Why do diminished chords have 3rd minor , not 3rd diminished ?
and what is the name of this chord : root + 3rd diminished + 5th diminished?
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In a diminished triad the diminished refers to the fifth of the chord and the third in understood to be a minor third.
In a diminished seventh chord the diminished refers to the seventh of the chord and the third is understood to be a minor third and the fifth is understood to be diminished.
The names are not a complete list of the intervals of the chords.
Think of it like the name major scale, not all the intervals in that scale are major, it's just a name, you still need to learn the interval structure.
To some degree the simple names work because the interval structure is understood relative to the diatonic scale. Major/minor triad refers to the third only and the fifth is assumed perfect, because all fifths are perfect except the one above the leading tone, and so the triad built on the leading tone is called diminished referring to the unique fifth in only that one diatonic triad.
In the case of non-diatonic seventh chords, the naming becomes more explicit, sort of listing out intervals. For example a minor major seventh chord is a minor triad with a major seventh added.
Why do diminished chords have 3rd minor?
Because the thirds of the triads built on the leading tone of diatonic scales, or the second degree on minor diatonic scales, are minor thirds.
A diminished chord has a diminished fifth, not diminished third https://en.wikipedia.org/wiki/Diminished_triad
Diminished third is two semitones https://en.wikipedia.org/wiki/Diminished_third
"root + 3rd diminished + 5th diminished" ... if you play that rooted on the first scale degree, it gives a Lydian'ish sound, because it's enharmonic to something like II 7 / I.
Let's say the root is F#, for example. The diminished chord would be F# A C. (If it were a diminished 7th chord, it would be F# A C Eb.)
The diminished third would be Ab. That would make the chord F# Ab C (plus Eb if you add the diminished seventh). That chord exists: it's the German sixth chord.
A minor third is unambiguous, and generally accepted to be consonant ("pleasant sounding").
By using a diminished third, we are dangerously closing in to a non-third-based chord, which are generally regarded as too dissonant for basic harmony (i.e. a different area or level of expertise).
A diminished third is more similar to a "second" interval.
If we define the triads of all degrees of any scale (minor or major) we have major thirds and minor thirds with perfect fifths: the terms major / minor refer to the lower third.
Only the seventh degree ti-re-fa (= B,D,F) in a major scale and the second degree of a minor scale se-ti-fa (= G#,B,D) has a diminished fifth - that's why this chords are named "diminished": the term diminished refers to the dim. 5th.
(In a chromatic minor scale the 3rd degree can contain 2 major thirds (do-mi-se = C,E,G#). This triad is named "augmented" referring to the augmented fifth (+5 or #5).
"Double diminished" would mean that both intervals (3rd and 5th) are diminished: double diminished refers to the third and the fifth! (so double means that both intervals are diminished).
There isn't always a completely logical reason why things are named as they are.
A diminished triad has minor 3rd and diminished 5th
A diminished 7th chord has minor 3rd, diminished 5th and diminished 7th.
I suggest you accept this and move on!
A diminished 3rd is theoretically possible. (So is an augmented 7th.) But they sound so like a major 2nd (and an octave) that there's rarely any point in spelling them that way. And I don't think you'll gain anything from worrying about such extreme cases.
Root, diminished 3rd and diminished 5th (C, E♭♭, G♭) will be heard as C, D, F♯, a bare-bones D7.
Basically there are two types of diminished chords, namely, full diminished and half diminished chords. A full diminished chord is also called a diminished seventh(dim7) chord. A half diminished chord is called a minor seventh flat five(min7b5) chord.
A diminished triad has three equally separated notes of a diatonic scale, the first, flatted third and flatted fifth (1, b3, b5). This is diminished triad. This answers your first question i.e. why does a diminished chord has a minor third and not a diminished third? Note that the diminished triad has three notes which are 1, b3, b5. The b3 is the minor 3rd. The minor third note is made by flatting a major third. So a diminished chord has a minor third and not a diminished third because the diminished triad has a b3 note which is nothing but a minor third.
Also, a diminished third is a flatted minor third. It is the note one semitone behind a minor third. For example, in the diatonic scale of A, the third note is C# which is a major third. Flatting it would take us to C which is a minor third. Again flatting it would take us to Cb or B which is a diminished third.
A full diminished chord is made up by adding a doubly flatted seventh note. So, a full diminished chord has following notes:
Note that the last note(bb7) is not a part of the diatonic scale. For example, the scale of A consists of the following notes:
A B C# D E F# G# A
The first note is A. The flatted third would be C. A flatted fifth would be Eb and the double flatted seventh would be F#. All these notes would make up a full diminished chord. However, to construct a half diminished chord, the notes used will be, A, C, Eb and G. Note that only the last note is different in both the chords.
A full diminished chord is not played as the incomplete tonic parallel chord in a diatonic scale because the double flatted seventh note is not a part of the diatonic scale. What makes the incomplete tonic parallel or the seventh chord of a major chord progression is the half diminished chord or the m7b5 chord because all of its notes are a part of the diatonic scale. For example, in the chord progression of A major scale the seventh chord would be a G#dim (G#min7b5) which contains the notes G#, B, D and F#. These notes are part of the a major scale. A G#dim7 i.e. G# full diminished chord contains the following notes,
G#, B, D, F. Note that the note F is not a part of the A major scale, so a G# full diminished chord is not played as the seventh chord of the a major chord progression. That would be G#m7b5 (half diminished).
I hope that you understood the theory behind diminished chords.
answering the second part of your question i.e. name of the chord: root, dim3, dim5. It is somewhat wrong. It should be root, minor 3rd, diminished fifth. This is not a chord. This is what is called a diminished triad. I have explained this above. Adding a bb7 or a b7 would make full diminished and half diminished chords respectively.
One more important thing. In a full diminished chord i.e. 1, b3, b5, bb7, all the notes are equally spaced so there is no distinct root. The root is ambiguous. All four notes can serve as a root. As a result, four different diminished chords (full dim) can be played with exact same shape and position. For example, the notes A, C, Eb and F# make the Adim7, Cdim7, Ebdim7 and F#dim7. dim7 is a notation used for full diminished chords. So, playing Adim7 is the same as playing Cdim7, Ebdim7 and F#dim7.
Notation used for half diminished chords is m7b5 or chord root followed by a superscript o with a strike through. Notation for full dim chord is dim7 or (note)°. For example, A°, B° etc.