Talk:Tritone/Archive 5

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April 08 edit

The tritone wasn't exploited after equal temperament (which doesn't really have a good date to speak of, anyway, if you leave it unqualified. 1600? 1920?). It really began to be exploited as a modulation in the Romantic period by composers like Chopin, Liszt, and Schumann (there are of course earlier examples, but they are less extensive). By the 20th century is was a natural part of the tonal language of composers like Holst, Vaughan Williams, and Rachmaninov. The reason it was used is for that "unexpected" quality of modulating to a key drastically unrelated to the previous. (How much of this should go in the article?)

Inverted at the octave, no interval except the tritone retains its identity. Why qualify it as the only one within the octave, when none without the octave do?

To say it has no "harmonic relationship" is false unless your definition of harmonic interval dictates that every equal tempered interval has no harmonic relationship. The ET-tritone has justifications; they are just more ambiguous than any other interval.

I've put the later points into a bulleted list, because they are more or less a random collection of facts about the tritone. Rainwarrior 19:30, 8 April 2006 (UTC)

By "no harmonic relationship" I mean that because the ratio of 1 to the square root of 2 is mathematically irrational, two vibrating bodies tuned to a tritone that start in unision will NEVER return to unison, whereas when they are tuned to, say, an octave they will be passing the zero point in the same direction simultaneously (which is what I mean by "move in unison" in this context) after the slower has gone through one cycle and the faster through two. This follows from the mathematical nature of the root of 2. --Hugh7 08:20, 8 June 2006 (UTC)

I don't quite understand what process you are describing. Two pitches at a tritone, which "start" in unison? Then they return to unison, while constantly moving in the same direction? Is the speed in this direction geometric or arithmetic in this case? I can't picture what you mean; I'm sorry. What is the "zero point" you are referring to? I'll respond though:

The perfect fifth in equal temperament has the ratio of 1 to the twelfth root of 128, which is no less irrational than the square root of two. Irrationality is not the issue with regard to harmonic relationships; the harmonic relationship is defined more by the whole number ratio it approximates. For instance, the twelfth root of 128 is very close to 3/2. Similarly, in most tonal usage, the (equal tempered) tritone is more or less a stand in for 7/4. The difference is that it is not as closely aligned to this simpler ratio as, say, a fifth to 3/2, or even a major third to 5/4.

It would not be used so extensively in dominant chords if it did not have a harmonic relationship to the rest of the chord. As a place to modulate it is definitely more "distant" than modulation by a fifth or third. My opinion is that it's actually more strongly numerically related to the original key than the semitone relationship (though most harmony books use the circle of fifths as a metric of distance, and will state otherwise; In modulation, however, the most important factor seems to be how many tones are held common, and how many or not). - Rainwarrior 15:35, 8 June 2006 (UTC)

I was using "start in unision" in the lay sense, not the musical sense, or the micro (within one vibration) not macro (measured over many vibrations) sense. If two vibrating bodies start moving from their neutral (unstressed) position (easiest to think of two strips of steel clamped in the same vice being twanged together, or reeds of a reed organ); If they are equal length etc, they will sound in unison and continue to move together. If one is twice as long as the other, it will vibrate half as fast (they will sound an octave) and they will move together past the neutral point after one vibration of the longer one, two of the shorter. However, if they are tuned in the ratio of 1: root 2 because the ratio is mathematically irrational, they will never move simultaneously through their neutral points again. Rainwarrior said "the harmonic relationship is defined more by the whole number ratio it approximates." but the ratio of 1: root 2 does not approximate to any other. I think we are at cross purposes here because I am looking at it from a mathematical/acoustic point of view, not a musical/harmonic one. But isn't "the perfect fifth in equal temperament" a contradiction, because a fifth in equal temperatment is not perfect? --Hugh7 23:08, 24 April 2007 (UTC)

