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The first natural sign means all E5s will be natural, so the first E5 and the second E5 will be E naturals.

But then the 8va sign comes in and the third E is actually an E6 but resides on the same bar level as the previous Es.

Is this E6 an E natural? Or E flat because technically it's in another octave?

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    Have you played it with/without? – Tim Nov 20 '20 at 12:00
  • Musescore treats this as a natural, but I never could completely trust it on that issue. – Dekkadeci Nov 20 '20 at 12:39
  • Have I played it with/without? Do you mean with or without the naturalized E6? – John Nov 20 '20 at 14:25
  • John, I suppose that @Tim does mean to ask whether you've tried it both ways. Nine times out of ten it will be obvious that one option is correct and the other incorrect. – phoog Nov 22 '20 at 1:13
  • I have tried it both ways and I like Tim's answer; that the E6 is a natural, also by looking at the surrounding context, I think – John Nov 22 '20 at 4:45
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It's still a natural. True, the notes at that part of the bar are played an octave higher, but because it's written on that top space, and a previous E was naturalised (?) in the same place, it retains its accidental.

A cautionary natural sign wouldn't go amiss, though.

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    Yes to the "cautionary" bit. WHen there's a chance of misinterpretation, always add the symbols to remove all doubt – Carl Witthoft Nov 20 '20 at 18:03
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    This contradicts the answer here. – guidot Jul 14 at 7:16
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    @guidot - technically, the dots are probably 'wrong' - in that it's not as crystal clear as it should be. However, anyone playing it would (should) realise that it actually needs to be E nat. to sound correct. Hence my last sentence. – Tim Jul 14 at 7:25
  • @guidot the answer there is really insufficiently specific to know what Elaine Gould would think about this question. Maybe she wasn't thinking at all about 8va notation when she wrote that rule. Maybe she covered 8va notation in the next sentence, but the quotation doesn't include it because it's not relevant to that question. – phoog Jul 14 at 10:54
  • @phoog Gould goes on to say that the same pitch with an octave sign (as in the example) requires an additional accidental. Other authorities probably say something different. The only way to avoid this kind of confusion is to always explicitly add these accidentals. – PiedPiper Jul 14 at 15:56
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EDIT:

I was wrong. The best published examples do not support my previous answer (see below).

In fact, it wouldn't even make sense if the rule were that an accidental applies to any note within the measure at the same vertical position, since a note could be on the same line or space but be a totally different letter name, if there were a clef change partway through the bar!

But neither can the rule be that an accidental applies to any note within the measure at the same pitch level.

Instead, the best rule (in piano music, at least) must be that an accidental applies to any note within the measure, as long as that note is at the same vertical position AND at the same pitch level. Anything else should have an extra accidental.

Examples:

Chopin Etude Op. 10 No. 1, Paderewski edition

Chopin Etude Op. 10 No. 1, Paderewski edition:

Here we see that the notes at the same vertical position but different octave get new accidentals. But we also see that the E♭ at the same pitch level but different vertical position gets a new accidental (the 3rd and 5th notes).

Chopin Nocturne Op. 9 No. 2, Paderewski edition

Chopin Nocturne Op. 9 No. 2, Paderewski edition:

Here we see that the A♮ on beat 3 does not apply to the exact same pitch level two eighths later in the left hand. The second A gets a new accidental.

Schumann Symphonic Etudes, appendix variation 5, Henle edition (This line is in the treble clef, with five flats in the key signature.)

Schumann Symphonic Etudes, appendix variation 5, Henle edition:

The C♭ under the 8va sign wouldn't apply to the C that comes later at the same vertical position. But the rule must also be that a note in the same measure which is either at the same vertical position or same pitch level, but not both, needs an accidental no matter what! Ambiguity must be avoided!

Similarly, the E♮ wouldn't apply to the E that comes later at the same pitch level. If that second E were supposed to be natural, a natural would be necessary. But we see that despite E♭ being in the key signature, a flat sign is still used here. Ambiguity must be avoided!

In conclusion, the composer/editor of the example in the original post has done wrong by neglecting the natural sign. It is ambiguous, and leaves it up to the performer to decide the best answer. I often have to do this and it is frustrating to my perfectionism. In fact, it's the reason that I'm reading and writing about this topic today.

ORIGINAL ANSWER

Quoting from Ultimate Music Theory:

An accidental in the Music/Measure only affects a note written on that line or in that space within that measure.

This is clearly referring to vertical position on the staff, not to pitch. So that second E that you've marked is natural because it is written on the same space.

Based on this understanding of accidental rules, which I find more intuitive for sight-reading, your opening statement is inaccurate: "The first natural sign means all E5s will be natural." In fact, if another E5 in that measure were written on a different line or space (because of a different clef or the use of an 8va marking) it would require another natural.

Many people on the internet feel differently. But this is supported not only by the website I've referenced and the increased intuitiveness of this method, but also by real published examples like the one you've shared. So thank you!

Another real published example can be found here: Do accidentals carry through 8va?

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    I'd like to challenge the statement. The referred source seems more on introductory level, where complications like the asked one may more confuse than help. I find the clear rule of "modifying one note in one octave for the remaining bar" more convincing than: "Clef changes terminate the accidental, except they are repeating the same clef with an octave modifier (or equivalent 8va)". The argument to check how it sounds does not help to clarify the notation. Published counterexamples have limited value, one can always find them and they may be deficient (here: lacking cautionary acc.). – guidot Jul 14 at 10:53
  • @guidot - Thank you! In my effort to rebut your challenge, I realized I'm wrong. I will create a new answer based on my findings. – Godwin Friesen Jul 14 at 14:25
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Flats and sharps in key signatures always apply to all octaves.

8va also doesn't affect accidentals, they always apply as usual.

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    Accidentals only apply to the actual notes they are written for. Never for all octaves. – Tim Nov 19 '20 at 16:15
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    Are you sure? It seems clear here that it still applies but I always heard that accidentals only apply to the octave where it appears (see Guido's comment here: music.stackexchange.com/questions/104284/… ) – Tom Nov 19 '20 at 16:16
  • @Tim synched ;) – Tom Nov 19 '20 at 16:17
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    @Tim, I get the point, but how do you call flats and sharps in the key sig, if not accidentals? – MMazzon Nov 19 '20 at 16:27
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    Yea, but what do you call the flats and sharps in the key sig? – MMazzon Nov 19 '20 at 16:28

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