Talk:Semitone/Archive 1

Dissonance

The article "Minor Second" says that it is considered "the most dissonant interval after the tritone," The "Major Seventh" article says that "The Major seventh is considered the second most dissonant interval after its inversion the minor second." 160.39.225.77 07:18, 20 December 2005

So, which is it?

Good question! Unfortunately, the answer seems to depend on who is doing the considering. As the article 'consonance and dissonance' points out, although there is a broad consensus on which intervals are dissonant and which are consonant, there is no single definitive way of quantifying dissonance, which, to some extent, has been culturally defined through the centuries and is subject to change. I think the safest bet would be to re-write those sentences, mentioning the intervals' dissonance, but without comparing them to other dissonant intervals. Fretsource 12:58, 20 December 2005 (UTC)

I vote that this is the most dissonant interval in the 12 tone system. Other than the reason that there are plenty of simpler ratios for the tritone (and other acoustic reasons), traditionally the tritone gets much more use harmonically than the minor second. They are an integral part of every dominant seventh chord, and even diminished triads are incredibly common. The minor second is rather rare, even as a suspension. Rainwarrior 04:26, 3 April 2006 (UTC)

I agree. The minor second is definitely the most physically rough. —Keenan Pepper 15:02, 3 April 2006 (UTC)

I think the consensus is that the major second is more dissonant than the tritone; at least, I can find cites to support that. Hence, I changed that sentence. Gene Ward Smith 14:02, 11 June 2006 (UTC)

Did you mean to say minor second? I would definitely think that the minor second is more dissonant than the tritone, but the major second seems disputable to me. - Rainwarrior 17:48, 11 June 2006 (UTC)

I have to agree with Rainwarrior. The tritone, to me, is at least as dissonant as the major second, if not more so. I wasn't aware of any such consensus claiming otherwise. Can we have a recount? Mark - 11 June 2006

Category listing

I removed this article from Category:Intervals because Category:Meantone intervals is a subcategory of Category:Intervals, and per Wikipedia:Categorization/Categories_and_subcategories it should not be listed in both (I did not think it justified any of the exceptions listed). -Big Smooth 16:55, 16 June 2006 (UTC)

It passes every listed exception. This is not only a meantone interval, but also a non-meantone interval. Putting it only in the meantone interval category denies this fact. - Rainwarrior 03:42, 17 June 2006 (UTC)

Ah, OK. That makes more sense, my mistake. -Big Smooth 18:29, 19 June 2006 (UTC)

Content from Semitone

The content below was at semitone. Semitone now redirects here.

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'The size of a semitone'

The terms tone and semitone are often used together, in abbreviated form, to express the pitch difference between the successive notes of diatonic scales. For example, the major scale can be expressed by T-T-S-T-T-T-S. where T equals one tone and S equals one semitone.

• The above is about the diatonic and major scales, not the semitone. Hyacinth 00:04, 16 August 2006 (UTC)

However, equal temperament systems other than 12-tone (derived by dividing the octave into various numbers of equal segments, such as 5-tone equal temperament) are often utilized in musical composition today. In tunings other than the equal temperaments, the octave is not divided into equal segments, so none of the segments might correspond to an equally tempered segment (or semitone, in the case of 12-tone equal temperament).

• The above is about non 12TET, not the semitone. Hyacinth 00:04, 16 August 2006 (UTC)

Rational expressions of the equal-tempered semitone'

Musical intervals are frequency ratios, often defined in terms of fractions (integer ratios). Fractions are therefore of significance in the discussion of semitones. It was long-believed that "true" musical intervals[1] are based upon integral ratios (in effect, harmonic ratios [2]).

• The above is about tuning, not the semitone. Hyacinth 00:04, 16 August 2006 (UTC)

Since it's first appearance, many musicians were reluctant to accept equal temperament[3], as it was a system based upon a mathematical formula, rather than harmonic ratios. It was therefore regarded as a compromise or contrivance, since all "true" intervals except the octave are violated. In equal temperament, the semitone is the basic building block of all intervals; therefore it's influence upon other intervals merits consideration. The major 3rd, for example, becomes about 1/7 of one semitone larger than the traditional 5/4 ratio of the major 3rd of just intonation; the major 6th becomes about 1/6 of one semitone larger than the traditional 5/3 ratio of the major 6th of just intonation. However, if equal temperament can be expressed as a system based upon exact, or virtually exact harmonic ratios, it may well be viewed as a "natural" tuning system, admittedly more complex, but indeed as natural as any tuning system in the history of music.

