Talk:Tritone/Archive 3

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Abbreviation for Tritone

All the other articles about intervals mention an abbreviation, such as P5 or M2. i can't seem to recall what the abbreviation is for augmented and diminished intervals, and I was displeased not to find the answer here... Luqui 19:55, 31 December 2005 (UTC)

a4 (or A4) or d5 are in common use. Frequently you will see them notated as x4 for the augmented 4th and ο5 for the dim 5th

Degree of dissonance of the tritone

I don't think this should be exaggerated, which I think the article did. Persichetti in Twentieth-Century Harmony has what I think is a more usual view:

It is difficult to classify the tritone or the perfect fourth out of musical context. The tritone divides the octave at its halfway point and is the least stable of intervals. It sounds primarily neutral in chromatic passages and restless in diatonic passages. The perfect fourth sounds consonant in dissonant surroundings, and dissonant in consonant surroundings.

That's more like it, I think. Gene Ward Smith 03:01, 11 June 2006 (UTC)

I just added a sentence in the article based on that book.--Roivas 21:50, 16 November 2006 (UTC)

The Tritone's usage in history

The tritone is a restless interval, classed as a dissonance in common practice music; even more dissonant than the minor second or seventh‹The template Talkfact is being considered for merging.› , and given the name diabolus in musica ("the Devil in music") by some from the early music era to the baroque period, likely because of its unwanted occurrence as F against B when two voices were singing a fifth apart‹The template Talkfact is being considered for merging.› . It was exploited heavily in the Romantic period as an interval of modulation for its ability to evoke a strong reaction by entering the key least related (retaining only two common tones, the least possible) to what occurs previously‹The template Talkfact is being considered for merging.› .

I'll try to provide some better information on this. Anyway, this paragraph needs to be improved.

Anyway, you'd think the "key least related" would have no common tones.

The secondary dominant and diminished seventh relations are what facilitates modulation in tonal music. Both of which contain tritones. Not sure what the paragraph above is trying to get at and I think it would just confuse someone who is trying to learn the basics of music theory.--Roivas 22:02, 16 November 2006 (UTC)

When taking 7 of 12 tones to build your scale, you only have 5 tones left over. So, if you want to pick the most unrelated scale of 7 tones, you have to reuse at least 2 (7-5 = 2). (Though the minor-second modulation also has only 2 common tones.) - Rainwarrior 22:55, 16 November 2006 (UTC)

Okay, two keys with roots a tritone apart have two common notes between them. I get what you mean now. I would make the paragraph a little more clear, that's all.--Roivas 23:54, 16 November 2006 (UTC)

As for the diabolus in musica stuff, a citation should be really easy to find. If my books weren't packed up in boxes right now I'd put one down for you, but if anyone else out there has, say, the Richard Hoppin Medieval Music, or even the Grout history, check the index for it. You'll find something. - Rainwarrior 23:01, 16 November 2006 (UTC)

I put in some clarification there--the Diabolus in Musica is not a medieval term (search for it in the Thesaurus musicarum latinum and you won't find anything). I also tried to start a little section on its use in the classical and baroque eras. I think that after the initial definitions, dividing the article by usage historically is important. (I also think that the list of pieces with tritones in them needs to be pruned carefully--naming works without prominent tritones is harder than naming works with them). --Myke Cuthbert 00:32, 28 November 2006 (UTC)

Yeah, I've never been too clear on this. I saw the term mentioned in Walter Piston's "Harmony" when I was flipping through it, but only as a passing comment. Agree on your other points as well.

An answer might be found in this great little collection of papers: Music in the Western World, a History in Documents. I can get it from a local library in the next couple of weeks.

