Talk:Undertone series

Missing references

...the undertone series is considered by many to be purely theoretical.‹The template Talkfact is being considered for merging.› 

I'm pretty sure Paul Hindemith makes this argument, so I'll probably have a reference when my university's music library opens tomorrow.

When describing the overtone series, it is conventional to start with C.‹The template Talkfact is being considered for merging.› 

This is pretty trivial, so we might want to just leave it out. The starting pitch doesn't matter.

During the experiments with temperament and tunings systems in the 1500s and onward, the discovery of the overtone series largely contributed to the rise in popularity of equal temperament and the major triad.‹The template Talkfact is being considered for merging.› 

This one doesn't even make sense to me. The overtone series would lead people to use just intonation, not equal temperament, right? This one definitely needs a source or it has to go.

Many people in this time used this as proof that equal temperament should be used because the physics of sound implied it with the major triad.‹The template Talkfact is being considered for merging.› 

Ditto for this one. How in the world is it a "proof that equal temperament should be used"? —Keenan Pepper 23:52, 2 September 2006 (UTC)

• Alright, it was definitely a bit stupid of me to put down things I was not completely sure of yet (that's what makes Wikipedia inaccurate often times) and so I will NEVER again do that. So my apologies. Now, as to what I was referring to. The book where I recall reading about what I vaguely described above and confused you with it was Stuart Isacoff's book called Temperament. I can't quite remember the exact arguement he makes in there but I will get ahold of that book as soon as I can and find it. I left my copy at my house so until I go back there I'll look for it in the university's music library. Thanks for your editing. SN122787 15:34, 3 September 2006 (UTC)
  • I have that book right here and it seems to say the opposite. On pages 179-180 it describes string harmonics, overtones, and beating, and then the next paragraph starts "All this was bad news for advocates of equal temperament." It doesn't have much to do with the undertone series anyway, so for now I think we must remove it from the article. —Keenan Pepper 19:06, 3 September 2006 (UTC)
    • Alright, thanks a lot! It looks much better now. I'm going to get rid of the phrase with starting on C. I also agree that it is useless. Sorry about all this, but then again it is my first article. My next ones will be better and I'll be sure to cite sources and everything as well. SN122787 04:42, 4 September 2006 (UTC)
      • No problem! I'm sure it's a lot better than my first article was. (Hmm, what actually was my first article?...) —Keenan Pepper 05:04, 4 September 2006 (UTC)

POV much?

I mean, if at least one expert in the field considers undertones "purely theoretical", isn't a bit POV to say they are naturally occurring? Of course, I assume good faith--perhaps this doesn't say what I think it does. But if the actual existence of undertones (beyond mathematical proofs) is contested in the academic community, I think both sides need to be given equal weight.

-- trlkly 07:42, 19 July 2007 (UTC)

It isn't a contested issue: undertones don't occur physically in the same way that overtones do. But the page does a very bad job of explaining what exactly is going on here. Mathematically, of course, you can describe a set of frequencies related to each other by the ratios of the harmonic series, and depending on whether you multiply or divide from your starting note, you'll get either the overtone or undertone series -- simply as a list of pitches. But the difference is that musical instruments (those capable of producing tones--that is, not percussion instruments) produce complex tones that are basically sums of the overtone series. A single middle C played by the piano is actually sounding the frequencies of {C4, C5, G5, C6, E6,...} all at once. The same is not true of the undertone series: playing a middle C doesn't also produce the set of frequencies of {C4, C3, F2, C2, A-flat1,...}. That's why people usually say that the overtone series is "naturally occurring" but the undertone series isn't. Of course you can define the undertone series and then play it as a sequence of notes (which is what the article currently describes doing), but you could do that with any series you can define mathematically.