• Same. Hyacinth 00:04, 16 August 2006 (UTC)

Despite the very close approximation of the ET semitone provided by the 196/185 ratio, the accumulative error over an 88-note keyboard could present a potential musical problem. If two keyboards were to be tuned with the lowest note, A, at 27.5 Hz, one tuned to true ET and the other to the 196/185 ratio, and both were to sound the highest C (top key) simultaneously, a "beat rate" of about 1.25 beats per second (equivalent to the beats of a metronome set to 75 or 76 beats per minute) would be heard. (A "beat rate" is the "wah-wah-wah" we hear when two notes of similar, but not identical frequencies are sounded simultaneously, such as in "Honkey-Tonk" piano tuning.)

• The above is not about the semitone, but about tuning and dissonance. Hyacinth 00:14, 16 August 2006 (UTC)

In 1895, the violinist Julián Carrillo[4] discovered he was able to create sixteen distinct pitches within the range of one whole tone. He named his discovery Sonido 13 (13th sound). In effect, semitones were divided into eight equal segments, and the octave, therefore, into 96 such segments. This smaller unit is now known as a triamu (written 3mu) [5]. Mu is an acronym for "MIDI unit", the 3 indicating that $2^{3}=8$ such units form one semitone. The term was introduced in 2003 by a group of music theorists. A 3mu can also be described as ${\sqrt[{96}]{2}}$.

The 1/8 semitone, or 3mu, is extremely well-approximated by the frequency ratio of 139/138. The semitone is thus expressed as $(139/138)^{8}$, which is less than 1/2000 of one cent shy of the equal-tempered semitone (an infinitesimal deviation, far beyond aural detection, even when accumulated over the entire range of human hearing [6], and for all intents and purposes can be considered equal to ${\sqrt[{12}]{2}}$.

• The above is about the triamu, not the semitone. Hyacinth 00:14, 16 August 2006 (UTC)

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Discussion from Semitone

below is the discussion from Talk:Semitone

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Smaller than a semitone

Musical intervals smaller than a semitone are common in Eastern music and also occur, though rarely, in some Western classical music since the beginning of the twentieth century. Such intervals are called microtones and the music containing them, microtonal. This doesn't include small pitch variations often encountered in tonal music, such as vibrato or guitar string bending, etc.

For technological, rather than musical, reasons, the semitone can be divided into one hundred cents, with around five cents (one twentieth of a semitone) being the smallest pitch difference discernible by the human ear. Cents are generally of interest only to audio engineers or instrument manufacturers - and are irrelevant to most musicians.

I removed the above section as it is not about semitones and was accumulating arguments. Hyacinth 10:40, 1 January 2006 (UTC)

Useless references

Prof.rick, thank you for finally trying to cite your sources, but keep in mind the references must be not only present, but useful. I found "Sympathetic Vibratory Physics" via Google, so that's at least a usable reference, regardless of whether I think it's complete crap (I do), but "(source: a 50th Anniversary Hammond Organ Pamphlet)" is totally useless to me. There's nothing I can use to look up this publication, so it doesn't count as a reference. —Keenan Pepper 04:12, 28 July 2006 (UTC

Useless criticism?

Keenan,. I wish you wouldn't call things "crap". This is NOT appropriate Wiki-talk. (If you have any knowledge of Ancient Greek Music History, references should not be necessary. These are NOT disputable points.) And please, don't be arrogant! YOU can look up the history of Hammond Organs for yourself, and find out when they stopped making the tone-wheels. These are well-established facts, and hardly need "proving" by an over-abundance of references!!!

(Please don't assume the utter stupidity of the average reader, who you infer can neither take established facts as presented, nor do their own research!) OF COURSE, DISPUTABLE INFO should be substantiated!!! Now, can we try a more mutually co-operative approach? (Prof.rick)

I corrected your formatting, hope you don't mind. What exactly are you claiming is indisputable? I have never heard 25/24 called "semitone minimus", and the only Google hits are Dale Pond's website and verbatim copies of it, so I don't think Wikipedia should call it that. My problem with the last sentence isn't so much that it's disputable, now that I think about it, but simply that it doesn't belong in this article. The article is about the semitone; what does that have to do with the year they stopped making tonewheel organs?

