In mixolydian mode, does the major chord on the subdominant lowered by a half step have any name? For example in E-mixolydian it is Ab-major and enharmonically, it would be the mediant. But what happens when viewing it this way?
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The mediant is a half step below the subdominant in mixolydian mode. What's the point of calling it flat four instead of natural three? Some will say "tuning," as in El Ectric's answer, but I doubt you'll find any actual music where that difference in tuning is significant. So why do you want to maintain this difference? – phoog Dec 20 '23 at 19:08
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Wait a sec, the mediant is also a half step below the subdominant in plain old major keys. – Dekkadeci Dec 21 '23 at 06:39
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If by "subdominant lowered by a half step" you mean the altered fourth degree, I doubt that having a "name" would really matter, as it would be altered even in the other mode that has the mediant a half tone below subdominant, which coincidentally is the ionian mode: writing F♭ in C. The fact that you want to write A♭ instead of G# makes me think that you are probably modulating or have some cadence temporary going beyond the established tonality/mode, and if that's the case, you should consider that chord in the context of what will happen next. – musicamante Dec 21 '23 at 17:29
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@Dekkadeci Indeed, I just wrote mixolydian to confirm whether it makes any difference. The motivation is this YouTube video: https://youtu.be/bgWruQVlY4c?si=HKQsvBfyR2WXHUGn. I was thinking that one could also write the second beat of the first measure as B - C - Ab - Gb in the upper voice and then continuing with a Db-minor chord. – HDB Dec 22 '23 at 01:35
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@HDB - People write C# minor chords, not Db minor chords (unless they're writing in some truly messed-up territory, possibly involving A flat minor in contexts where G sharp minor was probably the better idea). – Dekkadeci Dec 22 '23 at 07:49
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My first reaction to your question is simply you have used the wrong enharmonic spelling.
Without any other context, and working with the basic description that the music is in mixolydian mode, the choice is between a subdominant chord with all the chord tones lowered a half step, or a mediant chord with only the third raised a half step.
Mixolydian mode is a diatonic system with some chromatic alterations. With no other context to help guide enharmonic spelling choices, the basic thing to do is spell things to make the diatonic mode clear.
So, in E mixolydian, the simpler spelling, the spelling with fewer accidentals is a G# major chord using on accidental for B#. By contrast, lowering the diatonic submediant chord of A major requires accidentals on all chord tones A♭ C♮ E♭.
For example in E-mixolydian it is Ab-major and enharmonically, it would be the mediant [with a raised major third]. But what happens when viewing it this way?
What happens is the diatonic aspect of the mode becomes hard to read.
There could be a reason to spell the chord that way. Descending chromatic motion could be a reason. A sudden change in tonic could be another. But, with no context given in the question, it's hard to justify an A♭ spelling.
Michael Curtis
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The problem when viewing it this way is the assumption of G♯ and A♭ (for example) being the same pitch is only true in an extremely narrow set of circumstances, which eliminates any generalisability that this may have had.
In 12-TET, the major chord on the third can be written as A♭ major since it maps G♯ and A♭ to the same pitch. In most other tunings, though, you'd actually be referring to G♯ major (yes, with an F in the key), not A♭ major. Likewise, if you moved up a major third to A♭, you would have actually started from F♭ major (which has a B in the key signature) instead of E major.
000
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1G sharp and A flat are the same pitch in almost all circumstances. There are only very narrow contexts in which the pitch difference between them is significant -- and there are plenty of circumstances other than 12-tone equal temperament where there is no pitch difference. – phoog Dec 22 '23 at 00:03
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@phoog No, they're not. Most people think they are always the same because they've only ever been taught 12-TET and nothing else, and I'm glad to have broken out of that bubble. – 000 Dec 22 '23 at 00:09
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Then you may be interested in the motivation for the question, which I have now added as a comment above. There is even an enharmonic exchange (or maybe not) occurring in a held note. – HDB Dec 22 '23 at 01:42
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1@HDB - I looked at your sheet music, and my gut instinct was to not hold down the note and treat that curved line as a slur, not a tie. Don't try getting too clever with enharmonically respelling notes across ties. – Dekkadeci Dec 22 '23 at 07:56
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But they're also the same in the Werckmeister temperaments, Valotti, Young, etc., and every other 12-tone temperament. Conversely, in flexible-pitch just tuning, the frequency variability of a single named pitch is in the same ballpark as the frequency difference between G sharp and A flat. Further, enharmonic misspellings are also wrong in equal temperament. The correct spelling is about melodic and harmonic function; the choice of tuning follows from the function. The idea that enharmonic differences are significant because of pitch doesn't hold water in practice, nor even in theory. – phoog Dec 22 '23 at 12:00
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@Dekkadeci Same for me. But since it is the same key on the piano, it got me confused and the performer also does not repeat the note. I think one issue is that slurs and ties are denoted with the same symbol, which is a bit strange anyways. – HDB Dec 22 '23 at 17:24
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@phoog Well temperaments are also only a small subset of other tunings. In many ways they are also exceptions. – 000 Dec 22 '23 at 20:56