another theory question

[dveej]

I'm studying up for my Grade 2 Rudiments (RCM/RACE) exam, and I have a question:
What does an augmented octave invert to?
I don't know if it follows the rules for compound intervals (it's an octave, so it's not a compound interval, right?) or for simple intervals. If I treat it like a simple octave, then it doesn't seem to work: the note which should be on the bottom is above the other note, so the inversion is a "negative" interval, or something.
Thanks for any help.

[pianonut]

i've never heard of an augmented octave.  maybe a perfect. (fourths, fifths, octaves, if i remember correctly)

but, if you mean a minor 9th. it would be also a minor 2nd. 

[thalberg]

Augmented intervals by rule become diminished when you invert them.  Octaves invert to unisons.  So an augmented octave becomes a diminished unison. 

To play this on an acoustic piano requires incredible ability--only Martha Argerich has ever been able to do it, and it happened by accident right in the middle of Alborado del Grazioso. 

[pianonut]

she was humming at the time.

[BoliverAllmon]

Augmented intervals by rule become diminished when you invert them.  Octaves invert to unisons.  So an augmented octave becomes a diminished unison. 

To play this on an acoustic piano requires incredible ability--only Martha Argerich has ever been able to do it, and it happened by accident right in the middle of Alborado del Grazioso. 

please explain

[dveej]

Boliver Allmon, I think they're having a bit of a joke.

Of course there is such a thing as an augmented octave. And according to the rules I learned, it should invert to a diminished unison. But what would a diminished unison be? It would have to be an interval of which the bottom note is higher than the top note! So I still need help with this, because I don't get it. (There is probably some rule that explains this or takes it into account -- I just wish I could find it!)

[xvimbi]

I think you got it right.

Take an augmented octave, for example C - C#. The C is the bottom note, the C# is the top note 13 half steps away. Note, the interval is measured from the bottom note.

Now "invert" by raising the bottom note by an octave. The C# stays, the bottom C becomes the C an octave higher. The C# is now formally (!) the bottom note (because of the inversion), but it is higher in pitch than the C. The Interval from C# to C is a diminished unision. Voila!

It works the other way around, too, i.e. starting out with the C# as the reference note.

I think the trick is to treat all this in a formal way and not look at the staff until you got the hang of it.

Disclaimer: my theory knowledge has abandoned me in the past, so beware

[dveej]

Thanks, xvimbi.

It seems strange to have an interval where the "higher" note is lower than the "lower" note -- but I guess that's how it is.

[BoliverAllmon]

I think you got it right.

Take an augmented octave, for example C - C#. The C is the bottom note, the C# is the top note 13 half steps away. Note, the interval is measured from the bottom note.

Now "invert" by raising the bottom note by an octave. The C# stays, the bottom C becomes the C an octave higher. The C# is now formally (!) the bottom note (because of the inversion), but it is higher in pitch than the C. The Interval from C# to C is a diminished unision. Voila!

It works the other way around, too, i.e. starting out with the C# as the reference note.

I think the trick is to treat all this in a formal way and not look at the staff until you got the hang of it.

Disclaimer: my theory knowledge has abandoned me in the past, so beware

I think it abandoned you again. A diminished unison would have to be a note that would have a distance shorter than an unison. THat is impossible seeing that the notes are exactly the same. Remember unisons and octaves don't invert. They stay the same.

[xvimbi]

I think it abandoned you again. A diminished unison would have to be a note that would have a distance shorter than an unison. THat is impossible seeing that the notes are exactly the same. Remember unisons and octaves don't invert. They stay the same.

No, no, no, not this time!

An augmented interval is a semitone larger than a perfect (or a major) interval.
C->C: perfect unison (P1)
C->C#: augmented unison (A1)

A diminished interval is a semitone smaller than a perfect (or a minor) interval.
C->C: perfect unison (P1)
C#->C: diminished unison (d1)
(I agree, it sounds weird, but that's the way it is)

In other words, what do you call the intervals from C to C# and C# to C? Augmented unison and diminished unison! Remember, C and C# are both C's, so all intervals between them are called "unisons".

[BoliverAllmon]

but you always read an interval from the bottom up, therefore it would still be an augmented interval.

[xvimbi]

but you always read an interval from the bottom up, therefore it would still be an augmented interval.

Intervals are read from the top to the bottom as well. It all depends on the context. Let's look at the original question again.