Funny that Black Sabbath's "Symptom of the Universe" wasn't mentioned! That's a great tritone riff. Just kidding.--Roivas 07:05, 28 November 2006 (UTC)

I have the Weiss-Taruskin Music in the Western World sitting here, but I didn't see anything on the tritone (it avoids music notation for the most part, so discussion of elements such as tritones is sparse). The Strunk volume of documents might have something on this topic, but I also doubt it. --Myke Cuthbert 22:41, 28 November 2006 (UTC)

Btw -- Nice job on so many edits, Roivas. I want to move the part about the TT in "first inversion" chords from Jeppesen to between the Medieval and Baroque sections, since it's really dealing with 16th century music which played by different rules. We don't yet have a subsection on the tritone's use in Renaissance music. --Myke Cuthbert 03:23, 1 December 2006 (UTC)

Sounds good. Thanks!--Roivas 05:08, 1 December 2006 (UTC)

My suspicion is that people use it because it "sounds cool."--Roivas 22:50, 28 November 2006 (UTC)

"even more dissonant than the minor second or seventh‹The template Talkfact is being considered for merging.› "

This statement bothers me as dissonant and consonant are fairly subjective terms and the definitions for both tend to change drastically within short periods of time. I know that Hindemith qualifies what you are saying (for his own reasons), but I'm sure I can find sources that say otherwise. Can we make this statement less definite or, at least, qualify it with a source?--Roivas 19:30, 27 November 2006 (UTC)

Can someone...

Can someone translate the German regarding the tritone in the history section? Not all of us (including me) understand German. bibliomaniac15 05:26, 27 January 2007 (UTC)

I'm not so fluent, but since I did add the texts, I guess I have some responsibility.  :) I am surprised by the "angenehme" and really hope I didn't mistranscribe "unangenehme," but given other things Mattheson has written, it is entirely possible. Also, I added a little section explaining how mi-fa can be a tritone. --Myke Cuthbert 05:11, 29 January 2007 (UTC)

Occurrences

If folks would please refrain from adding more and more "tritone occurences" in popular culture, that would be great. It's one of the most ubiquitous musical devices in Western Music. Nothing special about it. Suprised no one has mentioned the English police siren! At least that's sort of interesting.--Roivas 22:08, 3 December 2006 (UTC)

Agreed. The newest Ravel example seems absolutely silly. A list of Ravel works without tritones would be more interesting. Or a discussion of how Ravel uses it would help someone who comes to this article expecting to learn something. The European siren usage would be a good mention though. --Myke Cuthbert 04:32, 31 January 2007 (UTC)

Sympathy for the Devil's Interval

I've always thought that the lowest note in the repeated bass line of the Rolling Stone's "Sympathy for the Devil" sounds like "the wrong note". Someone suggested to me recently that maybe they were using "the Devil's interval". Does someone know if this is true? If so, it seems like that fact would be a worthy addition to that article (and maybe this one). - dcljr (talk) 03:13, 21 March 2007 (UTC)

No, the wrong note is an 'A' beneath an E major chord. Usually a bassist should play a 'B' in this context, but here he plays an 'A'. The interval is a perfect-fourth which sounds wrong because it's not part of the harmony (producing other dissonant intervals, like a major second/ninth and a major seventh, but no tritones). - Rainwarrior 05:37, 21 March 2007 (UTC)

Hugh7 trivia

It seems to say more about Lillian Hellman or Dorothy Parker than about the tritone. Consensus to remove? --Myke Cuthbert 19:39, 23 April 2007 (UTC)

Yes, it's amusing and characteristic of Parker's wit but obviously pretty irrelevant here. Rigadoun (talk) 19:04, 24 April 2007 (UTC)

Isn't trivia always more or less irrelevant? Rather than delete, if you must, please move it to Dorothy Parker or Lillian Hellman --Hugh7 23:15, 24 April 2007 (UTC)

I might be more against trivia sections in articles than others, but I think this piece of trivia might not have a good place in Wikipedia. There are many great jokes about diminished fifths, but I don't know where they should go. Though I think an article on humorous statements made about music theory would be a fine addition to WP, and I'd definitely want to contribute. --Myke Cuthbert 01:06, 25 April 2007 (UTC)

function of chord

can someone please add a section to explain its function? Jackzhp 20:53, 7 May 2007 (UTC)

Function of what chord? Hyacinth 20:56, 7 May 2007 (UTC)

most dissonant interval?

isn't the tritone the most dissonant musical interval? 67.172.61.222 22:48, 9 March 2007 (UTC)

