The Diminished Square:, A little chord discovery I made... Options

[Benny G]

Hey,

*WARNING* If you are not interested, or do not know anything about chords, you may as well stop reading now.

A few days ago I was playing around on my piano, testing out a few chord inversions and I made a little discovery about Diminished 7th chords.

I wrote down the notes of 4 of the 12 dimished chords. I found that all the dimished chords were tightly related to one another. This is the image I created.

(IMG:http://i275.photobucket.com/albums/jj299/guruben/Dimished_example3.jpg)

As you can see from the image above, I have written out four Dimished 7th chords in a box form.
Each dimished 7th chord has 4 different inversions. Those 4 different inversions, in unison with the correct bass, it creates four different diminsihed chords.

The chord tonics are ringed in this next picture.

(IMG:http://i275.photobucket.com/albums/jj299/guruben/Diminished_example2.jpg)



The final picture (below) shows you that the tonics are the same on both the "X and Y axis" (if you like)

(IMG:http://i275.photobucket.com/albums/jj299/guruben/Dimished_example1.jpg)

If you found this incredibley boring please don't tell me.
If you found this interesting then please leave your comments below.

To everyone, best of luck in the future.
Ben (IMG:style_emoticons/default/smile.gif)(IMG:style_emoticons/default/smile.gif)

[fsharpminor]

Of course that happens, its really only one diminished 7th chord, not four.
There are in fact really only 3 different ones in practice.

[hello_cello]

i was expecting a picture of a triangle.

[kenm]

Of course that happens, its really only one diminished 7th chord, not four.
There are in fact really only 3 different ones in practice.

True on a piano, or any other instrument tuned in equal temperament with only 12 notes in each octave. The inversions and the other diminished chords on the same bass have different spellings, and in other tuning systems someone with a sufficiently acute ear for pitch (Boulez?) could hear the difference.

A# C# E G A# C# E
. . .C# E G Bb C# E
. . . . . E G Bb Db E
. . . . . . .G Bb Db Fb

[Czerny]

It's because all the notes are a minor third apart, so each diminished 7th is formed from a 'stack' of minor thirds. This is also why there are only really three of them, as fsharpminor said.

[des]

Of course that happens, its really only one diminished 7th chord, not four.
There are in fact really only 3 different ones in practice.

True on a piano, or any other instrument tuned in equal temperament with only 12 notes in each octave. The inversions and the other diminished chords on the same bass have different spellings, and in other tuning systems someone with a sufficiently acute ear for pitch (Boulez?) could hear the difference.

A# C# E G A# C# E
. . .C# E G Bb C# E
. . . . . E G Bb Db E
. . . . . . .G Bb Db Fb

you don't need that good an ear - in (modern) just intonation dim7s sound dreadful, and in authentic baroque tunings the thirds are all different sizes so it just sounds out of tune!

[kenm]

you don't need that good an ear - in (modern) just intonation dim7s sound dreadful, and in authentic baroque tunings the thirds are all different sizes so it just sounds out of tune!

That's very sweeping. How rough a chord sounds depends on other things as much as the pitches; specifically, the timbres of the component instruments,* the registers of the notes and their dynamics.

* Major triads are notoriously dissonant on bells of standard Hemony design, so composers for carillons use them very sparingly.

I wrote down the notes of 4 of the 12 dimished [sic] chords. I found that all the dimished chords were tightly related to one another. This is the image I created.

C Eb F# A
Eb F# A C
F# A C Eb
A C Eb F#

As you can see from the image above, I have written out four Dimished 7th chords in a box form.

What you have written is one diminished 7th (the third line is its root position) and its three inversions. The only diminished intervals in the inversions are A-Eb and F#-C, which are diminished fifths. Eb-F# is an augmented second; C-Eb, F#-A and A-C are minor thirds; C-F# and Eb-A are augmented fourths; C-A, Eb-C and A-F# are major sixths, not diminished sevenths.

[des]

you don't need that good an ear - in (modern) just intonation dim7s sound dreadful, and in authentic baroque tunings the thirds are all different sizes so it just sounds out of tune!

That's very sweeping. How rough a chord sounds depends on other things as much as the pitches; specifically, the timbres of the component instruments,* the registers of the notes and their dynamics.

* Major triads are notoriously dissonant on bells of standard Hemony design, so composers for carillons use them very sparingly.

fair enough! i was thinking of the harpsichords in our department (IMG:style_emoticons/default/blush.gif)

[sgudm]

It's because all the notes are a minor third apart, so each diminished 7th is formed from a 'stack' of minor thirds. This is also why there are only really three of them, as fsharpminor said.

Exactly. Only the spelling changes if you want to change wich note is the root of the chord. And you can handle the root as the 7th degree of a new key - use it to modulate to a new key.

[Composing Head]

Hardly Sherlock Holmes, since a diminshed chord is symmetrical (as someone indicated I think). And there are only 3 diminished chords, it's mathematically impossible to have four diminished chords of a different nature and quality.

[kenm]

you don't need that good an ear - in (modern) just intonation dim7s sound dreadful, and in authentic baroque tunings the thirds are all different sizes so it just sounds out of tune!

That's very sweeping. How rough a chord sounds depends on other things as much as the pitches; specifically, the timbres of the component instruments,* the registers of the notes and their dynamics.[...]

fair enough! i was thinking of the harpsichords in our department (IMG:style_emoticons/default/blush.gif)

Harpsichords have strong fifth and sixth partials, clavichords even stronger, as composers of the 18th C. knew very well (see Turk's late 18th C. clavichord method, in which he recommends composers to write very sparsely for it, because the sound is so rich). I don't think it's a coincidence that both the diminished seventh chord and the piano became popular in the early years of the 19th C. Equal temperament for keyboards came a bit later: middle of the 19th C. for pianos, second half for the organ)..