The article reads like a summary of this book by Graham Jackson (which, incidentally, looks to be basically self-published and not exactly notable in the worlds of academic music theory or acoustics); I'd try to clean it up if I had time, but this isn't really my area of specialty anyway.Masily box (talk) 00:27, 1 December 2008 (UTC)

I agree that being useful in music theory is not the same as occurring in nature. I placed the citation-needed tag in that sentence. -- Another Stickler (talk) 21:36, 9 December 2008 (UTC)

Well, I'd contest that it's even "useful" in music theory, but I suppose the point about a flute with equidistant holes is at least of some interest. Thanks for all the work digging up sources on it not being naturally occurring.Masily box (talk) 14:23, 29 December 2008 (UTC)

undertone series not naturally occurring

"While the overtone series, 1,2,3,4,..., is manifested, its reciprocal series, the undertone series, 1/2, 1/3, 1/4,..., is a kind of phantom in that it is implied by the first series but not present in naturally occurring sound." -- Jay Kappraff, "Beyond Measure" page 97. [1] -- Another Stickler (talk) 18:53, 10 December 2008 (UTC)

"However, this undertone series, though it can be produced by increasing string lengths, does not sound in outer, audible nature in the same way as the overtone series." -- Richard Bunzl, from "New View" magazine, Spring 2007, London, U.K., reviewing "The Spiritual Basis of Musical Harmony" (ISBN 978-1-55246-760-2) by Graham H. Jackson [2] -- Another Stickler (talk) 21:05, 10 December 2008 (UTC)

"In [Hugo] Riemann's earlier works he explained the nature of the minor triad in terms of the undertone series (frequencies = 1, 1/2, 1/3, 1/4, 1/5 etc.), which he claimed was a natural physical phenomenon like the overtone series — which it is not. But in his later works he tends to explain the difference between major and minor as a basic aesthetic principle of symmetry, although related to physical phenomena." -- page 195 "Layers of Musical Meaning" By Finn Egeland Hansen, Published by Museum Tusculanum Press, 2006, ISBN 8763504243, 9788763504249 333 pages [3] -- Another Stickler (talk) 22:16, 10 December 2008 (UTC)

Here's Alexander J. Ellis complimenting Hugo Riemann for dropping his claim after further study: "Latterly Dr. Hugo Riemann has given in his adhesion to this view, and in his lately published Musical Syntaxis has attempted to examine and establish the consequences of this system by examples from acknowledged composers. The application of this critical method appears to me very commendable, and to be the indispensable condition to advancing in the theory of composition." -- Alexander J. Ellis (translator), page 365, "On the Sensations of Tone" , by Hermann Helmholtz, Contributor Henry Margenau, Published by Courier Dover Publications, 1954, ISBN 0486607534, 9780486607535, 576 pages [longer title in other editions "On the Sensations of Tone as a Physiological Basis for the Theory of Music"] -- Another Stickler (talk) 22:23, 10 December 2008 (UTC)

Check the posting on subharmonics. They are essentially the same as undertones and as that posting points out can be created, though not occuring normally — Preceding unsigned comment added by Cboody (talk • contribs) 20:45, 25 April 2010‎

I think that it is not quite correct to say that undertones do not appear naturally. I think that sinusoidal excitation can produce undertones when fed into a non-linear resonator. I'm not sure but I think that AM radio is an example: aren't the audible (low-frequency) pitch and overtones we hear the result of a much higher frequency radio signal being fed through a non-linear amplifier (one containing diodes)? An example in the article of undertones is a string vibrating against an obstacle. This is an example of a non-linear system (the paper, analagous to the diodes in a radio) intermittently and suddenly colliding with the periodic oscillator (the string), isn't it?

Mcamp@cinci.rr.com (talk) 00:20, 3 October 2014 (UTC)

Cboody and Mcamp@cinci.rr.com, you folks are missing the point. Something close to the harmonic series occurs naturally in many, many acoustic bodies. The so-called subharmonic series never does. The fact higher and lower tones can arise as artifacts of distortion in measuring apparatus including the ear does not change that. 38.86.48.38 (talk) 02:14, 2 June 2015 (UTC)

The subharmonic series DOES occur naturally in acoustics; this isn't really disputable. It's just not terribly common or terribly loud in the most popular sorts of Western instruments. A large portion of this page is pointing out that it does occur naturally in things sounding boards, which really ought to be uncontroversial. Basically any acoustic technique which generate secondary tones of lower pitch than the main tone, and there are many of them, generates the subharmonics; it's related to echoes. In physics terms, you generally get them with *forced* oscillators (rather than the unforced oscillators which typically generate the harmonics). I may have to dig the stuff about this out of books on acoustics in physics, since the musical world seems to be very confused about the matter. 67.249.141.66 (talk) 03:08, 16 June 2015 (UTC)

Undertones and uppertones

I came to read this article almost by accident, because a link to it has been added in Riemannian theory. It seems to me that there is a confusion, both in the article itself and in the comments above, between two related but somehow independent phenomena.