No. Aside from dissonance being difficult to define in a quantitative way, it shows up in every dominant seventh chord. If anything is "most dissonant", it's probably the minor second. - Rainwarrior 17:33, 10 March 2007 (UTC)

I'll see your "no" and raise you a "yes for the following reasons"-

• since a tritone divides the octave in half symmetrically, there is no tonal center between the two notes, and it is always unstable regardless of context (from a purely clinical viewpoint, a lack of tonal center is a pretty concrete definition of dissonance, imo).
• a dominant seventh chord is relatively dissonant, but it doesn't always sound out of place, given certain contexts (i.e. building tension to resolve to the tonic or elsewhere or harmonizing with a scale).
• when you play the two intervals as one note after another, there is a pretty obvious difference. going from E to F sounds much different than going from E to A#.
• like all other imperfectly consonant intervals, a semitone can sound perfectly fine in certain contexts. listen to "From the Morning" by nick drake for example. A tritone never sounds melodic.
• finally, since an interval is roughly as consonant as it's inverse (i.e. a fourth and a fifth), a major seventh would have to be similarly dissonant. it is, of course, not especially consonant, but when you play a maj7 chord it's certainly not comparable to what a b5 chord (or a diminished chord) sounds like.67.172.61.222 00:23, 16 March 2007 (UTC)

Dissonance and consonance are culturally defined terms and don't obey particular universal rules. You could correctly say that an interval was "treated as the most dissonant interval by composer X" or "in period Y" or "in Noh theater" but your ear is not a good judge of universal dissonance. For instance, C->B# is considered a dissonance by most tonal theorists (augmented 7th). Similarly, a bare perfect 4th is definitely treated as a dissonance in nearly all common-practice harmony--it does not have roughly the same degree of consonance as the P5.

I think I remember reading somewhere that Dissonance can be measured as the least common multiple between two frequencies or something like that; C to B# is anything but dissonant as it is an augmented seventh which is enharmonic to an octave. It is the most consonant interval other than unison.

Your point about harmonic and melodic dissonances being different beasts is a quite good one. --Myke Cuthbert 01:03, 16 March 2007 (UTC)

What do you mean by "sounds melodic"? And what is melodic dissonance? Do you have a definition or is this an entirely subjective measurement? And as for "harmonic" dissonance, consider any chord with exactly two semitones and no tritones (either C D♭ D♯ E, or C D♭ E F) and and compare to any chord with two tritones but no semitones (C D F♯ G♯, or C E♭ G♭ B♭♭); which is more "dissonant" subjectively? Or similarly with three semitones (C D♭ E F G♯ A) vs three tritones (C D E F♯ G♯ A♯)? - Rainwarrior 06:53, 16 March 2007 (UTC)

Rainwarrior, how do you do the sharp and flat symbols? --63.25.21.154 (talk) 11:10, 7 March 2008 (UTC)

The easy way is to use the {{music}} template. For a sharp use # as the parameter, for flat use b, for double sharp ##, for double flat bb. It can do a lot of other things too. It's pretty handy. — Gwalla | Talk 17:48, 27 May 2008 (UTC)

Myke and Rainwarrior are right. There is no fixed definition of "most dissonant interval" in equal temperament. It's completely subjective. Maybe if User:67.172.61.222 took beats into consideration he/she may have something going for his/her argument. Even then it wouldn't matter.--Roivas 22:07, 18 April 2007 (UTC)

I'm inclined to agree with 67.172.61.222. In twelve-tone music, this is roughly how consonance and dissonance play out:

INTERVAL AND INVERSION:

Most consonant

0 (perfect unison) and 12 (perfect octave)

5 (perfect fourth) and 7 (perfect fifth)

4 (major third) and 8 (minor sixth)

3 (minor third) and 9 (major sixth)

2 (major second) and 10 (minor seventh)

1 (minor second) and 11 (major seventh)

6 (augmented fourth) and 6 (diminished fifth)

Most dissonant

I imagine the only thing that changes culturally is exactly how strict the definition of "consonant" is (in medieval times only 0/12 was consonant, gradually growing to include 5/7, 4/8, and 3/9)--but that still gives you a pretty clear indication of where every interval falls regarding sonance. Joe routt (talk) 04:40, 27 May 2008 (UTC)