Hugh7, Rainwarrior meant that the equal temperament tritone functions harmonically because of its approximation to just intervals such as 7/5. You're right that two sine waves at an irrational frequency ratio would only cross the middle line at the same time once if sustained indefinitely. I know what you mean. (In fact, they would cross each other an indefinite number of times, but only once per any vertical position.) The ear is forgiving though and tends to hear an interval "as it should be" rather than "as it is". That's why we can listen to someone strumming an out-of-tune guitar and still be able to identify the chord progression (I, IV, V, etc.). It's the same with equal temperament versus just intonation. The ear receives irrational relationships, but tends to overlook the errors and "hear" them as whatever just relationships they are close to. It's hard to prove that, since it deals with the personal experience of perception, but people like Helmholtz found good supporting physiological evidence. A fifth in equal temperament is not "perfect" but it approximates 3/2, which is perfect, so it's "convenient" in harmonic discussions (though false) to speak of a "perfect fifth" in equal temperament. It’s good to remind ourselves of the truth once in a while though.—76.94.244.40 (talk) 00:40, 1 April 2013 (UTC)

Rainwarrior, it's interesting to note that the reason the tritone retains its width when inverted depends entirely on the fact that it's tuned in equal temperament. The just intervals that the tritone approximates do not retain their width when inverted, and those intervals are most likely the original basis of harmony (not the tempered tritone). For example, 10/7 is wider than 7/5, 16/11 is wider than 11/8, 64/45 is wider than 45/32, etc. You can definitely hear the difference between a tritone and its inversion in just intonation. That means that all those "tritone substitution" progressions in chromatic romanticism and later in jazz are artificial. The F-to-B tritone in G7 is not the same interval as the F-to-Cb interval in Db7 in "real" harmony, so you can't substitute Db7 for G7 in "real" harmony, and you can't quickly modulate from the key of C to the surprise key of Gb by going CMaj G7 Db7 GbMaj. We can only get away with that kind of G7 to Db7 substitution because equal temperament confounds all the many kinds of tritone intervals into a single interval (just as it confounds many major seconds into a single interval, etc.). Equal temperament allowed a freedom of modulation (well, a freedom of “frenetic motility” at least, if you will) but only with the associated costs of a loss of musical accuracy and a restriction of the number of destinations to modulate to (only 12 possible tonal centers instead of potentially hundreds of audibly distinguishable targets). Fortunately, electronic technology makes pitch control easy, and lots of electronic composers are now exploiting the wider pitch resource with increased vigor, joining the much-harder-working acoustic folks (for whom increased pitch variety is always problematic). (By the way you said the tritone is a stand in for 7/4. I think you meant 7/5. 7/4 is a minor seventh interval.)—76.94.244.40 (talk) 00:40, 1 April 2013 (UTC)

Pi

Is it true that "a common symbol for tritone is π"? We used "TT" in school. Hyacinth 11:28, 23 February 2006 (UTC)

I have never seen pi used, so I'm removing it until someone provides a source. —Keenan Pepper 13:59, 23 February 2006 (UTC)

For goodness sake that's T.T (Tri Tone), which is commonly used, only someone wrote the two Ts too close together once and someone else interpreted it as a pi symbol.--JamesTheNumberless 15:52, 19 January 2007 (UTC)

Did you mean T.T. (not T.T) with a period after each T? I never saw TT or T.T. used anywhere myself, and I personally don't like either of them, because each requires more than one character to represent an interval, which gets confusing when strung together. Also the real word is "tritone", not "tri tone", so the second "T" in TT or T.T. capitalizes a letter found in the middle of the word, in which case, to be consistent, semitone would have to be abbreviated "S.T.", but whatever, it doesn't matter what any of us happened to see in some class or happen to use ourselves or happen to like. What matters is what can be backed up by citation, of which there still is none for "TT" or "T.T.". By the way, Howard Hanson used "t" (the single, lower case character) to represent the tritone in his "Harmonic materials of modern music: resources of the tempered scale", which absolutely could be cited (here's a peek http://archive.org/stream/harmonicmaterial00hans#page/10/mode/2up/search/tritone), but his system is a little different in that octaves are ignored, and all intervals are reduced to their smallest inversion. For example, a major seventh interval equates to a half step, so his system doesn't really apply in this article. Still, "t" would be so much more convenient than "TT" or "T.T." would be. Whatever.—76.94.244.40 (talk) 21:50, 31 March 2013 (UTC)

Most commonly, as far as I know, S = Semitone, T = Tone, TT = Tritone. The symbols used by Howard Hanson seeem to be quite unusual, difficult to remember, and hence only useful in the textbook in which he operationally defined them. Paolo.dL (talk) 23:09, 31 March 2013 (UTC)

"As far as I know" is nice for the talk page (I'm guilty of it too), but it doesn't help the article. Can you cite the use of TT?—76.94.244.40 (talk) 01:14, 1 April 2013 (UTC)

Free use of tritone started with Romantic period?