The first is the phenomenon of harmonic partials, those that arise when producing one single complex periodic tone. The very fact that this tone is called "complex" denotes that it consists in partial tones, as suggested by Fourier's theorem. Partial tones can only be equal or higher in frequency as the complex tone itself. It is a matter of a whole and its parts: no part of a whole can exceed the whole itself; or a matter of boundary conditions: nothing falling under boundary conditions can have boundaries outside those considered. Lower harmonic partials ("undertones") are impossible both from a logical and from a physical point of view. And if Riemann thought otherwise, it only is because he at first only had a confused idea of what harmonic partials were; he eventually recognized his mistake in this respect.

The second phenomenon is the production of sounds described as "subharmonics" by a variety of means, e.g. by specific techniques of violin playing beyond the ordinary limits of vibration. Such sounds always results from distortions creating abnormal sound productions. The article mentions them, but without enough stressing the distortive aspect. As it states, "there are '...only fairly restricted conditions' that will lead to the 'nonlinear phenomenon known as subharmonic generation'", the important word being "nonlinear". Differential tones, for instance (Tartini tones) result from nonlinearities. And the production of "undertones" on bowed instruments involves a high pressure of the bow forcing the string outside its normal modes of vibration. The article puzzingly indicates that the production of such sounds "has nothing to do with the undertone series", probably because by "undertone series" it still expresses a belief in the possibility of "subharmonic partials".

It has for a while been thought, I think to remember, that even harmonic partials resulted from nonlinearities: they were described as "subjective", because one believed that they arose in the nonlinearities of individual ears. If this were so, two different hearers would have heard them differently, and they might not have existed outside individual ears. It has been proved early in the 18th century, however, that harmonic partials are "objective" and that they exist even in the absence of any hearing subject.

Nonlinearities on the other hand, as needed for the production of undertones, are not "objective" in the same way. They may exist independently of human ears and of hearers, but they would still belong to specific conditions – the acoustics of a given space, for instance, within which the sound might be distorted up to giving an impression of undertones.

The article is not too bad in making this difference between overtones and undertones. It induces some doubt, however, when it says that "The overtone series can be produced physically in two ways—either by overblowing a wind instrument, or by dividing a monochord string." In this it forgets a third (or the main) way or producing the overtone series, by producing any periodic sound with a rich timbre.

I don't know whether anything can be done about this. The article also shows that a belief in the quasi natural existence of undertones is still very much among us. I trust that convincing references could be found to explain that harmonic partials are utterly different from undertones. This is so obvious to me that I am afraid I won't bother to seek these references ... — Hucbald.SaintAmand (talk) 21:35, 14 November 2019 (UTC)

Triads

In the "Triads" section, the following is stated: "Sotorrio, however, said that since this minor triad is not built under the fundamental (C) but under the fifth below (F), major and minor tonalities cannot be said to grow out of this 'polarity'.[3]" This really makes no sense, as the minor chord occuring in the C undertone series actually is built "under" the fundamental (C), thus forming an F minor triad. It would be more logical if he had said "built ON the fundamental (C)" which is, in fact, not the case. It is built on F (but "under" C, so to speak). Anyone agree or am I completely wrong because of complete confusion? Thanks for any comments! --kzuse (talk) 12:05, 2 February 2020 (UTC)

This statement indeed makes no sense – A♭ is the major third below C, not below F – and the Sotorrio "book" is nowhere documented (Google does not give a single answer). I remove the whole from the article. — Hucbald.SaintAmand (talk) 14:22, 2 February 2020 (UTC)

I discovered that there is another citation of this Sotorrio "book", again without page number. It reads as follows:

In addition, undertones can be made through the use of a simple oscillator such as a tuning fork. If the oscillator is gently forced to vibrate against a sheet of paper "it will naturally make contact at various audible modes of vibration." Because the tuning fork produces a sine tone, it normally vibrates at its fundamental frequency (e.g. 440 Hz), but "momentarily", it will make contact only at every other oscillation (220 Hz), or at every third oscillation (147 Hz), and so on. This produces audible "subharmonic spectra", (e.g. 1:1 yields A4 (440 Hz), 1:2 yields A3 (220 Hz), 1:3 yields D3 (147 Hz), 1:4 yields A2 (110 Hz), 1:5 yields F2 (88 Hz), etc.). José Sotorrio claims it is possible to sustain these "subspectra" using a sine wave generator through a speaker cone making contact with a flexible (flappable) surface, and also on string instruments "through skillful manipulation of the bow", but that this rarely sustains noticeably beyond the "sub-octave or twelfth".