Can you cite a source for this information per Wikipedia:Citing sources? Hyacinth (talk) 06:09, 27 May 2008 (UTC)

I'm afraid not, it's original research. I was making a point, not an edit.Joe routt (talk) 08:10, 27 May 2008 (UTC)

Huh? 4/8 is 1/2, aka an octave, which is consonant no matter what culture or time period you're looking at. 3/9 is 1/3, octave-equivalent to a perfect fifth. 0/12 isn't even possible (I guess it could mean a single string with no harmony at all). Not sure what you intended. — Gwalla | Talk 17:48, 27 May 2008 (UTC)

You misunderstand me. When I refer to 0/12, 5/7, etc, I am not referring to fractions. In this case 0/12 would be an interval of 0 semitones (unison), paired with its inverse, an interval of 12 semitones (octave). Likewise, I used 5/7 to mean a perfect fourth and fifth. Allow me to rephrase myself: In medieval times, only unisons and octaves were consonant, gradually growing to include perfect fourths and fifths, and major and minor thirds and sixths. I hope that clarifies things. Joe routt (talk) 00:06, 28 May 2008 (UTC)

Oh, I see what you mean. — Gwalla | Talk 19:43, 28 May 2008 (UTC)

World of Warcraft

I thought this info from the recent Blizzcast about the use of the Tritone in World of Warcraft was interesting:

"There's an interval in music known as a tritone ... it's been referred to in history as the "devil's interval" ... it's not quite consonant and it's certainly not terribly dissonant and it makes people feel ... uneasy without being harsh or jarring so I wrote all around that interval ... I'm playing a lot of games with that interval ... that interval and the cello ... that pallet of choir and harp ... blended all together to create the vocabulary for the Blood Elves." — Brower, Russell (2008-08-07). "How the music in World of Warcraft has evolved since the game began" (Interview). Interviewed by Nethaera, Blizzard Inc. Retrieved 2008-08-12. {{cite interview}}: Unknown parameter |program= ignored (help)

Since this is the only citable mention that I know of of the use of the tritone in modern media, I thought it would be worth including. -Miskaton (talk) 15:00, 12 August 2008 (UTC)

This really seems too trivial to mention. The only recent citable mention of the tritone? Really? — Gwalla | Talk 16:00, 12 August 2008 (UTC)

Hmm, I agree with Gwalla. There certainly isn't any shortage of modern writing and scholarship on the tritone. Now it would be wonderful if someone had the time do dig through them and expand the article, but that's a different matter entirely. :) --Blehfu (talk) 18:13, 12 August 2008 (UTC)

Car Horn

Aren't car horns deliberately tuned to the tritone? This must be the main reason why they're duophonic. Worth mentioning? —Preceding unsigned comment added by 81.109.11.182 (talk) 13:37, 8 October 2009 (UTC)

yes, i think i read that somewhere, car horns are usually tritones, and they're intended to be annoying, so someone's opinion is that this is the most dissonant interval —Preceding unsigned comment added by 170.170.59.138 (talk) 20:48, 26 October 2009 (UTC)

Some car sirens use tritones, but normal car horns are a single note. (Usually natural F) 151.103.236.139 (talk) 18:34, 27 September 2010 (UTC)

17:12

What about 17:12? It's only slightly sharper than the equal tempered tritone. 151.103.236.139 (talk) 18:40, 27 September 2010 (UTC)

What about it? Do you think it should be added to the article? Hyacinth (talk) 22:05, 28 September 2010 (UTC)

The ratios in just intonation

The tritone was never 5/7 in either Medieval or Renaissance music when the term first arose. In the medieval period, it was 512/729 (8/9 x 8/9 x 8/9 --three tones of a tritone) in the content of Pythagorean tuning. In that tuning, all other intervals are 2/3 and 3/4. 512/729 is startingly dissonant in that context. The so-called equal-tempered in so tame in comparison that most contemporary musicians cannot understand what the original fuss was all about.

In the Renaissance, it was 32/45. A whole tone (8/9) higher than than the just major third (4/5). Again, in the context of Renaissance just tuning, it seems rather jarring.