Augmented octave C to C# an octave higher
The C is the bottom note
The C# an octave higher is the top note

Inversion leaves the C# unchanged and elevates the C to the C an octave higher

The C an octave higher is the new top note
The C# is the new bottom note
The interval between the bottom note (C#) and the top note (C) is a diminished unison. I agree, it sounds weird, but that's the way it is.

It just happens that the new top note is lower in pitch than the new bottom note.

[BoliverAllmon]

I dont see it as such. We can agree to disagree though.

boliver

[Bob]

I would say an augmented octave would invert to a diminished unison too, and the idea of measuring intervals from the bottom up makes it.... useless?  Like a key signature with 8 or more accidentals.  Possible in theory, but it doesn't really happen.

Diminished intervals are really used for diminished seventh chords, that diminished seventh interval.  And the d5's in a V7 chord.   Otherwise, I can't think of other places diminished intervals are really used -- just d7's and d5's.  Can anyone think of more?  I might just be "mentally fuzzy" right now.

[xvimbi]

I dont see it as such. We can agree to disagree though.

boliver

I don't think you have a choice, unless you make up your own definitions. The interval from C to Cb is called a diminished unison - check out the theory books. "Diminished" simply means "a semi-tone lower".

Bob: Diminished and augmented apply to anything, for example seconds and thirds, not just fifths or sevenths. Any interval comes in diminished and augmented flavors.

[BoliverAllmon]

I don't think you have a choice, unless you make up your own definitions. The interval from C to Cb is called a diminished unison - check out the theory books. "Diminished" simply means "a semi-tone lower".

Bob: Diminished and augmented apply to anything, for example seconds and thirds, not just fifths or sevenths. Any interval comes in diminished and augmented flavors.

true, but you always read from bottom to top.

[xvimbi]

true, but you always read from bottom to top.

OK then, so you don't accept a diminished unison. Fine.

[abell88]

Dveej, I asked my "source" (RCM theory exam marker) about this, and he says he remembers when it came up in an examiners' meeting a few years ago. It was very contentious and they couldn't come to an agreement on it...so the decision was not to put it on an exam ever again. So, although it may be fun to argue about it, don't worry about it if you're doing an RCM/RACE theory exam.

I think a diminished unison is kind of like the mathematical term i, the square root of minus 1...can be useful, but gets your head kind of screwed up if you try to take it literally. 

[BoliverAllmon]

C to C is a unison. To diminish that interval you have to shrink the size of the interval. You can't shrink the same note.

boliver

[Floristan]

To play this on an acoustic piano requires incredible ability--only Martha Argerich has ever been able to do it, and it happened by accident right in the middle of Alborado del Grazioso.

she was humming at the time.

LOL    If anyone could do it, Argerich could. 

[xvimbi]

C to C is a unison. To diminish that interval you have to shrink the size of the interval. You can't shrink the same note.

boliver

This is the misconception. One doesn't SHRINK the interval, one SUBTRACTS one semi-tone from the interval. That is what "diminished" means. The distance from C to C is zero, so subtracting one semi-tone gives minus one semi-tone (e.g. C to Cb, or C# to C)

[cchamp27]

It's like math.

[thalberg]

Okay, all, I was totally joking about the diminished unison!!! They do not exist!!! And about Argerich!!!  I chose Alborado del Grazioso because with all those lightning-fast repeated G#'s, I figured one of those melodic unisons could come out diminished under Argerich's hand--she does the impossible in every other way.  Floristan and pianonut, I'm very glad you appreciated the humor!

I've never said this on this forum before--but I'm sensing you need someone you can trust, so I'll say it now--I have a doctorate in music.  So you can take it on good authority that diminished unisons do not exist.  The direction of the interval does not matter--C to C# is an AUGMENTED unison, and C to Cb is also and AUGMENTED unison.  Obviously a unison cannot be "shrunk," so the diminished unison doesn't exist from that vantage point.  And if you use the subtraction method, what do you get when you subtract a half step from a P1?  Nothing.  Numerically, it's zero.  Just because you go DOWN doesn't mean it's subtraction!  It's NOT subtraction to go down.  You just get an AUGMENTED unison, as I just said.  Neither shrinking nor subtraction can be applied to a unison.  A PERFECT unison is the smallest interval possible.

What you've discovered here with inverting the augmented octave is just a little musical paradox.  I was simply making fun of it.  I debated telling you that it produced a DEMENTED unison. 



 

[thalberg]

By the way, whenever you run into a musical paradox, the way out is usually enharmonics.  Want to modulate up a fifth from C# major?  Impossible!  G# major would have 8 sharps!  It's a paradox. But we still do it--we just write Ab major.  Enharmonics.