This again is a misconception. What happens with the tuning fork hitting a suspended sheet of paper is that if the sheet is pushed away by the first vibration of the fork and does not flap back in time for the second, it may be excited by only one vibration on two and vibrate at half the frequency; if it takes more than that to come back, it may be excited by only one vibration on three, etc. That is that the fork will act as some sort of intermittent bow; but the resulting vibration of the sheet certainly does not produce a "spectrum" of any kind.

Henri Bouasse describes a similar phenomenon somewhere, when he describes how one can make a church bell mounted on a wheel sound by pulling its rope: after a first pull of the rope, the wheel-bell ensemble oscillates at its own frequency. One may pull the rope again either at each oscillation, or at one on two, or one on three, etc., because it only is necessary to entertain the movement. The pulls must be synchronized with the oscillations, otherwise one would stop everything by pulling against the movement of the wheel. The pulses on the rope therefore must happen at entire multiples of the wheel oscillation – that is have a frequency submultiple of the wheel's own frequency. This might be described as "subharmonic", but certainly neither as subharmonic partials nor as subharmonic spectrum.

I suggest therefore to remove this citation completely: it is hardly understandable, and cannot be traced back to any traceable book. But as usual, I'll leave a few days for possible reactions before going to this "extremity". — Hucbald.SaintAmand (talk) 10:13, 16 February 2020 (UTC)

I removed the "Sotorrio" claim mentioned above. But I discovered another claim that hardly seems to make sense, footnote a, which says:

The statement that minor chords are not explained by overtones is wrong: In just intonation the minor triad is exactly the 10th, 12th, and 15th overtones (10:12:15); similar to the major triad 4th 5th and 6th overtones (4:5:6). If the "lost" 7th chord is taken to be 4:5:6:7 (the interval ​7⁄4 is the "lost" tone), then the minor third is a mis-tuned approximation of 5:6:7.

The minor chord is not explained by the 10th, 12th, and 15th overtones, but by the fact that all its intervals (fifth, minor third, and major third as difference between the fifth and the minor third) are consonances. One may express that saying that the 15th overtone of the fundamental, the 12th overtone of its third and the 10th overtone of its fifths are in unison, as are their multiples. The same is true of the major chord, which is not made of the 4th, 4th and 6th overtones [of its fundamental?]. But at least the article does not claim that elsewhere as forcefully as it does here.

I have no idea of what the "lost chord" is, nor of what the "lost tone" is, nor how this tone could be an interval, as the above statement says. Once again, I intend to remove all that, if there are no objections. — Hucbald.SaintAmand (talk) 20:40, 9 May 2020 (UTC)

As announced, I deleted footnote a. — Hucbald.SaintAmand (talk) 09:55, 21 May 2020 (UTC)

Resonance?

The subsection "Resonance" of this article includes a puzzling statement:

Helmholtz argued that sympathetic resonance is at least as active in under partials as in over partials.

Helmholtz indeed may seem to say that. He writes more precisely:

To make experiments [...] on the sympathetic vibrations of strings, [...] press down the key of the string (for c ' suppose) which you wish to put into sympathetic vibration. [...] The motion [of the c ' string] is greatest when one of the under tones of c ' is struck [...]. Some, but much less, motion also occurs when one of the upper partial tones of c' is struck [...]

But what does it mean to say that "sympathetic resonance is active in the partials"? Is that really what Helmholtz says? I don't think so. Sympathetic resonance, in all cases, occurs in the c ' string and there only. If the note struck is one of its undertones, then one of the harmonic partials of that undertone is in unison with the c ' string and causes it to vibrate by sympathy. If the note struck is one of the upper tones, then the corresponing upper harmonic partial of the c ' string is set in sympathetic vibration with it, but that effect is much less than in the first case because here only an upper partial vibrates.