The big problem of the tritone in the original choral contexts is that it tended to disorient the chorus and its sense where the correct pitch should be placed. 71.210.0.143 (talk) 16:35, 10 April 2009 (UTC)Robert Ross

The article didn't say that 7/5 was used in the Medieval or Renaissance periods, it simply listed 7/5 among just tritones. Hyacinth (talk) 08:12, 29 September 2010 (UTC)

I am unable to understand this article

Hello, I am kind of Newbie and I am aware a Simple English version is being written. Still I started reading this article and there were scores of words that do not form part of my vocabulary.

Love, Andreseso (talk) 10:39, 22 May 2009 (UTC)

Such as? See Help:Contents/Links. Hyacinth (talk) 21:09, 22 May 2009 (UTC)

Like much of the music theory at Wikipedia, it only makes sense if you already know what it means. —Preceding unsigned comment added by 72.94.170.222 (talk) 17:49, 3 April 2011 (UTC)

Could you be a little more specific? I notice, for example, that User:Paolo.dL has recently begun making some changes in the introduction. Do you feel these are improvements, or have they made the article less comprehensible for you?—Jerome Kohl (talk) 22:34, 3 April 2011 (UTC)

Toward an English version of this article

In an edit summary, User:Paolo.dL wrote: "Thank you Jerome Kohl. Would you mind if I delete the reference to German language? Knowing it comes from Latin is more than enough. Specifying how the tritone is called in other languages does not seem relevant in this article."

Please do delete the reference to the German/Latin term, or else I will be forced to list the Spanish, French, Albanian, Japanese, etc. terms as well. I'm not actually sure that the Latin term itself has any relevance in English. As a native English speaker myself, with graduate degrees in music earned at American institutions, I have never heard it used outside of musicological seminars and articles discussing Latin treatises.—Jerome Kohl (talk) 16:40, 4 April 2011 (UTC)

I found the statement in the first section of the article. It was inserted by someone else, and I just moved it in the introduction. I believe you misinterpreted it: it did not state that "tritonus" was the Latin term. It listed it as another term (sometimes) used to indicate the tritone. Obviously this meant it was used in English, as this article is in English and all synonyms listed in the intro are supposed to be in English, unless otherwise specified. Initially, before my edits, the text specified that the term was often used in German as well, but I thought it was irrelevant information, and deleted it. Later, it was you who wrote it came from Latin, and inserted again the reference to German language! So, I really cannot understand why you wrote: "delete the reference to the German/Latin term, or else I will be forced to list the Spanish, French, ... terms as well". However, I am glad that you changed your mind. It's good to know that, in your opinion, this term is so rare that it does not deserve to be mentioned. I agree: let's delete it. Paolo.dL (talk) 18:50, 4 April 2011 (UTC)

I did understand all that. When you moved that text, your edit summary questioned whether it was a German or Latin term. It is the term used in German (as well as in Dutch, Danish, Finnish, Norwegian, and Swedish, as can be determined from the interwiki links to the corresponding articles on the Wikis in those languages), though of course it is a borrowing of the Latin word. Your position is perfectly correct, in my opinion. I believe that whoever thought "tritonus" is used in English was wrong. My reference to adding other languages was a (poor) attempt at humour. My apologies for leading you astray.—Jerome Kohl (talk) 19:15, 4 April 2011 (UTC)

I am sorry for not understanding immediately :-). I appreciate your attempt. It is extremely rare to find editors with friendly attitude. Thank you again for your contribution, and please do not stop with your attempts at humour. Paolo.dL (talk) 19:30, 4 April 2011 (UTC)

Evident logic fault in the introduction

I recently rearranged the article to make the introduction more complete and more easily understandable. However, I did not fix a logic fault which was already present in the text (I just moved it from first section into the intro). The logic fault is easy to detect: the "traditional" definition, given in the first sentence of the intro, clearly implies that both the augmented fourth (A4) and the diminished fifth (D5) are tritones, as both span 6 semitones (3 tones). However, the article states that D5 should not be called a tritone (unless A4=D5, as in 12-TET). There are three possibilities to solve this fault:

1. We accept the definition as it is: simple and clear (tritone = 3 tones = 6 semitones), and as a consequence we also accept that both A4 and D5 are tritones (even in tuning systems which assign them different sizes).
2. We make the definition stricter. In short: tritone = A4. For instance: "the tritone is an interval spanning 3 tones, and encompassing ONLY 4 staff positions".
3. We say that both definitions 1 and 2 are used by different authors. In this case, we also need to decide what one we give first, according to some valid criterion (the most commonly used one?).