Really? What is the definition of "free" being used here?

The works of J. S. Bach, A. Vivaldi and others from the Baroque era are full of tritones.

Granted, it's not "free" use in the sense that the melodic theme of a piece will begin on a tritone.

216.31.219.19 (talk) 16:53, 17 June 2013 (UTC)

This certainly needs qualifying, doesn't it? I suspect the intended distinction is that in Bach, Vivaldi, et cie the tritone always requires resolution, whereas at the very beginning of the Romantic period (which as everybody knows occurred 11 May 1782, about 10 o'clock in the morning) this compulsory usage was abolished to general rejoicing. However, correcting this problem must start with a reliable source, and this may prove to be a tall order. Let us start by closely examining Persichetti's 1961 book, cited in its entirety at the end of that passage. If as I suspect it does not verify this claim, then a source must be demanded.—Jerome Kohl (talk) 17:09, 17 June 2013 (UTC)

Ridiculous Encyclopedia entry: “Tritone”

With so many sensibilities affected on reading the absurd Wikipedia node, Tritone (music theory), I'm nearly as troubled for indecision over which notion needs addressing. I'll concede, to focus on that which is likely most relevant.

The resource, Wikipedia, is an Encyclopedia. An encyclopedia is a collection of knowledge essentials. The article, although accurate perhaps, is grossly overstated in this context. If Wikipedia were an Encyclopedia of Music Theory then, maybe, this node-- as authored, in its current state (20110711) -- is an appropriate candidate.

Analogizing the notion: the reader who seeks a general understanding of the “tritone”, who happens upon this /article/, is likely to become confused about it. If he or she was not already confused, there's a good chance that this article will accomplish it. Likewise, if the reader is confused, seeking a better understanding, this article is not going to satisfy.

In the context of a contemporary encyclopedic resource, to provide information beyond the scope of general education is in high risk of self deprecation of the greater volume, as is clearly illustrated here.

Jsabarese (talk) 10:39, 11 July 2011 (UTC)

Wikipedia contains articles about any topic. There's no restriction. We have detailed separate articles about any musical interval.

In my opinion, this article is quite exhaustive, useful to both beginners and experts, and cannot be made significantly clearer. Unfortunately, the definition of the term tritone is particularly complex, as it needs to refer to a crazy naming convention, which is widely accepted and commonly used in music theory. The article explains everything, but cannot be completely understood unless you study that naming convention, for which a link is provided. We did not create that naming convention, but we are forced to use it. If you don't like it, you can either ignore it, and accept you won't ever be able to understand music theory, or study it.

Paolo.dL (talk) 12:07, 11 July 2011 (UTC)

While I disagree that the article, itself, is ridiculous, I do agree that the text of the article could only be understood by those who already know the subject matter.

I reject the assertion that this article is useful to beginners. I was talking with a friend of mine and he was helping me understand keys and scales and such and he mentioned the Devils Chord. So I came to WP to look it up and did not recognize anything he was telling me. I knew what I was looking for and I was still unsure if I had the right article until the Historical Uses section.

I also reject the assertion that it cannot be made more clear. We have articles on here explaining any number of things, to think that any concept can't be made clear is contrary to Wikipedia's mission. To paraphrase an old addage "It is a poor teacher who blames his students" Padillah (talk) 18:29, 7 November 2012 (UTC)

Since this point has been debated already at considerable length, and the article has been edited with this complaint in mind many times since it was first lodged quite a long time ago now, it would be useful to have some examples of just what you find still could "only be understood by those who already know the subject matter". Without such guidance I am at a loss to know where to begin. For example, you name something called "the Devils Chord", a term I have never before heard and which I cannot find in this article, either, although I do find a reference to the expression "diabolus in musica" as a nickname for the interval of a tritone. Are you trying to say that "chord" here would be clearer to the layman than "music" or "interval"? As far as I can see this would increase, rather than decrease confusion. As another old adage has it: "It is a poor critic who waves his hand at the world and says, 'something isn't right here, fix it'".—Jerome Kohl (talk) 19:05, 7 November 2012 (UTC)

OK, how about the first sentence?