By "under tone" of a note, Helmholtz means the tones of which the note considered is an upper harmonic partial, and certainly not a "lower harmonic partial" of this note (Helmholtz merely does not believe in lower harmonic partials). But that does not seem to be the meaning of the title of our article, Undertone series or, at least, the article is ambiguous on this point. The general definition in the lead, "a sequence of notes that results from inverting the intervals of the overtone series," might describe undertones as Helmholtz understands them, as resulting from a mere inversion of intervals. (What is meant by that is not entirely clear.) But the article soon raises the question whether these undertones could be subharmonics of some kind.

Our article is about undertones: Helmholtz' quotation is therefore justified, if its particular meaning is more clearly explained. What is not justified, on the other hand, is that it appears under the heading "Resonance," which has nothing to do with our article (and is not at all mentioned in the Cowell book quoted after). And this particular meaning, which implies that undertones are no subharmonics, redounds on the article as a whole, which might perhaps make this distinction clearer from the start.

I really don't know what to do. Advices would be welcome. — Hucbald.SaintAmand (talk) 12:18, 30 December 2021 (UTC)

Cowell and Garbusov

I had a closer look at what Henry Cowell has to say of this otherwise unknown Nicolas Garbusov, who would have built an instrument producing undertones. He writes this:

Professor Garbusov shows that such resonating chambers [e.g. the violin "wooden sound-chamber"], under certain circumstances, respond only to every other vibration of the original sounding body, or that some part of the resonator will respond only to every other vibration, even if part of the resonator responds normally to every vibration. The part of the resonator which responds only to every other vibration is then vibrating at one-half the speed of the fundamental and produces a tone one octave lower. Under other circumstances a part of the resonator will vibrate at one-third the speed of the fundamental, thus producing a tone one octave and a fifth below that of the original sounding body. Through an extention of this principle, a series of tones is formed downwards from the fundamental which contains the same relative intervals as the overtone series, counting downwards instead of upwards, and which produces a different actual set of tones.

It is however improper to call the body of a violin (or of other similar instruments) a "resonator" or a "resonating chamber". Such bodies, on the contrary, are built so as not to resonate because that would undily reinforce some notes of the instrument. It is difficult to imagine what Garbusov (or Cowell) refers to when speaking of "some part of the resonator" responding to only some of the vibrations. This may remind the experiment where a tuning fork hits a sheet of paper which is pushed away and comes back to touching the tuning fork only at one vibration on two: the sheet then vibrates at half the frequency of the tuning fork. But it would be utterly impossible to produce a series of undertones in this manner: the sheet cannot vibrate at several frequencies at the same time.

Similarly, it seems utterly improbable that Garbusov's "resonator", or different parts of it, could vibrate at a series of undertones. Cowell continues in the obvious conviction that undertones are subharmonics. He writes:

just as the first four overtones form a major triad, the first four undertones form a minor triad, the necessary contrast to the major. Chords formed from the undertones are generated downwards; therefore their roots are really the top notes!

This apppears similar to what Hugo Riemann believed, in his "dualist" conception, until one shew him that he was wrong. But once again, I don't know what to do. I think more and more that the article needs a thorough revision. — Hucbald.SaintAmand (talk) 18:29, 30 December 2021 (UTC)

"Y-shaped strings"?

"The tritare, a guitar with 'Y' shaped strings" ... I've been playing and teaching guitar for 40 years and I have never seen, or heard of a "Y-shaped" guitar string. Is the cross-sectional area y-hshaped? Is the string split into three branches that instersect at a point? Are multiple strings tied together?

This phrase needs some explanation or perhaps an illustration. — Preceding unsigned comment added by 74.95.43.253 (talk) 23:27, 18 March 2022 (UTC)

Another way to think about the undertone series

Each note in the undertone series has the original note as their common harmonic

in mathematical terms, the frequency of the original note is the common multiple of the frequency of the notes in the series

that is if you played two or more notes in the undertone series and pressed down the original note, the original note should sound Ziconium (talk) 16:45, 14 November 2022 (UTC)

On the piano Ziconium (talk) 16:45, 14 November 2022 (UTC)

The combined undertone and overtone series, from 1/16 to 16, makes up jazz minor scale. Thus you can find a augmented third in this combined series. Ziconium (talk) 06:08, 17 November 2022 (UTC)

To say that "each note in the undertone series has the original note as their common harmonic" already somehow was said by von Oettingen, who called this state of affairs the Phonicität, as opposed to the Tonicität (Arthur von Oettingen, Harmoniesystem in dualer Entwicklung, Dorpat and Leipzig, 1866, pp. 27-35). Von Oettingen (note 2 of p. 32) refers to Helmholtz who would have said (p. 76, probably of the 1st edition, 1862) that "One names harmonic undertones all the tones that have a given tone as upper tone." This does not seem to be found in the 4th edition (1877), nor in Ellis's translation (1875).

