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I know what intervals are, and what constitutes a major, minor, perfect, etc. What I don't understand is the significance of intervals. When I'm practicing a piece, I don't take note of or think about the intervals that are on the paper. There must be a reason that intervals are part of music theory; I just don't know what it is.
 

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Think of it like a language.

Letters-words-sentences-paragraphs.

Notes-intervals-scales(or arpeggios)-melodies.

The theory is simply used to explain the sound.
 

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There are more than a few ways that intervals, per se, are useful.

One place to consider is the relationship of a pair of pitches to a given tonal center : for instance, how does the interval G up to C relate to a tonal center of C? (by "relate", I mean how strongly or weakly is the interval pair a part of that tonality).

But, how does the same interval pair G to C relate to a tonal center of Db ?

Big difference.

To the tonal center of E ? Different from C but less of a difference.

This aspect of interval study is closely related to the sonic implications of overtone series.

Also, stand alone interval pairs have built in implications. A minor second interval "wants" to resolve to the upper pitch. Same notes in reverse, a major 7th interval, "wants" to resolve to the lower pitch.

It's a fascinating study and great for ear training as well, not to mention arranging and composing.
 

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I think the reason why intervals are important in relation to studying music theory is due to their ability to be a stepping stone to understanding more complicated things.

When you don't know things at first, you can use intervals to help you work things out.
For example, working out a scale or a type of chord.

There are plenty more things you can use intervals to help make sense of things that at first can appear challenging.
 

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If notes were letters intervals would be syntax IMO
 

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From the physics standpoint, intervals are ratios between frequencies, the most obvious being a ratio of 2 between octaves. The way we perceive sounds is sensitive to those ratios, they "mean" something. A lot of harmony understanding is based on typical ratios, like the II-V-I progression, which our ears interpret as a resolution. Therefore, the music language relies a lot on those ratios / intervals, both for the underlying chord and the melody itself.
 

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If you didn't have intervals , you'd just have one long note . . . good for your embouchure but gets pretty samey .
 

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One of the most obvious effect of intervals in music is the difference between major (4 semitones) and minor (3 semitones) thirds. A major third sounds bright and happy, a minor third dark and pensive. A typical example is "No More Blues" (Chega De Saudade): it starts in Fm, and after 32 bars, it switches to F major. The first 32 bars are full of "saudade" (brazilian word for blues, melancholy), the melody is based on minor scales; the second part of 32 bars sounds happier, and is built on major scales.
Mozart's Piano Concerto #21 Adagio also uses the same effect, switching back and forth between major and minor in the same key.
The intervals define the mood or the color of the story the music is telling.
 

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One of the most obvious effect of intervals in music is the difference between major (4 semitones) and minor (3 semitones) thirds.

The intervals define the mood or the color of the story the music is telling.
Very true! i was going to make both of these points, but you beat me to it, lol. A lot of beginning improvisors get confused about major vs minor vs diminished, etc. And especially harmonic minor, melodic minor, natural minor and so on. The difference is in the intervals, starting with the fact that the minor 3rd is the definition of a minor key. That's what makes it minor.

But ultimately it's all about the sound. Once you can hear all the intevals, it is much easier to know what is going on in the music and to come up with ideas. If you can identify and use a tritone interval, for example, then you 'own' that sound and can use it at will. Each interval has a sound all its own. Sit down at the piano and listen to the difference between say, a major 7th and a minor 7th, or a perfect 5th and a diminished 5th, both as separate notes and played together. It's huge. And of course you can play those same intervals on the horn, as separate notes.

You may have noticed how changing just one note in a musical phrase (or even a scale or chord) will often give a totally different sound. That's due to a different interval. So don't underestimate the importance of intervals.
 

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Intervals are the distances between two notes. Think of all the notes as if they are part of a number line.

If you spend 10 dollars to open up a lemonade stand and at the end of the day you've got 50 dollars then you made 40 dollars -- that's the net difference, the interval. That interval is the same if you spent 50 to make 90. Spend 1 to make 41. Interval is distance, difference, or the way of counting the keys of a piano between any two notes.

If you play an A and you move up to an F#, the difference/distance/interval is a Major 6th. D to B is also a Major 6th, as is C to A. They will all sound relatively the same in the corresponding context of music. Like the money example, you just start at a different spot.

Here's an example: "Somewhere over the Rainbow". Do-Do-Ti-Sol-La-Ti-Do. 1-8-7-5-6-7-8.
No matter where you start your melody, if you follow the formula and the interval pattern you will end up with the same melody.

Here's an easier one: "Mary Had a Little Lamb". Mi-Re-Do-Re-Mi-Mi-Mi. 3-2-1-2-3-3-3. This melody is comprised of whole steps (intervals).
Play this simple melody in all keys and you will begin to have an answer to the questions you have about intervals. This should answer 'WHY'.
 

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Interval is distance, difference, or a systematic way of counting adjacent keys of a piano between any two notes.
1 key = minor 2d
2 keys = major 2d
3 keys = minor 3d
4 keys = major 3d
5 keys = p4
6 keys = aug 4/dim 5
7 keys = p5
8 keys = minor 6
9 keys = major 6
10 keys = minor 7
11 keys = major 7
12 keys = octave

...easy to understand why we have '12 keys' ? Hopefully this answers 'WHAT'.
 

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One of the best things that happens to you when you start to hear intervals is it gets a lot easier to figure out music by ear. For example when you know what a minor third sounds like, even if its not being played from the root of the scale/chord, you will hear it and play it right away instead of farting around rewinding and trying to figure the passage out using trial and error. This becomes a real freedom later on.
 

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Intervals will only be practical once you internalize them. The ability to hear an interval and either consciously or reflexively think "that is a major sixth" or "oh, that's a tritone!" will help you enormously when learning melodies and chords by ear. For example, it is possible with a basic aural grasp of intervals to start recognizing and naming intervals immediately in passages like Miles Davis's solo on "So What". This translates to having a more precise and quicker grasp on learning those solos by ear.
 

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I had a number of years of ear training at junoir and four year colleges. Eveything from interval and chord recognition to dictation of four part harmony. It was helpful to be in a classroom situation and hear other points of view. Also the discipline and structure made for better concentration. I still practice whenever I hear music.
 
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