P.S. Please don't take anything I say personally, I'm just devoted to the encyclopedia and sometimes forget about the people who help write it. —Keenan Pepper 05:48, 28 July 2006 (UTC)

Also, I started out with a well-defined mathematical sequence, the continued fraction expansion of ${\sqrt[{12}]{2}}$ (17/16, 18/17, 89/84, 196/185...) and it seems like you just ignored that sequence and added in some other numbers that don't fit, which upset me. —Keenan Pepper 05:52, 28 July 2006 (UTC)

Keenan- No, I appreciated your re-formatting. I have changed the contribution's basic nature to include various definitions of the semitone (by frequency ratios). You might find the numbers I've added do not relate to the sequence approaching E.T., because I have tried to provide a view of the various "sizes" of semitones which have been used historically. Sorry if these "other numbers" upset you! DON'T TAKE IT PERSONALLY..my first concern is the completeness and accuracy of the encylopedia. (Certainly you were already familiar with the ratios I presented, if you at least own or have access to copies of Oxford Dictionary of Music, and/or Groves Dictionary of Music and Musicians. I don't mind removing the terms "semitone minimus", "semitone medius", and "semitone maximus". The NAMES are irrelevant; it is the actual musical sounds which matter. (However, you might wish to change YOUR approach, looking at historic definitions of the semitone, as well as the semitone of ET. You probably focussed on ET in response to my original contribution, which related the semitone only to ET.) I find this surprising, since on your "personal" page, you expressed a desire to break free of equal-temperament (which I have done, in 5-TET, 7-TET, 8-TET, 9-TET, 10-TET, 24-TET, and 36 TET, and other microtonal compositions, although I still love 12-TET. I have decided NOT to provide references for the fact that tone-wheel organ production was discontinued in 1975. This is an undisputed historic fact, and needs no "proof". (Sometimes I think you would like me to prove that 1 + 1 = 2.) My final statement is as relevant as your reference to Hammond Organs! As a new-comer, I especially do not want a "Wiki-War". This is not a contest of wills, nor of views, but a sincere desire to present as much factual, historic, and scientific information as possible. As a compromise effort, and worthwile endeavor, I would like to encourage you to re-organize the numbers we have both presented, if necessary, presenting two distinct categories: a) historic semitones b) approximations of the ET semitone in terms of integral fractions. Unless we reach a compromise, we are destined to mutually self-destruct on these matters, like matter and anti-matter. This would deprive the public of the knowledge we both have to offer. -Prof.rick

Talk page etiquette

Prof.rick, please never remove text from talk pages, and never edit other people's comments. If you want to take something back, strike it out like this. When talk pages get too long, they are archived, not deleted. Also, don't change section titles, that breaks links to the section from other pages. —Keenan Pepper 10:00, 4 August 2006 (UTC)

Thanks for tips. Sorry, Keenan. I appreciate the tips, and will try to follow them in future. -Prof.rick

Further improvements

Keenan: I think the article now expresses the essence of what we each had to offer, namely two excellent rational approximations of the ET semitone. I am open to further discussion on how the article might be improved. (I know you would like to change the TITLE back to it's previous form, and I am agreeable to this...or else create new links.) Any further changes you feel should be made? -Prof.rick 4 August 2006

Okay, here goes:

Musical intervals are frequency ratios, often defined in terms of fractions (integer ratios).

The first sentence is vague and unclear. What does it mean "often defined in terms of"? An interval is either just (rational), or not. If it's not, it can't be "defined in terms of" fractions.

Fractions are therefore of significance in the discussion of semitones.

This statement doesn't logically follow from the previous one.

For millenia, it was believed that "true" musical intervals[7] must be based upon integral ratios.

This reference has nothing to do with the statement it is supposed to support.

For centuries, musicians were reluctant to accept equal temperament[8], since it was based upon a mathematical formula...

Again, the reference does not support the statement.

(The major 3rd, for example, becomes about 1/7 of one semitone larger than the traditional 5/4 ratio of the major 3rd of just intonation[9]; the major 6th becomes about 1/6 of one semitone larger than the traditional 5/3 ratio of the major 6th of just intonation.)

This article is about the semitone, not the major third or major sixth. This material is redundant with other articles.

However, if equal temperament can be expressed as a system based upon exact, or virtually exact harmonic ratios, it should indeed be viewed as a "natural" tuning system, admittedly more complex, but indeed as natural, as any tuning system in the History of Music.

I strongly disagree with this statement. I think any rational numbers complex enough to closely approximate equal temperament are too complex to be grasped by the human ear and mind. Since it is possible to disagree with this statement, it needs a source that directly supports it.

However, as harmonics, a limited number of these are within the range of human hearing.

