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I am going to start with something that may seem very obvious, but I don't think it is.

First, a statement: it is very clear that a simple tritone will show up with equal frequency as an aug4 or a dim5.

But how many other intervals have two spellings that are approximately as common?

I have my own ideas about this but would be interested, first, in what other people assume. The question may be more complicated than it first seems to be.

Technically, wouldn't the aug4 br written as a 4th with a sharp on the top note (except for the B-F one), and a dim5 be written as a 5th with a flat on the top note? Or are you not concerned with how it's spelled but rather how it's defined?

What gets a little dicy is the parameters of what you're asking. For example, what springs to my mind is Maj2 and min7. Same interval, two very common (but different) spellings and applications.

But maybe you'd say no, that's not the same interval, that's just an inversion... but so is your example of aug4/dim5, so maybe that is what your asking?

No. Bb-E is an aug 4th. No sharp. C#-Fx has a double sharp on top. But perhaps you mean simply to raise the 4th? I may misunderstand what you are saying!

OK. I think we are trapped in terminology. In a chord, #4 means to raise a P4 1/2 step. b5 means to lower a P5 by 1/2 step. Is that what you mean?

Let me clarify that first...

Yes, that is what I mean. A aug4 will always be written as a 4th altered somehow, and a dim5 will always be an altered 5th.

Not exactly.

C-F# is an aug4
C-Gb is a dim5

They are not inversions of each other.

C-F# and F#-C are inversions. But it is true that the tritone is the only interval that inverts yet retains the same sound.

What I had in mind was this:

Aug2, m3. Which is more common? At first glance it seem obvious. But in scales, the aug2 is very common - the harmonic minor scale is the most obvious example, but it can happen anywhere, since scales generally are formed alphabetically, when possible.

[cross-posted with the world...]

Morodiene, couldn't an augmented 4th be written with a flat on the bottom note, and a diminished 5th with a sharp on the bottom note? For example, Ab-D, or G#-D. But I think Gary is disintinguishing between interval meaning sound, as determined by two physical keys on the keyboard, or two notes played on an instrument, or sung one after another; and comparing/contrasting that with interval meaning the name we give it.

For example, C-F#, C-Gb, B#-Gb, Dbb-E## all press the same two notes on the piano and sound exactly the same. But they all have different names: augmented fourth, diminished fifth, doubly-diminished sixth, quadrupally augmented second . OK, some of these might practically never appear in music (although Gary is a walking encyclopedia of astonishing notation examples, so I will wait with eagerness for an example with my exotic Dbb-E##). But it shows that intervals as sound is a different thing from intervals as names.

Legal Beagle, is it fair to consider inversions as the same interval? They have neither the same sound nor the same name under any naming convention. They do contain the same two pitch classes, but surely there must be a name for "two pitch classes, considered as pitch classes rather than pitches" that doesn't overload the already overloaded word "interval"? Aug4/dim5 isn't just an inversion: both names can be played by the exact same two physical notes: middle-C and F# above, or middle-C and Gb above. Same sound, regardless of what you call the notes.

Yes, of course. Didn't mean to confuse the issue, just playing a little Devil's advocate with the boundaries. Sorry, carry on and don't mind me.

I've been hung up on the idea of how intervals are taught for quite a while. Intervals have two sides to them: what they are (what we hear - the distance between notes), and how they are named. If we distinguish between the two from the beginning, I suspect that things will be a lot clearer.

The "IS" --- An interval is a distance between two notes. When played together a given interval will have a particular quality of sound such as a "minor second" setting your teeth on edge, the "minor third" being smooth and sad, while the "major third" is smooth and happy (this is subjective of course). If we want to measure that distance, like when using inches on a measuring tape, we might think of semitones or tones.

The "NAMING" --- Involves the actual written notes that appear on the page. For this we count how many notes are involved (CE and CEb are some kind of 3rd because they involve three notes C,D,E). The qualities for each interval are called "major", "minor", "diminished", or "augmented" and these come from what got invented via a major scale and the diatonic notes of that scale - everyone here knows that part.

