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Many places say that doubling the bass of major chords in 1st inversion is bad practice. So using C major as an example the bass note E would not be doubled.

please see the following from a harmony and voice leading text book.

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Now you can see in the 1st example that shows common soprano lines over a chord progression and the I6 chord has the bass and soprano both with the 3rd of the chord. Can someone please explain. Is there any kind of standard or do people just not agree and you can actually double everything except a leading tone?

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Voice leading gets tricky; parallels are more difficult to avoid nicely.

https://musictheory.pugetsound.edu/mt21c/VoiceLeadingFirstInversionTriads.html

The link points out that one can double the third (of the chord) in first inversion diminished chords D-F-B-D. Of course, doubling the third of minor chords is good.

Another possibility is with parallel first inversion chords, one often doubles the third and root of every other chord. And it's possible to double the third if these are approached in contrary motion.

Another point (that may not apply in Renaissance or Medieval or other non-CPP styles) is that a major chord in first inversion with doubled third sounds like a Neapolitan Sixth. While normally the N6 occurs on the flat scale step (bII6 or Db63 in the key of C) it may be a "secondary Neapolitan Sixth" and may occur on any scale step.

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  • To the first sentence, I would add "and it's harder (for non keyboard instruments) to tune" I would also note in the next-to-last paragraph that the example in question employs contrary motion.
    – phoog
    Commented Aug 24, 2021 at 15:02