This doesn't make sense. What is 196/185 "as a harmonic"?

...a cent being 1/100 of one semitone.

There's no need to say this if there's a link to Cent (music).

(A "beat rate" is the "wah-wah-wah" we hear when two notes of similar, but not identical frequencies are sounded simultaneously, such as in "Honkey-Tonk" piano tuning.)

There's no need to say this if there's a link to Beat (acoustics). One of the benefits of an online encyclopedia with clickable links is you don't have to say anything in more than one place.

An even more accurate rational approximation stems from the area of microtonal music. In 1895, the violinist Julián Carrillo[10]...

I don't understand why this external link is here, and this entire paragraph doesn't belong in this article, because it's about Julián Carrillo rather than the semitone.

The 1/8 semitone is extremely well-approximated by the frequency ratio of 139/138.

So? What does this have to do with the semitone? This article is not about the sixteenth-tone. I don't understand why this rational number is mentioned at all. It's a convergent to the sixteenth-tone, but it's not a convergent to the semitone. The approximation to the semitone resulting from this construction is 139353667211683681/131532383853732096, which is a very complex fraction, and given its complexity it's not really that close to the twelfth root of two. 7893/7450 is much simpler and much closer.

196 / 185 = 0.999 996 569 of one ET semitone

$(139/138)^{8}$ = 0.999 999 719 of one ET semitone

This math is wrong! 196/185 is 99.9940603 cents, and (139/138)⁸ is 99.9995131 cents. Where did you get these figures?

Ironically, the Perfect 4th consists of a 4/3 ratio, but in common-practice harmony is considered a dissonance in certain contexts [11].

You realize answers.com is a Wikipedia mirror, right? Also, what does this have to do with the semitone?

Similarly, harmonic 7 is regarded as dissonant with all other harmonics except it's own octaves, as are harmonics 11 and 13, to name just a few. Yet many larger frequency ratios are distinctly consonant, such as 192/5.

"Regarded as dissonant" by whom? I regard 7/4 and 7/6 as sweet consonances, but that's just my opinion, so if I were going to put it in a Wikipedia article I would find a source to cite (which is easy for me in this case, because Harry Partch agrees with me and I've got his book right here).

Do we "hear" these higher harmonics? We certainly hear them...

Says who?

Nonetheless, we respond subjectively, according to neuroscience [12], particularly when the harmonic in question is a prime number [13] such as 139. Prime numbers indicate the "first occurrence" of specific pitches in the harmonic series [14]

These links make no sense. The last link is especially inappropriate, because that article is about a different meaning of the phrase harmonic series: it's about harmonic series (mathematics) rather than harmonic series (music). Also, I don't think 139 being a prime number has anything to do with the consonance or dissonance of the interval 139/138, so if you want to claim that you have to cite a source that agrees with you.

By traditional definition, the semitone is a discord. Yet for centuries, even the Major Third [15] was classified as dissonant. Certainly, the Art of Music and Aural Perception are in a state of continuous evolution, making such definitions strictly relative...

Finally something is both relevant and correctly referenced. See how easy that was? Why are you capitalizing all these words though? This isn't the eighteenth century, we don't just capitalize words whenever we feel like it.

Okay, I'm done. Now, what do you want to do about all these problems? You can either have a long, pointless discussion with me about how none of them are really problems, you can fix them yourself, or you can let me fix them. It's up to you. —Keenan Pepper 09:10, 6 August 2006 (UTC)

Keenan, you've said a lot. I will address each point.

And why do you assume the discussion would be long and pointless??? (Yes, it could prove to be long, but in the interest of the encyclopedia, hardly pointless, unless an arguement is our ultimate goal.)

If it didn't result in any improvements to the article, it would be pointless. I admit I simplified the options for the sake of rhetoric.

I hesitate to "let you fix them", since upon my last invitation for you to do a re-write, you excluded ALL relevant material I had introduced, and wrote a pointless article, failing to equate integral fractions with harmonic ratios, and side-tracking to claims regarding the Hammond organ (citing reference which not only failed to substantiate your claims, but proved your them UNVERIFIABLE). I will therefore endeavor to make any revisions myself, but I will certainly weigh carefully every arguement you present.

You're acting as if you own the article. Please stop. Also, if you're going to accuse me of adding unverifiable claims, be specific. I said only that ratio 196/185 appeared in the tuning of the Hammond organ, and the Tonalsoft article I cited clearly supported that.