So we get CD# and CEb, one being called an augmented 2nd, the other a minor 3rd, but on the piano they both create the same quality, use the same piano keys, and the distance we measure between the two keys or pitches is the same. The thing that's the same is the "is" part. It's what we hear.

The Tritone is the only interval name that actually refers purely to the "is" --- refers to what we hear whether it is called an aug4 or a dim5. Of course somebody can turn this around and say "The tritone can have two spellings".

I don't know if this is useful to anyone else, but to me it seems important.

Legal Beagle, I'm familiar with what you wrote. We learned a handy rule of 9: CA is a maj6, AC is a min3, 6 + 3 = 9, and each interval type when inverted because the opposite of the other (major becomes minor etc.) They fit together, but they are not the same interval. A maj6 does not sound like a min3, while an aug2 does sound like a min3.

For tritones, I understand that they are written either as 5ths or 4ths - the 5th is lowered a half step, and the 4th is raised a half step. I can't imagine a funky tritone with a double sharped third (CEx) - so is it safe to say that in the real world tritones are ONLY written as 4ths or 5ths?

Btw, as I understand it, the aug4 and dim5 are the only intervals that DO stay the same interval when reversed --- they both remain tritones, whereas for example a major chord, inverted, becomes a minor chord etc.

That is exactly correct.

For those who have not stumbled over these notation problems, I think we have a good analogy in English:

Thair could be a phonetic spelling for three words, including a contraction. In internation phonetics there is a symbol for "th", for the vowel sound in air, stare, pear, and so on. There is also a symbols for the final "r" sound, and that might be omitted for the British pronunciation.

There are obvious advantages in have precise, phonetic symbols. We avoid the problem of there, their, and they're. However, we also lose grammar, and we lose the ability to trace back in time where these spellings came from.

Music notation is not nearly so bizarre as English spelling, but its rules are complex. Notation provides us with additional information, so the various choices we have about how to spell a chord (or other things) have many potential advantages.

The disadvantage is that we lose the pure sound. When we are in equal temperament, a tritone is what it is. It is a sound. Spelling does not even enter our minds when an interval is isolated.

In contrast, a M7 written any other way than the standard way (C-B, F#-E#, Bb-A) is very unusual. I do have an example of when it is written in this manner: E-Eb, which becomes a diminished octave. That happens in a variant spelling of a sharp 11 chord, which I would be interested in discussing in another thread, at another time.

At the moment I can't imagine a tritone being written other than in the two standard ways. Now, having said that, I may run into an exception in the next 24 hours. Usually intervals go no farther than diminished or augmented, and that is generally true for 3rds, 6ths and 7ths. So, for instance, finding an alternate spelling for a P5 will be very hard, or an example of this happening will be hard to find. Generally you can simply rule out doubly diminshed or augmented intervals as so bizarre that you simply don't ever have to worry about them unless you run into one - at which point you will probably do what I do, curse the composer!

It is useful to mentally link inverted intervals, but to use the same name for them will usually result in great confusion. We could, for instance, say that a M2 is an inversion of a m7, or vice versa, but we don't want to give them the same name. Now, having said that, what happens when we invert a G7 chord? Well, the G-F, m7, becomes a M2, if we do not consider open voicings. So the concept of inverting intervals is useful.

Exactly!!!

Sorry for my mix-up. That's why I checked!

I agree.

Yes, but what I would like to get away from, for sound only, are the names you just used. I would prefer half step, step or whole step, etc.

m3 could be called a quarter octave, just as a tritone is a half octave.

M3 could be called a double-step. I would prefer to teach intervals by number only, 2 3 4 5 6 7, as an intro and use the other descriptions I just used to fine tune the exact distance. I need this for my very young students, but it also helps my older students.

I will GET the standard names later. I am not really doing anything new, but the order in which I present things is reversed.

And perfect - p4 and p5, for example.

Which is why I am suggesting "quarter octave". For those who already know the conventional system, this may be annoying, but I know for a fact that it helps beginners.