With the augmented octave, I say just treat it as a minor ninth like piano-nut said at the beginning.    That's just my opinion. People will still disagree on this, but as mentioned above, it won't be on an exam!

[xvimbi]

I've never said this on this forum before--but I'm sensing you need someone you can trust, so I'll say it now--I have a doctorate in music.  So you can take it on good authority that diminished unisons do not exist. 

Oh what irony. You've got a doctorate in music, and then you make a statement like that Bernhard will be delighted. I hope you are not teaching music theory...

Diminished unisons do "exist" just fine, but one has to accept that the analysis of notes and their structures can be done using certain mathematical algorithms. In fact, negative intervals in general "exist" just fine. They don't even violate the doctrine that intervals are measured from the bottom up. Now, if one can't get their heads around the idea that "bottom" does not necessarily mean "lower pitch" than that is not a limitation of the formalism - doctorate or not

Eric Clapton also plays diminished unisons , not just Martha (there was a similar discussion on a guitar forum), and he is not just humming, but outright moaning.

[BoliverAllmon]

thank you for telling that there is no diminished unison. I learned that in my beginning theory class.

[abell88]

Hmm...I learned in middle school that you can only take a square root of a positive number...then four or five years later they told me that you could take the square root of minus one...but it's "imaginary". I really think you guys are fighting about the same kind of thing...definitions. 

So, here is another way  to think about it: what is the definition of inversion? Does it simply mean to put the lower note up an octave, or the upper note down an octvave? To invert, in the ordinary non-musical sense, means to turn over. When you put that lower note up an octave, you haven't actually turned it over and created a new top/bottom note...you've just shrunk the interval to an augmented unison. To actually turn it over, you could put the bottom note up an octave and put the bottom note down an octave...giving you a diminished 7th. 

Hmm...actually the same would apply to any compound interval (greater than an octave). Does a major 10th invert to a major 3rd, or to a minor 6th? Or...a negative major 3rd???    Do we have to stick to the rule that inverted intervals add up to 9, or can they add up to 16 in the case of compound intervals (and if was a 17th we started with, would it add up to a 24th when inverted??)?

[thalberg]

xvimbi, you seem like a good person and I bet you are a fine musician.  However, there is no irony in my having a doctorate and "making a statement like that." 

Diminished unisons do not exist.  It is not an apt comparison to liken these to imaginary numbers in math.  And we are not "arguing definitions." If you want to get really philosophical about this, you can, but no one in academia will join you on this one.

[xvimbi]

With the augmented octave, I say just treat it as a minor ninth like piano-nut said at the beginning.    That's just my opinion. People will still disagree on this, but as mentioned above, it won't be on an exam!

I am surprised to hear such a suggestion from a PhD in music. A ninth is completely different from an eigth. After all, you don't go around saying that the F# in the G major scale is a Gb. And it wouldn't make things easier anyway, because the inversion of an diminished ninth is formally a negative augmented second!

Look, the whole thing is really not that dramatic. There is a formal (theoretical) way of looking at music, and there is an informal (practical) way. The formal way can lead to results that are a bit difficult to grasp, such as key signatures with 678 sharps, or negative intervals. There is nothing magical about it, and there is certainly nothing paradoxical about it. The formal way, however, is often not very useful, although strictly absolutely correct. That's why we introduce conventions, such as "let's measure intervals from the bottom up". Those are conventions, not strict logical requirements. Conventions are arbitrary and can be debated, the formalism really can't.

So, abell88 is perfectly correct. Those are two different ways of looking at the same thing. I thought I had made that clear throughout the discussion by putting the word "formal" everywhere I thought is was helpful.

And if you use the subtraction method, what do you get when you subtract a half step from a P1?  Nothing.  Numerically, it's zero.

This is the first time I hear that 0-0.5=0   I'll have to tell this to my accountant.

thank you for telling that there is no diminished unison. I learned that in my beginning theory class.

Well, I learned in my first class that there IS a diminished unison, and it was even explained to me WHY that is.

[BoliverAllmon]

Well, I asked our composition teacher about this. he says it inverts to a chromatic second. He says that using that jargon is easier to grasp than negative intervals or anything of the such.

boliver

[xvimbi]

Well, I asked our composition teacher about this. he says it inverts to a chromatic second. He says that using that jargon is easier to grasp than negative intervals or anything of the such.

boliver

So finally, we are coming to the consensus that I offered before: if you don't want to use the term "diminished unison", fine. If you want to use "augmented unison" or "chromatic second", your choice. They are different words meant to describe the same thing.