In my opinion, the definition tritone = 3 tones = 6 semitones is correct. But since I trust the editors who wrote this article, I suspect that the strict definition tritone = A4 is ALSO true. So, I would discard solution 2, and suggest either the first or the third solution, but this is only my personal opinion, and I do not have enough information to take a decision. I need the opinion of an expert to fix this logic fault.

Paolo.dL (talk) 17:09, 5 April 2011 (UTC)

As with so many terms (not just musical ones!), the exact definition for "tritone" varies with time and, perhaps, place. The definition in the lede paragraph is the usual modern definition, which refers to the absolute distance in an equal-tempered conception (not necessarily in 12-EQ tuning) of the pitch space. This definition also applies in many historical periods where an equal-tempered conception is not applicable. When a distinction is made between "tritone" and "diminished fifth", it implies one or both of two things: (1) the absolute distances are different (because of the non-12-EQ tuning system used), or (2) the interval species are different. The latter condition is of particular concern to music theorists of the 16th century, when tuning debates were also of considerable importance. At that time, a sharp distinction was made between the tritonus which, as its name states, contains three consecutive whole tones, and the semidiapente (diminished fifth) whose name means "short of a (perfect) fifth" (not "half of a perfect fifth") and which is divided as semitone–whole tone–whole tone–semitone. This matters because of the types of semitones and whole tones used in just intonation, where two semitones do not necessarily add to exactly a whole tone. (Zarlino explains in his Istituzioni armoniche that in this context "semi" does not mean "half", but rather "less than", "falling short of". Therefore "semitone" means any interval in the range of half of a major second.) Interestingly, for earlier theorists (in the 13th century, for example) this was not an issue and they, like most modern theorists, used the word "tritone" to describe both the augmented fourth and diminished fifth. I qualify "modern theorists" with the word "most" because, although this applies to beginning-theory textbooks and most other contexts, just intonation has again become a concern for a certain number of theorists in the 20th and early 21st centuries, who make the same distinctions as their 16th-century forebears. This article as it stands does not appear to go into this (and of course some proper references would need to be found), but it is already so complicated, turgid, and technical that it has earned cries of "I can't understand this!" This poses a bit of a problem: how not to sacrifice accuracy while at the same time reducing the complexity. This seems to be something of a specialty for you, Paolo—or at least, I have seen you tackle this kind of problem before in music theory articles—so, for the moment at least, I shall stand back and watch the master at work.—Jerome Kohl (talk) 17:55, 5 April 2011 (UTC)

Thank you for this outstanding and extremely interesting contribution. I will study interval species ASAP. In the meantime, I am not convinced (yet) that there is a difference between tone+tone+tone, semitone+tone+tone+semitone, or semitone*6. This is because "tone" is as ambiguous as "semitone". I mean, they are not specific intervals, but just "interval types" (the 12 semitones, 12 tones, 12 intervals composed of 3 semitones, etc.). In other words, unless you use 12-TET or a similar tuning system, there's no such a thing as a semitone, tone, "1.5-tone", "bi-tone", "2.5-tone", and tritone of fixed size... As explained in Pythagorean tuning, 1/4-comma meantone, and 5-limit tuning, every interval type in these tuning systems (except for unison and octave, of course) has at least two possible sizes. I'll think more about this. It may take time before I understand 100%. Paolo.dL (talk) 18:45, 5 April 2011 (UTC)