In classical music from Western culture, the tritone is traditionally defined as a musical interval composed of three whole tones.

So a tritone has three notes in it? Is that what you're saying? You mean to say you honestly think some person off the street will understand this with little to no problem?

And the second sentence doesn't help:

In a chromatic scale, each whole tone can be further divided into two semitones. In this context, a tritone may also be defined as any interval spanning six semitones.

So is it three notes or six notes? Why introduce a level of complexity in the second sentence before clarifying the first sentence?

And there is the interchangeable use of "tone" and "pitch" as in the beginning of the second paragraph:

Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones...

And I am not confusing the use of tone within the word tritone with the word tone. I am pointing out that in the first sentence the word for what a layperson would call a note or sound is "tone" ("musical interval composed of three whole tones"). The word used then switches to pitch in the very next paragraph. This is confusing for a layperson. As a layperson I'm telling you - it confused me. Surely you must know that laypeople don't refer to C as a tone. We call it a note. This may upset some people that spent years learning the difference but that is what we call it.

I'm not going to bother mentioning the introduction of different types of scales as being not understood to any degree by laypeople.

Then there's the third paragraph:

In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details).

This means nothing at all to a layperson. I don't even know what to object to, it's that obscure. Having followed the link to the Staff article (a violation of WP:EGG BTW) I realize that you have now used three distinct terms to reference what, to the layperson, is the same thing. Believe it or not this does not make the article easier to understand.

The sentence after that:

The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening and narrowing by one chromatic semitone the perfect fourth and fifth, respectively.

Do you really think that someone that knows what it means to "widen" or "narrow" a perfect fifth doesn't know what a tritone is? And how does introducing the narrowing of a perfect fifth to a diminished fifth help the layperson understand the concept of tritone? (Always assuming that's what was being mentioned, I don't know)

It's content like this that makes us complain that this article is not understood by the layperson. With obstacles like these in mind is there value in trying to edit this article for the layperson? Is this one of those exceptions that just isn't going to be understood until you understand what these other things are first? How do we note that in the article? Is there something that can be added to the heading to warn people this is a technical article and requires detailed knowledge to use? Padillah (talk) 15:32, 9 November 2012 (UTC)

Thank you for the specific examples. I immediately see a recurring problem, which is the confusion of "tone" with "note". This has got one very unfortunate aspect, which has to do with the difference between American and UK terminology ("note" in the UK is often used where Americans say "tone", and from what you say I assume you are from the UK). However, I think the problem is larger than this, and should be addressed. As for using "three distinct terms to reference what, to the layperson, is the same thing", I believe this may have been done in a desperate attempt to address exactly the problem of which you are complaining: explain one and the same thing in three different ways, in the hope that, if the first two fail to get the information across, the third one is bound to accomplish the task. I really don't know what to say about using "widen" to explain "augment", and "narrow" to explain "diminish". Perhaps a little meditation on this passage will produce some alternative that is more meaningful to the layperson. Perhaps in the end you are right: the layperson will never be able to understand such a complex and sophisticated a subject as different degrees of pitch difference.—Jerome Kohl (talk) 17:05, 9 November 2012 (UTC)

Carefully reading over the lede paragraphs, I have to admit they make my brain hurt. In addition, there is at least one statement that is flat wrong: a diatonic scale may well be said to contain only one tritone, but it is not necessarily an augmented fourth—it depends on which of the two notes you start from (e.g., in a C-major scale F up to B is an augmented fourth, but B up to F is a diminished fifth). I am also inclined to agree that all that stuff about different kinds of scales does not really belong in the lede, which should only summarize the main points. The opening sentence, however, is strictly accurate: the traditional definition of a tritone does in fact include the idea that it is composed of three whole-tone intervals, and therefore "includes" four (not three) notes. However, strict accuracy and a useful summary are not necessarily compatible, either, and the distinction between a distance and the way that distance is built up within a scale, while important to understanding musical interval terminology, is probably best delayed until later in the article. This quickly becomes way too complicated for a lede, since it entails explanations of "species" of an interval, such as whether that distance is (virtually) divided into three whole steps, or a half step (semitone, or minor second), two whole steps (whole tones, or major seconds), and another half step. Now my brain is starting to hurt again, so I will stop.—Jerome Kohl (talk) 18:51, 9 November 2012 (UTC)