But your point is correct. I would add that the octave is almost 100% standard in its spelling, and both the P4 and P5 are very close to 100% standard. And we can take care of m2 and M2 with 1/2 step and (whole) step. The problem intervals for using just one name are the 3rds, the 6ths and the 7ths.

You may see a double aug fourth (AA4th) -- here is my example pulled from the aug6 chord thread:

"Due to some of the awkward voice leading (and confusing accidentals), you'll sometimes see the perfect 5th in a Ger6 chord spelled as a double augmented 4th .. and yes, it is ironic that doing so makes things simpler lol. So.. a ger6 chord built on Eb .. you get Eb, G, C# and A# (instead of Bb). We do this because, in voice leading, there is a general rule that raised notes should resolve upwards and lowered notes should resolve downwards.

In the example of the Ger7 chord spelled Eb G C# A#, it resolves to a V6/4 - V5/3 (in this example - D). When you spell the Bb enharmonically as A#, the A# resolves upwards to B. This is preferred to a Bb resolving upwards to B."

I just looked this up in my book, and one of the examples is Beethoven's Piano Sonata op. 57, second movement, mm 6. The last eighth note of the bar spells a ger6 chord as Bbb-E (there's your AA4th) -G-Db. Yikes. I don't think it's terribly common though.

Please see my clarification with Gary's post afterwards. I was just saying that anything called an aug4 would have to be some form of 4th, and I had given the sharped top note as an example of such, but not the ONLY way something could be aug4.

Now I understand. Thank you, Morodiene.

I see what you are saying. I am thinking, however, of calling these things: whole step & half step; semitone & tone (Cdn, British); tritone; quarter octave & octave --- as descriptions and/or measurements. I am thinking of reserving the word "name" for the "grammar" part. We WILL run into terms like "minor 3rd, augmented 2nd" etc., and they do have their place because of the grammar in music. I think it is important to be aware of these two aspects so that we don't mix them up (or get mixed up by them). What do you think?

The on the "what it is / what it sounds like / pure measurement" side, the idea of half octave and quarter octave has tickled my imagination. I suppose that the "major third" could be seen as a 1/3 octave if it had any use which it probably does not. It is, however, another division of the octave which we can see when we think of the augmented chord.

I'm not sure I'm following this thread and the recent threads fully. Though I'm enjoying them as heck, so keep them up! (And thanks Gary and Keystring mainly (and morodiene) for offering so much already).

Now...

I'm not sure this came up, but even if the aug4th and dim5th are the same pitches, in effect they are used differently and for different reasons. This applies to most tonal music. (eg. C-F# would be resolved to B(b)-G, while C-Gb would be resolved to Db-F(b)). The spelling is there to help with the grammar. An exception to that is always voice and melody leading... As mentioned in the example by LadyChen!

I will agree that it tends to be confusing for beginners, and I will also confess that in my own writings, drafts and scores I very rarely will use this system. I have my own numbering (1-12... duh!) and it serves me fine for my purposes!

Finally, I should note that in other instruments (strings, for example and winds... everything else anyhow) things are NOT equal. An aug4 will tend to sound a bit different than a dim5th! We pianists are so content to have our very specific and set keyboard and forget what's going on outside this! I know for a very fact that if I write a melody as such: C - F# - E the F# will be a bit off for most violinists and it will be my will to do so (especially in a very complicated music). If, on the other hand, I go C - Gb - E the Gb will come closed to the F and thus to the E eventually...

Gary, something I missed in my previous post (dang this forum software!!!)

Actually in contemporary music it's far from uncommon! Apart from the reasons I mentioned above in other instruments, I frequently run into trouble with my bizarre chords. E-Eb might be uncommon, but... E-Eb-Bb-D is another issue altogether. Because if you go for D# instead of Eb, then you have to go to A# instead of Bb and then there's this bizarre A#-D coming your way which is not fine. In which case you put a natural on the bottom note (the natural E) to make sure nobody's confusing it as an octave and you're done!