1. "Musical intervals are ratios...etc." Whether a ratio is measured as a rational fraction an irrational number (e.g. the major 3rd of equal temperament is a 1.25992105.../1 ratio). An interval IS a frequency ratio. SOME intervals (i.e. Just) are generally expressed as integral fractions.

You don't have to use capital letters. I already know this.

2. "Fractions are therefore of significance..." Obviously, fractions are of significance in expressing Just Intervals. They CAN be of significance in approximating the intervals of ET. Since the semitone is an interval, this holds true, particularly since this article is an attempt to find reasonable integral approximations of the ET semitone (and therefore, of ET).

No, this is a general encyclopedia article about the semitone. You're the one trying to make it "an attempt to find reasonable integral approximations". I disagree with your assertion that all these fractions are significant to the semitone, so you need to provide a source.

3. "For millenia, it was believed...." We both know this to be the case, and was the source of prolonged resistance to ET. If you like, I will find an alternate reference to support it, although that might take a day or two.

I would like that.

4. "I strongly disagree. I think any rational numbers close enough to approximate..." What is the basis for your arguement? The FACT that Carrillo was able to distinguish 16 distinct pitches within one whole tone indicates that the human ear and mind CAN comprehend such differences of frequencies (and thus, fractions which approximate them). If we can comprehend 1/8 semitone, surely we can comprehend 8/8 semitone. This certainly justifies links to "Carrillo".

I think you misinterpreted what I said, or else I'm misinterpreting your argument here. Simple fractions have a special aural effect, but I think that special effect disappears when the numbers in the fraction are greater than 20 or 30, especially when the interval is small. For example, I don't think 35/33 is aurally distinguishable from the equal-tempered semitone. Anyway, it doesn't matter what I think, unless I can provide references which support it, and it doesn't matter what you think either unless you can provide references.

To suggest that this has nothing to do with the semitone is akin to saying that electrons, protons, and neutrons are irrelevant in a discussion of "the atom". I am approaching the semitone from a point of view of dissection...into smaller, but equal component units.

This analogy makes no sense to me. By adding something to the article, you are implicitly making the claim that it is significant, and you need to provide references for that claim.

Yes, I believe the human ear, and music, are "evolving" to a state of greater complexity, like life itself. If you compare yourself, a human being, to an amoeba; both are "natural", but one is obviously one has evolved toward greater complexity (I would hope!)

This is another claim you're making entirely on your own.

5. The reference to the P4 is intended to point out that the simplest of harmonic ratios do no necessarily result in consonance (i.e. the old assumption, probably creditable to Pythagoras, who permitted ratios involving integers only as high as 4, is obvious erroneous). (Interestingly, some of the "mean tone" semitones involved some rather high integers!) Similary, the higher harmonics need not necessarily be regarded as dissonant, simply because they are higher!

I don't think it's "obvious erroneous". In medieval organum, one of my favorite genres of music, the concluding sonorities only include ratios involving integers at most four, exactly those Pythagoras found consonant.

6. "Do we hear higher harmonics....says who? We certainly do! As you know, harmonics are simple multiples of a given frequency, e.g. 1f, 2f, 3f, etc. As long as these harmonics are within our range of hearing, we hear them! (This largely accounts for differences in timbre between various instruments.) In fact, I have seen some old pianos, with extremely hard hammers, which produce very strong higher harmonics (on the verge of the limits of human hearing)...so high and so strong, they are painful to the ears. The question is not, "why assume we hear these higher harmonics?", but "why assume we do NOT hear them?" (We know they exist, and being within the range of the frequencies detectable by the human ear, we do hear them!) Consider the Pythagorean semitone, expressed as ratio 256/243.

Okay, I'm considering it. Were you going somewhere with this?

7. I have classified Harmonic 7 as a dissonance, based on Helmhotz's definitions of consonance. (You may find it "sweet", and I might find 13/5 "sweet", but this enters into the area of contemporary thinking about music, rather than traditional views, to which this article refers.) Having Harry Partch's book "right there" does nothing to strengthen your arguement!

The title of this article is not "Traditional views of the semitone", it is simply "Semitone". On Wikipedia, having a published source that directly supports my argument is all I need. (BTW, Helmholtz's graph of dissonance has dips at 7/6, 7/5, and 7/4, so I wouldn't be so sure he's on your side.)

8. References to Neuroscience and Prime Numbers are admittedly of remote significance in this discussion. However, the fact that prime- number harmonics represent the FIRST OCCURENCE of certain pitches seems relevant. I WILL MODIFY THIS PARAGRAPH, AND IT'S REFERENCES. However, I have NOT claimed that prime-number harmonics are necessarily consonant!