I think the root problem this article has is an assumption of familiarity. Even when I mention that most laypeople call them notes the assumption is that I'm referring to tones and therefore I'm British. Well, as much as I'd love the accent, I'm not. What I'm trying to communicate is that you are using "tone" in an effort to be accurate while most laypeople, true laypeople, use the colloquial note to refer to the sound of a key pressed on a piano. They believe there is a reason flats and sharps are "different" than regular notes. When someone hits a key on a piano they don't ask "what tone is that". They ask "what note was that". When a layperson, and even some professionals, are singing they don't say "I can't hit that tone", they say "I can't hit that note". So, to explain the concept of an interval we must first start by explaining the correct concept and use of "the individual sounds instruments make are technically referred to as tones". Then maybe we can approach the explanation of their distance from one another on the staff, etc.

I don't mean to imply that the layperson is unable to understand the concept of different degrees of pitch. I mean to say that explaining all the terms and technicalities needed to then explain the different degrees of pitch may be too extensive and intricate for a single article in WP. I don't mean to diminish music theory at all. In fact, quite the opposite. I understand that there are reasons for all these intricacies and special terms and that is what may make this article too intricate... not too complicated. Padillah (talk) 19:23, 9 November 2012 (UTC)

Well, if you really want to go into the issue of accuracy, when we (by which I mean literate musicians, not necessarily laypeople) use the word "note" in a context like this, we often mean "that black spot written on a sheet of staff paper" (for efficiency, let us call this "note₁"), and not "the sound of a key pressed on the piano" ("note₂"), though the word can mean that. This does have implications for this article, since when I hit one of those black keys in the middle of the keyboard, and then hit a white key a little further up, at a distance corresponding to the interval we are speaking of here, I may not think about whether the notes₂ in question are the notes₁ E♭ and A, or D♯ and A or—much worse—D♯ and B, in which case the interval is not a tritone at all (in spite of sounding exactly like one) since it is now a doubly diminished sixth. As for the difference between "intricacy" and "complication", we laypeople use those terms interchangeably ;-)—Jerome Kohl (talk) 20:10, 9 November 2012 (UTC)

Isn't "tone" being used here to mean "major second", not "note"? Double sharp (talk) 05:22, 30 January 2014 (UTC)

A straight fence with n sections has n+1 posts.

While definitions are not supposed to be circular, there is no requirement that they be self-evident. If they where no one would need them.
An editor claimed that the article, and editors in discussion, assume too much.

• In response to "In classical music from Western culture, the tritone is traditionally defined as a musical interval composed of three whole tones," the editor wrote:

"So a tritone has three notes in it?"

This is called the Fencepost error. It is fairly common when learning about music (people are often confused by the difference between interval names and semitones; a minor second (posts) = 1 half tone (section)). However, an explanation of the fencepost error here would assume both that one doesn't know the meaning of the word "interval" in all contexts ("a space between"), not just music, and that one couldn't look it up.
A definition of a musical interval including a definition of "music", "interval", and an explanation of the fencepost error is too much. I believe there is a guideline against this (perhaps Wikipedia:Summary style).

• In response to "In a chromatic scale, each whole tone can be further divided into two semitones. In this context, a tritone may also be defined as any interval spanning six semitones", the editor wrote:

"So is it three notes or six notes? Why introduce a level of complexity in the second sentence before clarifying the first sentence?"

I added the following explanation to the article: "(1 = 1/2 + 1/2)", but I hardly think this is complex. However, I have gone through and changed all uses of "semitone" to "half tone".

• In response to "Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones...", the editor wrote:

"And there is the interchangeable use of "tone" and "pitch" as in the beginning of the second paragraph...I am not confusing the use of tone within the word tritone with the word tone. I am pointing out that in the first sentence the word for what a layperson would call a note or sound is 'tone' ('musical interval composed of three whole tones')."

First, the paragraph in question only uses the word "pitch" and does not use the word "tone" or "note". Second, while the sentence could read "12 tones" (and this is common), previously "tone" is used for "whole tone", thus this may not be used in this context. While the sentence could read "12 notes", this is unnecessary and overly broad as a musical scale does not contain duration, while notes do. One may call an apple a carrot, but this does not make "carrot" the correct term for an apple. If one insists on being confused by this, there is little that Wikipedia can do. Third, in the first sentence the word "tone" is not used for what a layperson or anyone would call a note or sound. The word "tone" is part of the term "whole tone", and refers to an interval (two pitches), not to a single note. If one doesn't know what a note/pitch/sound is and what a musical interval is, and is unwilling to ever find out, such as by clicking the internal links provided to those terms/concepts, then there is little that Wikipedia can do to help. However, I have gone through the article and checked the use of "tone", "note", and "pitch" and altered them in a few cases.

