:}[54]\\
fis^4{:}[45]\\
~ ~~d^4{:}[36]\\
~ ~~a^3{:}[27]\\
~ ~~d^3{:}[18]\\
~ ~~d^2{:}[9]\\
\end{matrix}\right\}\cup
\left\{\begin{matrix}
:\\
~ ~~b^4{:}[60]\\
gis^4{:}[50]\\
~ ~~e^4{:}[40]\\
~ ~~b^3{:}[30]\\
~ ~~e^3{:}[20]\\
~ ~~e^2{:}[10]\\
\end{matrix}\right\}\cup \cdots
Since any given klang always contains other klangs, it definitely exists as klang in the own compound and can be klang in compound of other klang with pitch always perceived as lower and always corresponds to the frequency of the stimulus that Integer multiple of stimulus frequency with sensation of given klang pitch. In other words every klang is an overtone of some undertone,
Helmholtz 1865, p. 76:
«It's easy to make sure using the tests in these properties resonators. You hold the such to ear and let play a polyphonic piece of music any instrumentswhere often occurs the natural tone of the resonator. So often as this tone is given, armed resonator ear hears its penetration through all other sounds of the chord.
Weaker, but often heard such marked in lower klangs, namely as showing the first detailed study of this when klang given with one of harmonic overtones at own tone of your resonator. These lower klangs are called harmonic undertones of resonator tone. These are klangs, whose period of oscillations in 2, 3, 4, 5 etc. times more than the resonator tone. If such is, for example, c, you can hear its sound when a musical instrument playing: c', F, C, As, F, D, C~~etc. In these cases resonator sounds from one of overtones in external airspace of the specified klang. But it should be noted that klangs of certain instruments do not always contain all harmonic overtones, and that they have a different power. In the sound of the violin, piano and harmonium first 5 or 6 most clearly present. On overtones of strings followed a more detailed description in the next chapter
()»
including case when klang is the first overtone of its first undertone, ie of itself.
The same ip range as before has now created this piece of gibberish. Although it is marginally better sourced than the previous "Musical Notation" section, it is still of very poor quality. The English is near-incomprehensible and the "mathematical" notation is highly non-standard. I have thus taken the liberty of removing it. SBareSSomErMig (talk) 11:48, 12 October 2016 (UTC)
replies: text: Elementary operations of set theory can not be "mathematical" notation is highly non-standard. This is a weak excuse to remove the entire section. Another excuse (The English is near-incomprehensible) is also far-fetched. In the section there is a tag for the language experts calling to take part in improving the text. The section has a right to exist in the article and thus was restored. --46.201.181.89 (talk) 12:58, 12 October 2016 (UTC)
replies: text:  Qouting from the section: "The score clearly show an essence that with involvement of alphabetic notation of Helmholtz looks more clear perhaps as math expression". I don't know what that is supposed to mean, and I don't think a "language expert" would either. The first, and maybe even the second paragraph might be relevant to the article after some some considerable sandbox work (at present it is barely readable and reads like a textbook). The rest of the section is plain nonsense and looks like the same original research that we previously reached consensus for removing. The set expression is a very poor communication of the idea that it tries to convey (indeed, as written, it is completely vacuous). SBareSSomErMig (talk) 13:20, 12 October 2016 (UTC)
replies: text:  Leave alone the material, for you personally (well and else for such within half percent out of daily readers) not understandable, as it turns out. The harmonic series is a natural system, and the systems are described by language of set theory, which has long and successfully implemented in general educational process. If it frightens you personally, it does not mean that out of 400 readers per day it frightens everyone. Frightening only you signs are recovered with small addition. Let at least 40 different readers will explain here why these inscriptions are so offensive that must be removed. May be then will happens consensus... --93.76.25.155 (talk) 23:36, 12 October 2016 (UTC)
replies: text: Personally attacking another editor is not a persuasive tactic. Linking to a google search result is not convincing either. Without a link to a reliable source using this infinitely extended union of sets in an acoustical or music-theoretical context, that section is original research, which does not belong on Wikipedia. Just plain Bill (talk) 00:00, 13 October 2016 (UTC)
thread_title:  At it again 

message: Several of these harmonics do not match their counterparts an octave lower. How can this be correct? For the octaves to match each other:
* 26 should be A, same as 13
* 28 should be B, same as 14
* 30 should be B, same as 15
Burninthruthesky (talk) 09:34, 21 March 2017 (UTC)
thread_title:  Johnston's notation 

