If we need to find the note which is a major 6th below G (which is a Bb), do we find it using the major scale of the top note (in this case G) or the major scale of the bottom note ( in this case Bb). Is it the bottom note (Bb) as G major does not have a Bb whereas Bb major has a G?
My understanding is you:
- first get the interval number (second, third, fourth, fifth, sixth, seventh, etc.) by stepping through pitch letters (A, B, C, etc.)
- then determine the interval quality (major, minor, augmented, etc.) and this determined by the number of semitone between the two notes.
- you may need to enharmonically re-name the second note to maintain the correct pitch class and lettering.
Example, a minor third above G: the third letter above G is B, a minor third is 3 semi-tones - stepping up we go G#, A, A# - we want a B and so enharmonically name A# to B flat.
That is a tedious process. So, probably most people use more practical shortcuts. On some instruments like keyboards and fretboards people probably end up learning 'shapes' of intervals. In a major key DO to MI is a major third, lower it a semitone to get a minor third. On a piano two white keys with two black keys in between will be a major third. Eventually you do this without really thinking about it. Sometimes you need to stop and think through intervals like augmented second, augmented sixths, or the use of double sharps and flats typically when the scale tonic is a sharp or flat (like c# minor or D flat major.)
Generally speaking, we number interval from the lower note - it's easier counting upwards. Every interval has its inversion, though, so knowing this will help make sense. Major 6th has its inverse as minor 3 - maj changes to minor, and the numerical part is 'rule of 9'. So an upside down of major 2 is minor 7, and so on.
The perfects stay as such - P4 changes to P5, and vice versa. Augmented change to diminished, and vice versa.So, an aug.5 (C>G#) becomes a dim 4 (G#>C).
The 'major' part of intervals is the datum point, but that gets thrown out when looked at logically - C>D = maj2, but C>Db is called min2 (and Db isn't in the C minor key!!) The intervals called 'major' are, unsurprisingly, all found in the major scale, along with P4 and P5, basing the counting up from the root/tonic of that major scale. Hope that clears things somewhat for you. The other potentially confusing part is when you consider two names for the same note: C>Gb = dim 5, C>F# = aug 4.
EDIT: thus, to actually answer the question - given the higher note, and asked for the lower note of an interval, count up to that note, an octave higher than it should be. Then work out its interval, and invert it ! Here - G>Bb = m3. Inversion = M6. Answer = M6 ! (Often becomes one of the longest car parks in U.K...)
You don't use scales at all for that. You use the universe of all pitches (of which all scales are subsets). Specifically, you count the 'natural' pitches and then adjust the variant of the target note until the distance is correct.
In the example, to find a sixth below G you count G - F - E - D - C - B, so the result is some variant of B. Because you want a major 6th, it's a Bb because that corresponds to the correct distance (9 semitones).
If you wanted the same interval below a G#, it would also be a form of B (B natural), and if you want a major sixth below Gb, that's still a sort of B (B double flat). In particular, it's not an A - the interval between an A and a G above that is always a seventh of some kind. All these calculations have nothing to do with which notes are contained in the scale named after either of the notes.
I think you are going about it a bit backwards. Whatever this note should be it should be six scale degrees below the G. G(1),F(2),E(3),D(4),C(5),B(6).
So for this interval to be a sixth it has to start on a B (of some sort) and go to a G. So you have a choice between Bb and B. B Major has a G sharp for its third sharp. Bb Major only has a Bb and an Eb, so that fits.
Remember a Major interval will have a note that forms part of the Major scale of the root note
Yes, bottom note and use its Major Key and you are correct then.
I'm no expert but the topic is interesting. One thing not mentioned yet is that Intervals, traditionally, form unique important ratios (and music students were taught by music literate people who knew this complexity. That is why (correct me if I'm wrong), they are traditionally an upward span. A ratio with a larger number on one side than the other, such as 3:2, doesn't invert when one plays the two notes in different orders. There is clarity on the lower pitch ALWAYS BEING THE BOTTOM.
You can say, incorrectly, to "go down" a P5 or go up a P5, and confuse the hell out of a music student. But the P5 is already a pair of things, i.e. a ratio. One does not go from one ratio to another. You CAN descend or ascend by a ratio. If someone who may be unclear about this says to go down a P5, i do assume they mean to find the inverted P5 (which is a P4 [found 7 half steps below]). I.e I assume they mean to take the starting pitch as the interval top, and find the P5 bottom for that starting pitch. This can be thought of as going down a P5 span, which u can do (and then from the original pitch it is an inverted P5, and from the pitch arrived at as the new interval bottom it is an univerted P5 of course), but it is not the same as finding the P5 of a chord key or scale - which DEPENDS ON the interval bottom BEING the CHORD ROOT or TONIC - and i can see how these approaches get intermingled in a student's mind.
So if one knows one wants a Perfect, Maj, min, aug or dim interval, and one knows one is starting on its top and want to find its bottom (in 12-ET), one is best to memorize the half step formulas. And count them up to find the interval top (usual case) or down to find the interval bottom.
In the past musicians cared about ratios. So the idea of the ratio being a pair of 2 pitches was obvious. Composers named music after the Key (Concerto in F Major) because every key had distinct ratios. With 12-ET equal tempered tuning this complexity carries on. One learns it to have a grasp of enharmonics and the correct naming/composing of notes (where a G## is correct for instance)
Wikipedia has an interval article which includes some "newfangled" conventions which can't help but lead to more confusion. My advise is stay simple at first. A melody with a Maj 2nd is just 2 notes that distance apart. It can have them in either order and it is still a M2.
Finding the bottom of (or going down by, but certainly not going down to) a M6 (to find the bottom of a M6 interval) actually does just require going down 9 half steps. Naming the note may be the more complex process & depends on the context. But the meaning of the term M6 is approximately a ratio. (In 12-ET it is a complex ratio with large integers, formed by successive multiplication of the 12th root of 2 [ 1.05946...]). The Just & Pythagorean 3:2 ratio is less awkward as a ratio.
When i saw Wiki describe directed and ordered intervals, etc, which deal with less common subjects (and I'd say confusing ones), i thought I'd take an overview of the traditional usage of Interval and I hope its helpful.