Okay, that's a start...

9. Your issue in reference to the capitalization of words certainly has nothing whatsoever to do with the semitone. I am a Canadian, and to some extent, we have followed this tradition. (I assume you are are from usa, which has evolved it's own "rules" regarding the written English language.) This matter hardly merits attention.

You're right, that's a trivial issue and I'm sorry I brought it up.

10. "By traditional definition..." (references to Major 3rd and Major 6th). Since all intervals of ET are constructed of equal semitones, to ignore them as "irrelevant to the semitone" would again be akin to discussing the atom without reference to molecules! We are discussing "building blocks".

The wording in the article did not convey that impression, but you have a point.

11. I have included "a cent being 1/100 semitone", for the benefit of those readers who simply don't bother checking links. Nonetheless, the link is provided for those who seek further information.

Fair enough.

12. I have treated 196/185 as a harmonic ratio, because I see NO OTHER REASON to even mention it! (If it is beyond the range of human hearing, it is purely theoretical, and therefore irrelevant.)

I don't understand what you mean here. Range of hearing ususally refers to absolute frequency limits, as in "The range of human hearing is about 20–20,000 Hz.". 196/185 is an interval, and its two pitches can be played in any register. Are you using the phrase range of human hearing with a different meaning in mind?

13. I will remove the link to "beats".

I didn't mean for you to remove any links, I meant for you to remove the redundant content and replace it with a link.

14. "The 1/8 semitone..." is indeed of relevance to the semitone! So are reference to Carillo, who established it's use. First, it demonstrates a PRACTICAL application of the 1/8 tone (which has since been imitated by other composers). It also "dissects" the semitone into equal, virtually rational components. It is is GREAT importance!

15. "So, what does this have to do with the semitone?" (Answer: no more than electrons have to do with the atom.)

Electrons are indivisible particles. Musical frequency is a continuum that can be divided in unlimited ways. If 1/8-semitones (which, BTW, are sixteenth-tones, not eighth-tones) belong in this article, then why not 1/3-semitones and 1/5-tones and all the other divisions? By your reasoning, we should scrap Tritone, Major third, Minor third, Major second, and Semitone and discuss them all at Octave, because in equal temperament they are all equal divisions of the octave. Obviously this is a bad idea, and I say talking extensively about sixteenth-tones in the article about semitones is an equally bad idea.

16. My math is not "wrong"! It depends if we want to regard our rational approximation of the semitone as fractions of semitones, or fractions of cents. Since this article is about semitones, let us focus on that area, thus, as simple ratios:

Let fundamental frequency = 1.0

(196/185)/12th root 2 = 0.999 996 5 ((139/138)^8 / 12th root 2 = 0.999 999 71

(These are simple ratios. If you wish to define them in cents, this could easily be changed in the article (but remember, this article is about semitones, not cents!) Seriously, I feel the above expressions would be better understood by an average reader, than the "cents" expressions. Nonetheless, I have edited this point according to your wishes.

Ah, I see. Your math wasn't wrong, and I'm sorry for saying so. You were simply comparing them arithmetically instead of geometrically. Well, think about this: is the perfect fifth 3/4 of an octave because (3/2)/2 = 3/4?

17. Your reference to the 7893/7450 ratio is POINTLESS, since these harmonics lie far beyond the range of human hearing, and are therefore purely theoretical.

Again you use the phrase range of human hearing in a way that confuses me. I agree with you that it's pointless, but it's no more pointless than your own (139/138)⁸ = 139353667211683681/131532383853732096. That's the point I was trying to make.

I hope I addressed all you "concerns". (In future, it would make things much easier, for any reader, if you were to NUMBER you comments, as I have done in my response.) I am now going to revise the article, but ask you, please, do NOT remove material which is still under debate! I am prepared to LISTEN with an open mind (although I may not pay much attention to irrelevant criticisms, such as capitalization.)

In conclusion, Keenan, I am a newcomer to Wikipedia. Wikipedia suggests SUPPORT of newcomers. (I have found your comments and editting at times to be anti-factual and personally annoying.) Your criticism are often far-fetched. It is interesting that I have placed articles on several other pages, but have encountered no such conflicts.

Same to you. I find your use of capitals and scare quotes personally annoying.