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thread_title:  External links modified 

message: The article's "Terminology" section states that "An overtone is any partial above the lowest partial"
This is technically incorrect -- it implies that the lowest overtone is the fundamental, which is untrue. According to this definition, ANY partial (other than the lowest one) is an overtone.
I believe it should read, "An overtone is any partial above the FUNDAMENTAL."  — Preceding unsigned comment added by 98.109.98.74 (talk) 03:27, 5 July 2018 (UTC)
thread_title:  Definition of "overtone" inaccuracy 

message: what is harmonic singing 102.89.3.217 (talk) 18:20, 18 January 2022 (UTC)
thread_title:  harmonic  singing (music) 

message: The section "Harmonics and tuning" currently says, "...small whole-numbered ratios are likely the basis of the consonance of musical intervals." The closest thing it has to a supporting citation is a reference to Hindemith's The Craft of Musical Composition, Book 1, pg. 15. I have that book on hand and I can't find a statement quite like that; the closest he comes that I've found is his statement that "[the ear] hears simple ratios as beautiful and correct sounds" in Ch. 2, Sec. 4, "The Triad." That's not nearly as strong a statement as "basis of consonance," and considering that he basis the whole discssion in that section first and foremost on the actual overtone series I don't think he means to imply that either.
Even if he had said that, though, it would just be his opinion (he doesn't offer any further evidence even for "simple ratios" being "beautiful" except to suggest that the reader go listen for themselves). Research on human subjects has been done which shows that the perception of dissonance between two pure tones is typically greatest when their interval is 25% of the critical band in that area of the spectrum, and the perception of consonance is greatest when their interval is 100% of the critical band. Perceptions of dissonance and consonance between complex pitched sounds is the result of the ratios and loudnesses of their partials on that basis. (See Plomp and Levelt's Tonal Consonance and Critical Bandwidth.) Models for predicting and quantifying the consonance of a set of notes together have been developed  from this that work quite well; William Sethares provides such a model (with source code for a computer program) based on Plomp and Levelt in Tuning, Timbre, Spectrum, Scale.
If this kind of material was accounted for in that section it would change the presentation considerably. It's worth remembering that some quite-dissonant intervals correspond to small whole-number ratios (7/4, 9/8) and some intervals that are widely considered more consonant correspond to ratios with larger whole numbers (19/16, 21/16, 25/16, 27/16). However, evaluating their consonance based on critical bands and acoustic beating as described above accounts for this. It's true that many consononant intervals do correspond to small whole-number ratios, and it's not a coincidence, but it doesn't tell the whole story just to say that.
Of course, an in-depth exploration of this topic probably belongs better in Consonance and dissonance, so to some extent there's not much reason to go into it very far here. Given all this, I sort of think that if we're going to talk about tuning in this article, it might make more sense to discuss things like  spectralism than just comparing the "harmonic intervals" to 12-TET and briefly mentioning just intonation.  Most popular forms of just intonation (e.g. 5-limit) aren't perfectly in-tune with the harmonic series any more than 12-TET is, but there are composers out there who write music explicitly based on the harmonic series, or on one instrument's overtones or the like. I don't think it would hurt to mention the bugle on that basis, speaking of, and it would probably also make sense to take about things like choirs and string ensembles instinctively tuning their chords harmonically. I do find the chart there really useful, but not so much the parts relating to 12-TET; I like the column giving the best-fit diatonic interval, because it helps you get an idea of the sound if you have that background, but those interval names are not specific to 12-TET (they're just diatonic). Mesocarp (talk) 07:34, 29 November 2022 (UTC)
thread_title:  Speculation about origins of consonance not supported by research