It is a confusing topic, to be sure, and this may be an argument against even mentioning it in this article. However, in Zarlino and other 16th-century theorists, there is never any question of arbitrarily mixing the different sizes of tone and semitone. The specific use of the major and minor varieties of each is determined by the tuning system used—in Zarlino's case, what he calls the Ptolomaic syntonic diatonic. In any case, in most varieties of just intonation, the minor semitone is chromatic, and the major semitone diatonic, and the species of the semidiapente includes only major semitones, for example.—Jerome Kohl (talk) 19:00, 5 April 2011 (UTC)

Yet we do have a problem. We did not explain the reason why D5 is not a tritone. Everybody can easily understand or read elsewhere that D5 spans 6 semitones, so it clearly meets our definition of tritone. Even if we simply use tritone = 3 tones, rather than tritone = 3 tones = 6 semitones, everybody can see in our main article about intervals (Interval (music)#Main intervals) that both A4 and D5 span 6 semitones. By the way, both are classified as tritones in that table. The notion that 1 tone = 2 semitones (which implies 3 tones = 6 semitones) is too elementary and intuitively appealing to be ignored altogether. Paolo.dL (talk) 19:22, 5 April 2011 (UTC)

I see your point. The problem, then, is "our definition", which is drawn from a 12-tet conception of pitch space (according to which, properly speaking, our interval under discussion should be simply called "6"). It is becoming increasingly clear why beginners are finding it difficult to understand this article. "Tritone", on the surface of it, ought to mean "three tones", yet we are debating whether it is four or five, and whether four and five are really the same thing. Have we gone completely mad?! Well, no, because of course the entire terminological apparatus was devised under a system in which there were not twelve divisions of the octave, but seven (even if unequal ones). The conflict is between the historical (etymological) explanation, and the one that assumes common music-making practice of today. Surely the latter ought to be the starting place, if for no other reason than this is what most beginners coming to this article will be assuming. Explaining why D5 might not be regarded as a tritone is a more refined level of explanation, and should be relegated to a later part of the article, in my opinion. Under this plan, the lede and main definition can, I think, ignore this distinction, though the beginner still may find it confusing to learn that the interval may be either a fourth or a fifth. Quite apart from the historical issues, we need to keep in mind that musical context still may make a distinction, especially if we are going to keep on describing the tritone as a "dissonance". Even the beginner is entitled to know why a G4–C♯5 tritone resolves to F4 (or F♯4)–D5, whereas as G4–D♭5 tritone resolves instead to A4 (or A♭4–C5.—Jerome Kohl (talk) 21:46, 5 April 2011 (UTC)

Ok. Thank you for the info. Everything you wrote makes sense. Let's try to be consistent with the intuitively appealing definition of "whole tone" = 2 semitones = either M2 or d3 (where M2 includes, in just intonation, both lesser and greater M2). Fortunately, this is also the definition given in Whole tone. In this case, are we sure that there is no simple way to write the strict definition of "tritone"? Can we write, for instance, that a tritone is, for some authors, just an interval composed of 3 whole tones (= 6 semitones), i.e. either a4 or d5, and for others, more strictly, an interval composed of three M2's, i.e. an A4? (I am just guessing; I did not verify: does this imply an A4?) Paolo.dL (talk) 15:26, 6 April 2011 (UTC)

At first glance, this formulation ("for some authors") seems to cover the situation admirably. The only thing I might suggest is that "composed of three whole tones", taken literally, could still be read as meaning "three M2s", the parenthetical equation to six semitones notwithstanding. What we are struggling with here is the contrast between an absolute distance (half an octave, measured on a logarithmic scale of course) as opposed to an interval measured by counting along the steps of a diatonic scale segment. In the former case, it would be just as true to say "3 whole tones (=8.5 seventeenths of an octave)", or any other mathematically equivalent expression, though this would hardly be helpful to anyone. In the latter case, we must assume that the reader already knows (or can discover this from a link to Diatonic scale) that the equivalent of a tritone can only be found as TTT or STTS, and not (for example) TSTS, or SSTT.—Jerome Kohl (talk) 19:43, 6 April 2011 (UTC)

This is again an insightful contribution. Thank you. Clearly I was missing something. I need to digest everything, and this may keep some time. I do not dare to edit yet. Paolo.dL (talk) 20:28, 10 April 2011 (UTC)