• In response to "In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details)", the editor wrote:

"This means nothing at all to a layperson. I don't even know what to object to...you have now used three distinct terms to reference what, to the layperson, is the same thing"

If one doesn't know what the problem is with something, there is little chance anyone would be able to fix it, and there is little point in demanding it be fixed. The editor does not say what the "three distinct terms" are, leaving other editors only to guess. I can only assume that the editor means "tritone", "diminished fifth", and "augmented fifth", even though those terms are not included in, or mentioned before, the quote provided. Using more than one term is not inherently confusing. Most people aren't confused that a car may be called an automobile. While saying two-and-a-half inches is a half-inch wider than two inches, and then saying it is a half-inch less than three inches (or saying "one more than five", and then saying "one less than seven"), may not provide more clarity, it isn't confusing.

• In response to "The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening and narrowing by one chromatic semitone the perfect fourth and fifth, respectively", the editor wrote:

"Do you really think that someone that knows what it means to "widen" or "narrow" a perfect fifth doesn't know what a tritone is?"

I really think that someone who knows what "widen" ("to increase the width, scope, or extent of") and "narrow" ("to decrease the breadth or extent of"/"to lessen in width or extent") means, can figure out what it means to narrow an object or "a space between" two objects.

To summarize, just because a complaint exists, doesn't mean that it may be taken seriously (like if I complain to the city council that the sky is blue), or that a change should or could be made. Hyacinth (talk) 05:47, 4 December 2012 (UTC)

Major and minor fourth?

These seem to be being used as names for the perfect and augmented fourths respectively (in Italian: "4^ta min", "4: mag:") in a chart of intervals in Thomas Attwood's theoretical and compositional studies with Mozart. The diminished fourth still has this name ("4: dim:"). (The diminished and perfect fifths seem to be labelled the "false" and "true" fifth respectively: "5: falsa" and "5: vera".) In fact in the margin there is a note saying "why can't the fourth have a Superflua [augmented version]". Double sharp (talk) 14:40, 15 March 2014 (UTC)

Historical uses

Wouldn't it be helpful to make it clearer that the equal tempered tritone familiar today is not the same as most older tritones? Close, perhaps, but I wonder how easy it is for people today to hear an equal tempered tritone and think that is the sound that was supposedly the devil in music and something to be avoided. I know I had that misconception for a long time. As I understand it, there are multiple types of tritones found in early temperaments and in just intonation, none of them quite the same as that familiar today. If so, it strikes me as potentially misleading to talk about the tritone's history as if it is a single thing. Does this make sense? Also, I have the impression that the "devil in music" thing was less about how dissonant it was and more about how the tritone's existence made it clear that a "perfect" system of harmony and tuning was impossible. Pfly 06:53, 12 March 2007 (UTC)

There is no historical evidence for the Devil in Music idea going before the middle Baroque era, so it's hard to say how temperament would have affected this idea. The tritone's unacceptability in certain contexts comes from the way it substitutes for a perfect fifth (or, more rarely, P4) and not from the way one tritone differs from another. Discussions of the size of the semitone, tone, major and minor thirds, etc, are common. Discussions of the size of the tritone are rare. --Myke Cuthbert 16:12, 12 March 2007 (UTC)

I just looked up what I could find on the topic in W.A. Mathieu's "Harmonic Experience" book, checking pitch names at the Scala site [1]. Again, I am still too unsure about this to edit the actual page but thought I'd put some thoughts here. Interestingly, Mathieu makes almost no mention of the word "tritone" in his book. His main theme seems to be showing out equal tempered intervals (ET) "stand in" for the more consonant justly tuned ones (JI) described with ratios; and that one ET pitch can "stand for" more than one JI pitch, depending on the harmonic context; and that the context can set up an ET pitch to stand for one JI pitch but then continue as if it was a different JI pitch. He makes a good case for this being one of the major strengths of ET and something composers have exploited, knowingly or not, for the way it can "trick the ear" (ie, sound cool). So in this framework, the term "tritone" is not very useful since it does not indicate what JI pitch is involved. So instead he almost always refers to tritones as "augmented fourths" or "diminished fifths", or less frequently, other terms. In ET an augmented fourth is the same pitch as a diminished fifth; the tritone. But in harmonic progressions, the two have different functions. Anyway, I made a list of the various JI pitches for which the ET tritone could "stand for" harmonically. Some are purely theoretical (too hard to set up a harmonic progression that would bring it out), but at least 4 are heard as different, maybe 5:

