I understand that the #5 and b6 or b13 are the same interval and that the terms are used interchangeably.
This got me to wondering: since the 4 is the avoid note and we use a #4 to get a more pleasing sound in many chords and scales, why do we name the interval in a minor ii chord a b5 rather than a #4; e.g. why is it a minor 7 b5 and not a minor 7 #4?
Is there a connection here, or am I just seeing relationships that aren’t important?
Great question here Wendy!
I feel that the best way to explain this is by looking at the associated chord scale or mode for -7b5 chords.
The Locrian Mode
The related mode for -7b5 chords which is the Locrian Mode or the 7th mode of the major scale, here is the C Locrian Mode:
If we look at the scale, we can see that the 11 or 4 is natural, and all other notes in the mode are flatted except C. In comparison to the major scale, we have root, b2, b3, natural 4, b5, b6, and b7.
The b5 could also be viewed as an ‘essential chord tone’ because it is specifically that note which distinguishes a minor chord and a -7b5 chord. To summarise:
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For regular minor chords, we would think 1-b3-5-b7
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For -7b5 chords, we would think 1-b3-b5-b7.
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If we thought of the latter as 1-b3-#11-b7 it might become a little confusing.
Next… the 11th is an extension
Another important point is that the natural 11th can be played on both -7b5 chords and regular minor chords to produce some really hip-sounding extended voicings.
For following -7b5 voicings are derived from the Locrian Natural 2 or the Locrian Natural 9 mode which is the 6th mode of the Melodic Minor Scale:
Again we can see that the 11th is natural, but now we have a natural 2nd degree too, which is why the scale is called “Locrian Natural 2” or “Locrian Natural 9”.
And here are the voicings:
Voicing 1:
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Here we have the root and b5 in our left hand, and then a major triad built from a whole step below the root in our right hand, or right hand plays b7, 9, and 11.
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If we thought of this voicing as root and #11 in our left hand, and then b7, 9, and natural 11 in the right hand, it might become a little confusing.
Here is the voicing played on its own, and then in context of a 251 in Bb Minor:
Voicing 2:
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Here we have the root and b7 in our left hand, and the b5, b7, 9, and 11 in our right hand.
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When I play this voicing I always look for the b5 to place my thumb in the right hand. We can also look at this as a major #5 chord built off the b5. See we have a Gbmaj7#5 in our right hand. I’m not sure if that makes it easier or harder to visualise
but it’s an option.
Here is the voicing played on its own, and then in context of a 251 in Bb Minor:
I hope that helps Wendy. Any further questions on this let me know 
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