(And this is why my proof readers do NOT like me! )

Yes, but there is extreme chromaticism right in the middle of tonality. There is a mixture. I posted a reference to a famous Chopin Etude where tritones are used chromatically, and I assure you there is no connection between the choice of aug4 or dim5 and pitch, and there would not be on any instrument.

Then there is another factor: even if we consider a temperament that is extremely different from equal temperament - for example mean tone - the unevenness of intervals has nothing to do with their spelling but rather their position on the keyboard. The dreaded "Wolf Tone", Ab - Eb, is equally horrendous no matter how you spell it. Minor thirds change in each key. Everything changes in every key. So if you play a harmonic minor scale in all keys, the aumented 2nd will be flat or sharp according to the blasted tuning, not according to our preference.

In contrast, stringed instruments have no fixed pitch. There you can "tweak" your intervals as you wish. However, I would argue that adjusting intervals is an intuitive thing, and when it is taught through a system, it won't work well. C F# G will make an intuitive musician want to do something to F#, not because it is written as an F#, but because it tends to sound better a bit sharp.

C Gb F, because of the half step down and the nature of the perfect 5th, is very likely to cause the player (or singer) to shade the tritone a bit lower.

More: what if these intervals are not notated? Then we listen, we play. We shape pitch intuitively, and the possible spelling when notated has nothing to do with it.

Now, what if we learn to blindly use notation to shape our pitches? My argument is that puts the cart before the horse.

One other thing: the amount that pitch is shaded or adjusted in a full orchestra is greatly ruduced by the time you get to very chromatic Romantic music. There you have many instruments, and each wind instrument has certain notes that are sharp or flat to the tempered scale - built in idiosyncracies. The same pitch can be and is flat on one instrument, sharp on another. Trying to make all these different instruments conform to any theoretical non-tempered tuning system when modulation is going on continually would produce chaos.

On the other hand, if something is in a fixed key and very diatonic, no modulations, THEN different tuning tweaks can be used, and some conductors insist on them.

On piano I want my young students to absolutely nail major and minor triads, root position, in all keys. I want them to have dom7 chords in all keys, root position. For me major and minor serves as "home base" for three note chords, not counting doublings, and the dom7 is home base for all sorts of seven chords. For something like a dom7 chord, the moment we start talking about how to write them, I stress four lines or four spaces. That seems logical to me. The trick is to add flats or sharps, never both, until what is notated produces the sound we already know.

This sounds easy, but of course it is not for most students - as you know!

I teach two things at the same time: playing chords by feel/ear, independent of musical notation, and playing notation, following it exactly. To me these are two extremes. One is creative, intuitive, free, exploratory, and when exploring no sound is wrong.

The other involves paying attention to as many composers as possible, subconsciously absorbing their styles and their notational preferences.

In a perfect world those two extremes would merge, and we would have complete musicians. If only that happened more often.

This idea is very different for winds. Wind instruments are in tune with themselves, and adjusting is called "lipping". There are weird problems, and I will give you just one:

On trumpet an E is played 12, second line in the treble. A very good player will be able to lower the pitch, because 1 and 1 together are sharp, but there is a limit to how fast the first valve slide can be moved. In faster passages, the note is sharp. End of story. The octave above is played open, and it is flat, about 13 cents. Yes, you can lip it up, but in faster passages you will always hear that it is flat. Without perfect pitch you can tell what key a trumpet player is in by these idiosyncracies. A medium speed scale sounds radically different in each key, and that quality is fixed, as in non-tempered piano tunings.

Woodwinds have similar problems. If you listen carefully, you will hear that the greatest clarinet players in the world play a little flat in their low register. High notes inevitably go sharp. Ask a trumpet player hitting high G to shade the pitch down, and he will laugh at you. That note is so high, just hitting it every time is a major accomplishment. Even on trombone, where most positions can be shaded, if a first position note is flat (because of the nature of harmonics) there is nothing to be done except to lip it up unless the player is using an unusual system in which he plays ALL positions a bit further out and compensates by pushing in the main tuning slide.

That just skims to surface.

I would say that stringed instruments and the human voice have the greatest pitch flexibility. There a soloist can and often does bend everything, in much the way you described!