You write that each note in the undertone series has the original note as "their common harmonic," but that does not seem to make sense. To what does "their" refer? And to say that "each note in the undertone series has the original note as 'its' common harmonic" would raise the question of what you mean by "common." You probably mean that any harmonic of a given note in the undertone series also is a harmonic of all the other notes in the series, but it is not what you say. Note also that all the upper harmonics of any note that is an upper harmonic of an undertone also are upper harmonics of the same undertone.

This all is true from an elementary mathematical, abstract point of view: any multiple of a multiple of x also is a multiple of x ... But this does not prove the existence of submultiples! — Hucbald.SaintAmand (talk) 10:30, 17 November 2022 (UTC)

least common harmonic = the note having the least common multiple of its numerical frequency of all the notes in the series 98.97.33.243 (talk) 15:00, 17 November 2022 (UTC)

Agreed – then, it was the word "least" that was missing. It would perhaps be more correct to say that notes sharing a least common harmonic form a series that corresponds to the inverted harmonic series (i.e. the series of whole numbers). This concerns the description of a series of submultiples, but does not entail that undertones exist. One could say, for instance, that the least common harmonic of a fifth is the 12th of the lower note and the octave of the higher, which would not mean that the two notes of the interval are undertones (unless only numerically) of the least common harmonic. – Hucbald.SaintAmand (talk) 15:35, 17 November 2022 (UTC)

lower note 1/3

higher note 1/2

LCH = 1 98.97.33.243 (talk) 15:40, 17 November 2022 (UTC)

when you combine undertone and overtone

C--------------------------------------------D------------------E----------------F------------------G------------------Ab-----------------Bb

root(1st overtone and undertone)| 7th undertone| 5th overtone| 3rd undertone| 2nd undertone| 5th undertone| 7th overtone

The jazz miner scale

(not 1/16 to 16) 98.97.33.243 (talk) 15:57, 17 November 2022 (UTC)

correction: G is the 3rd overtone 98.97.33.243 (talk) 16:04, 17 November 2022 (UTC)

Thus Ab C E, a augmented chord is explained Ziconium (talk) 16:09, 17 November 2022 (UTC)

not jazz minor scale, that page has a problem. it is actually melodic major ascending 98.97.33.243 (talk) 18:18, 17 November 2022 (UTC)

descending Ziconium (talk) 18:23, 18 November 2022 (UTC)

f natural minor/melodic minor ascending has the same notes as C melodic major descending Ziconium (talk) 17:59, 20 December 2022 (UTC)

Undertones, or subharmonics?

The first sentence of the article appears to say that the terms "undertone" and "subharmonic" are synonyms. The section on terminology is mainly devoted to the meanings of "subharmonic", but includes odd statements, e.g.

• subharmonic does not relate to the subharmonic series
• In a very loose third sense, subharmonic is sometimes used or misused...

The article is a redirection of "Subharmonic:" it seems probable that a former article "Subharmonic" appeared misleading in that the term often was misused for "Undertone," but that parts of this article were copied here without change. The section on terminology appears to be about the meaning of "subharmonic;" it fails to explain the difference between the two terms. Further in the article, there is frequent mention of an "undertone series," probably coming from mentions of "subharmonic series." But there is no particular reason why the undertones should form a series...

Is it possible to read the discussion that led to the fusion of the two articles? — Hucbald.SaintAmand (talk) 12:32, 20 November 2022 (UTC)

A way to reconcile the two views about the minor triad

Let x be the original frequency

subharmonic x/4, x/5, x/6 form the minor triad

now let’s multiply by the denominators’ Least Common Multiple, 60

now they are 15x, 12x, 10x

Multiplying by 60 is multiplying by 2^2 * 3 * 5

multiply by 2 is to go up P8

multiply by 3 is to go up P5 + P8

multiply by 5 is to go up M3 + 2P8

together to go up M7 + 5P8

interlocking suharmonic and harmonic? Ziconium (talk) 15:50, 21 December 2022 (UTC)