I managed to grasp everything you wrote, and it makes sense. Just a note: as far as I understand, it is not always true that "the equivalent of a tritone can only be found as TTT or STTS". I guess this is true only if you use a diatonic scale (such as the Ptolomaic syntonic diatonic used by Zarlino, I guess), or a similar scale with less than 12 pitches, and if you accept an important condition:

• always decompose the tritone into existing incomposite intervals,

where existing means that they are formed by notes which exist in the scale (for instance, C# does not exist in C major diatonic scale, so the semitone C-C#, and the tone B-C# do not exist in that scale), and incomposite means that the notes are adjacent (so an interval is incomposite if no intermediate note exists in the scale between the two ends of the interval). Using the above mentioned conceptual framework:

1. If you use a diatonic scale, there's only one interval which can be decomposed into 3 adjacent existing and incomposite tones; this interval is an A4 and the 3 tones are all M2. (notice that without your help I would not have been able to deduce this from the article)
2. However, interestingly, if you use a chromatic scale, the most complete decomposition of a tritone is into 6 semitones. The decomposition into 3 tones is possible and legitimate (because the 3 tones exist in the scale), but it is an incomplete decomposition, as all tones are "composite" intervals in a chromatic scale, and can be further decomposed into semitones. As a consequence, you can always find 12 "tritones" (intervals formed by 3 tones) in a chromatic scale. 6 of them are A4, and the other 6 are d5.

So, depending on the scale you use (diatonic or chromatic), the definition TT = T+T+T takes different meanings. Do you agree with this conclusion? Am I still missing something? — Paolo.dL (talk) 16:43, 12 April 2011 (UTC)

Yes, both accurate and elegantly put. The terms "existing" (not a usual music-theory term, but useful in this context) and "incomposite" (a usual theory term but ordinarily found only in discussions of ancient, medieval, and Renaissance tuning theory) will need to be defined for the benefit of the beginner. Equally for the beginner's benefit, it should probably not be mentioned that the identical sounding interval cannot be called a tritone if the existing intervals are drawn from, e.g., 10-equal or 14-equal tuning, where the octave divided exactly in half is made up of interval of 5 and 7 "semitones", respectively.—Jerome Kohl (talk) 23:42, 12 April 2011 (UTC)

Fixed

I dared to edit, according to Jerome Kohl's explanations and advices. I believe this fixes the logic fault. I did not use the words "existing" and "incomposite". If you think the article can be made clearer by using these terms, feel free to edit. I used the terms "strict" and "broad" to identify the two interpretations of the classical definition: TT = T+T+T. I don't know whether these terms are also used in the literature or not, but they seemed to be appropriate. I am sincerely grateful to Jerome Kohl for his outstanding contributions. I believe his historical review deserves to be published in the article. Paolo.dL (talk) 14:54, 13 April 2011 (UTC)

Mi contra Fa

Historically, the interval of the tritone was Mi-contra-fa, not Si contra fa; Si against Fa implies solfeggio thinking, while Mi against fa is hexachordal thinking. Please read the 18th century theoretical examples before changing this again. --Myke Cuthbert 02:15, 20 February 2007 (UTC)

how exactly does "mi against fa" describe a tritone and not a half step? Der Elbenkoenig (talk) 01:41, 8 October 2011 (UTC)

The question might just as well be asked the other way around, in which case the answer is "only when mi and fa are in the same hexachord". Mi in the "hard" hexachord (the one running from G up to E with all natural notes) is B, while fa in the natural hexachord (the one from C up to A) is F. The notes F and B form a tritone when F is the lower tone, and a semidiapente (diminished fifth) when B is the lower note. Of course, if you take the hard and soft hexachords (the "soft" hexachord is the one from F up to D with a B-flat), mi in the former is (once again) B, but fa in the latter is B-flat (hence a chromatic semitone, or augmented unison, which in the rules of 16th-century counterpoint is a very different thing from a minor second); the reverse puts A against C (a minor third). The dictum does not really apply to derived (transposed) hexachords, where virtually any interval at all may result from mi against fa.—Jerome Kohl (talk) 02:43, 8 October 2011 (UTC)