I therefore propose, if these issues are not resolved to our mutual satisfaction, we agree to to "truce", in order to avoid more radical processes. Can I have your agreement on this? -Prof.rick Aug. 6, 2006

Depends what that means. If it means being lax about core Wikipedia policies such as verifiability and no original research, then I heartily disagree. —Keenan Pepper 06:27, 8 August 2006 (UTC)

Keenan, WP:AGF. Prof.rick

Edits

Keenan, I have ammended the article, in consideration of your arguements. (I doubt if I've covered them all, since you find so many, whether justified or debatable.) If you still find any of the content objectionable, unsubstantiated, or irrelevant, please, SAY SO. I will then examine your comments further. -Prof.rick Aug. 6, 2006

If you are not satisfied with my editing, please comment further.

Keenan: If you are satisfied to accept the intervals of ET as simple multiples of the ET semitone, then why not consider the possibility that the ET semitone itself can be defined by multiples of a smaller unit (notably, powers of 1/2), especially when the human ear can recognize such fine distinctions as 1/8 semitone?

You are hardly in a position to criticize my data (or references) when your own reference for the Hammond Organ semitone disproved your claim, proving it non-verifiable. (And you did so at the expense of deleting my article, and all it's relevant information.) The inadequacies of the Hammond organ are hardly relevant, especially when misquoted!!! (Technological shortcomings cannot justify false semitones.)

I don't know what you're talking about. Please explain yourself. —Keenan Pepper 06:31, 8 August 2006 (UTC)

As I have suggested before, let's work together on this article, polishing it, improving it, and refining it. It is NOT a war!!! The purpose of the article, obviously, is to provide reasonable integral (but audible) ratios to account for the ET semitone.

No, that is not the purpose of this article. The purpose of this article is to provide an encyclopedic description of the semitone that follows Wikipedia policies. I don't know where you got any other idea. —Keenan Pepper 06:31, 8 August 2006 (UTC)

Now, can we discuss points of agreement? -Prof.rick Aug. 7, 2006

Internal links added

I have added a section for "See Also". (I have also added links from other pages to "Semitone" page.) If anyone wishes to add to the listings, it would be appreciated if you keep the list in alphabetical order. Thanks! -Prof.rick August 5, 2006

Combining Articles

May I suggest that Articles 1. and 2. of this page be reconciled into one Article. (One deals with "chromatic" and "diatonic" semitones strictly within the equal-temperament context, while the other also refers to mean-tone tuning.) It seems the two Articles could present a broader, more comprehensive viewpoint if combined. -Prof.rick CORRECTION: Sections 1. and 2. (not articles)

Back to "further improvements"

OK, Keenan. I've done a lot of revision. If you still have "beefs", please let me know on the Discussion page, and I will do my best to either defend them or change them. (But please, be reasonable. It IS possible that some people may know more than you!) -Prof.rick

Keenan, I wrote a thorough response to your last comment, but somehow it was not saved. I also removed the paragraph pertaining to the Perfect 4th and harmonic 7. However, rather than spend several hours rewriting it now (and it does address your concerns), I would suggest we try a "cooling off" period, of at least several days, without comments and without editing. (I believe this is the meaning of "truce" in the Wikipedian sense.) Are you agreeable? -Prof.rick Aug. 8, 2006

FOOTNOTES:

Already, I am feeling much better about the whole thing, and I am prepared to make major compromises. I have had the day free for further research, and I hope you will appreciate the results, which will be presented in a few days. As you can see, I have already responded to many of your valid points. Yes, your Math (pardon the capitalization...I'm Canadian, eh?) was correct...in fact, mine agreed, until I hastily attempted to convert to semitones, rather than cents. I also concede that my references to Harmonic 7 (or 11 or 13)have not been established as relevant to a discussion of the semitone.

AGAIN, ONLY PART OF THIS EDIT WAS SAVED.

Keyboard

Why does the opening paragraph (which attempts to define the semitone), make reference to the piano keyboard, naming "from a white key to it's neighbouring black key", "from a white key to the next white key, when there is no black key between them", but fails to mention "from a black key to it's neighbouring white key"? (A semitone does not have to start with a white key!) Anyone agree?