To be honest, I personally find the negative intervals perfectly OK, because they tell me exactly what is going on. With respect to the original question, only "diminished unison" is really unambiguous. The diminished octave truly inverts to an augmented unison. One can't have two different intervals (diminished octave and augmented octave) invert to the same interval (augmented unison/chromatic second). One would have to specifically name the notes to resolve this, which might be more cumbersome. Likewise, if I take an interval and invert it, and then invert the resulting interval, I should end up with the original interval. So, if we say an augmented ocatve inverts to an augmented unison, and the inversion of an augmented unison is a diminished octave, than we have an interval that is clearly different from the original interval. Again, one can resolve this only by exactly specifying the notes involved. Sounds very cumbersome to me.

[BoliverAllmon]

cumbersome indeed

[abell88]

Well, I went googling. Here's something that disagrees with my ideas:

https://www.thecipher.com/inversions-intervals_2.html


This one claims that the unison cannot be diminished:

https://www.musictheoryresources.com/members/FA_intervals.htm

So,  if we deal with compound intervals in general, the material I've found suggests that you simply subtract an 8ve, so that a major 10th "inverts" to a major 3rd. This seems rather specious to me; nothing has actually been inverted (turned over), only made smaller. The means (raising the lower note an 8ve) has been confused with the end (switching the relative positions of the notes). What do you think?

[BoliverAllmon]

so aug. octave becomes aug. unison?

[abell88]

so aug. octave becomes aug. unison?

Yes...but I don't like it!

[xvimbi]

Well, I went googling. Here's something that disagrees with my ideas:

https://www.thecipher.com/inversions-intervals_2.html


This one claims that the unison cannot be diminished:

https://www.musictheoryresources.com/members/FA_intervals.htm

So,  if we deal with compound intervals in general, the material I've found suggests that you simply subtract an 8ve, so that a major 10th "inverts" to a major 3rd. This seems rather specious to me; nothing has actually been inverted (turned over), only made smaller. The means (raising the lower note an 8ve) has been confused with the end (switching the relative positions of the notes). What do you think?

If you look for "diminished unison", you'll find many sites that describe them as perfectly valid intervals. One of them has the disclaimer that "Some theorists do not allow the diminished unison because the C flat lies below the C natural and this breaks the rule that all dyadic intervals are named from the lower note" (https://www.dolmetsch.com/musictheory12.htm), indicating to me that those folks are in the minority.

The first site you presented explains how to invert a compound interval by raising the bottom note by an octave (so far so good), but they don't FORMALLY treat this note as the new top note (so they don't really invert). The site then acknowledges the shortcoming of this treatment saying that the reverse cannot be applied, i.e. double inversion does not lead to the original interval. Yet, it should, if the concept of intervals is to make any sense at all.

The site https://emozi.com/MusicTheoryIntervals.htm has a table of names for the different intervals. A diminished unison is "d1".

So, I guess, if you want to be precise and accurate, get used to negative intervals. If that's not required and/or you get headaches, then just use the simple way.

[thalberg]

Okay, so Robot, Jenni R, and xvimbi see two people enter a house.  Ten minutes later, three people come out!! Robot says. "self must have counted wrong initially."  Jenni says, "ooh, they must have gotten married and had a child!"  Then xvimbi says, "Now, if one person will go back INTO the house, it will be completely empty!"

(this was just supposed to be funny--no offense to anyone!)

[xvimbi]

Okay, so Robot, Jenni R, and xvimbi see two people enter a house.  Ten minutes later, three people come out!! Robot says. "self must have counted wrong initially."  Jenni says, "ooh, they must have gotten married and had a child!"  Then xvimbi says, "Now, if one person will go back INTO the house, it will be completely empty!"

(this was just supposed to be funny--no offense to anyone!)

This is funny But don't get me started on virtual particles and Heisenberg's Uncertainty Principle. Then, our heads will be spinning thinking about negative mass and energy. Of course, physicists don't have any problems with that, but then again, they don't seem to have any problems with anything.

But you know, there could have been someone already in the house

[BoliverAllmon]

This is funny But don't get me started on virtual particles and Heisenberg's Uncertainty Principle. Then, our heads will be spinning thinking about negative mass and energy. Of course, physicists don't have any problems with that, but then again, they don't seem to have any problems with anything.

But you know, there could have been someone already in the house

geez my head is already spinning.

[Torp]

geez my head is already spinning.

a sure sign of being possessed by the devil.

[BoliverAllmon]

ah yes, the dr. of demonology has spoken. LOL