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The other day I stumbled onto a chord I played between Bb and Dm. It is a Cadd9 but I also played the upper C, so the 8th. So it's C E G C D. The 8th is important to how it plays with various bass options.

So the question is, is there any way to name it to indicate the 8th is included, or is it just Cadd9?

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It's just called Cadd9. Chord symbols describe the pitch(-class) content of a chord, but not the doublings or other voicing considerations (with the occasional exception of the bass note in the case of slash chords). When a specific voicing is required, the chord must be notated explicitly. (That said, you could invent some suitable indicator such as Cadd9[add8].)

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The options for naming notes contained in a chord are limited to 'flat something', 'sharp something', 'slash something'(Lowest note), 'add something' and 'no something'. Apart from the obvious '7', '6', '13', et al.

Since none of those essentially maps the voicings, as long as the appropriate notes are played, that's about all we have. Particularly when it comes to doubling, and particularly doubling the root, or tonic.

That's where chord names have a weakness. The only sure-fire way to ensure the notes played are what you want is to write the dots on the stave.

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