Please sign your posts on talk pages per Wikipedia:Sign your posts on talk pages. Thanks! Hyacinth 05:25, 13 August 2006 (UTC)

Sorry! -Prof.rick Prof.rick 05:37, 14 August 2006 (UTC)

Removed: Discord

By traditional definition, the semitone is a discord. Yet for centuries, even the Major Third [16] was classified as dissonant. Certainly, the Art of Music and Aural Perception are in a state of continuous evolution, making such definitions strictly relative. It is impossible to classify any given interval as "concordant" or "discordant" without reference to its musical context. In much 20th and 21st century music, the semitone (or Minor Second), along with its inversion (the Major Seventh) and its compound form (the Minor Ninth) have been utilized in contexts of "virtual concord", sometimes as satisying resolutions, even in final cadences, particularly in Jazz and Blues contexts.

Even the major triad can be interpreted as a relative discord, depending upon its context.

As the Art of Music and Aural Perception continue to evolve, concord and discord must be regarded as "relative" terms, determined both by musical context and personal taste.

I moved the section above, titled "Semitone: concord or discord?", from the article as it is not about semitones. Hyacinth 05:25, 13 August 2006 (UTC)

Thanks. I wasn't really sure where it should go. Appreciate the help! -Prof.rick Prof.rick 05:37, 14 August 2006 (UTC)

I think you will find this information conveyed in the article on Consonance and dissonance. Hyacinth 23:06, 15 August 2006 (UTC)

Removed: Harmonics

The human ear can hear frequencies as high as 20,000 Hz or so. Neither fundamental tones nor harmonics which exceed this limit are audible. This places harmonics 139 and 138 within audible range of an approximate fundamental frequency of D (middle line of the bass clef), and, of course, all audible lower frequencies. Do we "hear" these higher harmonics? Amongst a spectrum of other harmonics, we probably do hear them. Although we generally cannot selectively isolate a specific harmonic and "listen" to it, we no doubt hear it subliminally, and respond accordingly.

I removed the above as it isn't about semitones. Hyacinth 23:08, 15 August 2006 (UTC)

Editing

OK, Keenan. I am placing this note both here and on my own Talk page, to be sure you find it. I hope you find the revisions to be reasonable, particularly the new references to 3mu and prime-number harmonics. However, I am not finished editing. The earlier part of the section needs some revamping, and I can probably find better references. I will be waiting to hear your feedback (provided it can be presented without a demeaning tone). I did find some of your more recent comments to be nicely presented. -Prof.rick Prof.rick 05:23, 14 August 2006 (UTC)

I have removed a statement regarding the relative strength of prime-number harmonics, since the arguement relied upon reference to the yet-unproven Reimann Hypothesis, and the Zeta function. Prof.rick 11:51, 14 August 2006 (UTC)

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Removed: Diatonic vs chromatic

The minor second is synonymous with the term "'diatonic semitone'"; the other type of semitone, the "'chromatic semitone'", is technically the interval called the '"augmented prime'".

This conflicts with the information from semitone so I removed it. Hyacinth 23:52, 15 August 2006 (UTC)

Chromatic semitone

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In music, a chromatic semitone is the amount by which any note is raised by the addition of a sharp, or lowered by the addition of a flat. In equal temperament, it has a value of exactly 100 cents, or one-twelfth of an octave, and is enharmonic with the diatonic semitone, which is also 100 cents. The two are therefore often conflated, and given the common name of semitone or, in North America, half-step.

However, in any meantone tuning aside from equal temperament, the chromatic semitone is smaller than the diatonic semitone. In Pythagorean tuning it would be larger, equal to the apotome of 2187/2048 rather than the Pythagorean minor semitone, or limma, of 256/243, which corresponds to the diatonic semitone. Hence using it in connection with Pythagorean intervals invites confusion, and the word is not generally used in such cases, being reserved for meantone intonation.

For intervals in just intonation, the just chromatic semitone of 25/24 the septimal chromatic semitone of 21/20, and the major chroma of 135/128 can be equated with the chromatic semitone.

The chromatic semitone is the amount by which any minor interval needs to be expanded to become major, or by which any major interval is reduced to become minor . For any perfect interval, sharpening of its upper note (or flattening of its lower note) by a chromatic semitone leads to the augmented version of the interval. Similarly flattening of the upper note (or sharpening of its lower note) leads to the diminished version of the interval. For this reason, the chromatic semitone is sometimes called an augmented unison or augmented prime.

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The above is from Chromatic semitone, which now redirects here. Hyacinth 23:54, 15 August 2006 (UTC)

To do

• Incorporate appropriate information from #Content from Semitone.
• Incorporate appropriate information from #Chromatic semitone.
• Include more citations for ratio and historical claims.
• Archive old discussions on this talk page.

Above is a to-do list for this article. Please discuss and add more. Hyacinth 00:20, 16 August 2006 (UTC)