The augmented fourth F# 45:32 (590 cents) (aka "diatonic tritone"); the "diminished fifth Gb 64:45 (610 cents) (aka "2nd tritone"); the eleventh partial "undecimal semi-augmented fourth" F# 11:8 (551 cents); and the "grave fifth" or "wretched pa" 40:27 (680 cents), this last being more a "bad fifth" than a "tritone". Other intervals in the tritone range that are more theoretical than useful include "acute fourth" 27:20 (520 cents, more a bad fourth than tritone); the 555 cent quarter-tone; "classic augmented fourth" 25:18 (568 cents); "septimal tritone" 7:5 (583 cents); and the "Pythagorean tritone/augmented fourth" 729:512 (612 cents).

Anyway, I was just curious enough to look up stuff this far, but must attend to other things. Unsure whether this stuff is useful for this page (though its been very useful to me personally in understanding harmony), but thought it worth posting here. And yea, the "devil in music" comment was something dredged up from memory and probably just a theory I randomly speculated about once rather than something real. I'll keep my eye out for references on the topic though! Pfly 19:46, 12 March 2007 (UTC)

Sorry I jumped too much on the "devil in music" aspects of your first post. I think there is a lot of important information in your second comment which should be added to the article. A couple of thoughts though. Many of the names of pitches on the Scala site are not conventionally used, and are in fact historically inaccurate ways of attributing a name to an interval. The one that jumps out at me immediately is "wretched pa" -- "pa" is an Indian note name (from Sa to Pa is basically from Do to Sol), so no pre-ET European composer would have been thinking of their interval as any form of "pa". The terms I recognize are "septimal tritone" and "Pythagorean tritone". In many tuning systems, there would be all sorts of different A4s and d5s in use which don't have conventional names. I don't know which parts of the article would need to be modified to take account of these subtleties, but I suspect not all of it.

The part of the article which I think needs the most work is the use of the tritone from about 1650-1800, and as this is the period where ET thinking became dominant, it may be the best place to add your information. --Myke Cuthbert 19:22, 14 March 2007 (UTC)

Hmm, having research a bit more, I've gotten the impression that of the JI pitches I mentioned only the 45:32, the augmented fourth, is what the word tritone has mostly meant, historically. Maybe the diminished fifth as well, but I'm not so sure anymore. Today in ET I'm sure both are often called tritones due to enharmonic equivalence. The 11:8 pitch is supposedly a possible "blue note" rather than a "tritone", if it is harmonically used at all. The 729:512 Pythagorean tritone might count as a historically significant tritone, if it is not merely a theoretical construct. The rest of the pitches I mentioned seem better called fourths or fifths rather than tritones, except perhaps the 7:5, which as I understand is probably a modern theoretical invention anyway.

So I take it back! Except perhaps something about the difference between an aug4 and a dim5 and how they relate to the term tritone historically, and how once upon a time, the pitch of the tritone was different from that in common use today (the "wretched pa" term comes from Mathieu's book -- he uses Indian sagram terms for prime factors, "pa" being equivalent to "Pythagorean", but a lot easier to say and write!) Pfly 23:51, 14 March 2007 (UTC)

7/5 is far from modern. Offhand, I remember that Giuseppe Tartini wrote about them and a notation for them, but I'm reasonably certain there are earlier references to it. The tuning of a dominant seventh as 4:5:6:7 is really quite natural, and can be frequently heard in vocal quartets, string quartets, and small brass ensembles. It would be strange to think that this extremely stable chord was not used before modern times. - Rainwarrior 05:41, 15 March 2007 (UTC)

FYI I added a brief note about how the tritone is called "The Devils Interval". Jaydubya93 (talk) 12:21, 22 March 2014 (UTC)