# Enharmonic Equivalents



## tdc

Occasionally I come across statements like this I saw on another thread: "He insisted to keep the notation of enharmonic notes in the score. (for example, C# should not be replaced with Db.)"

Other than for notational purposes why would that make a difference? I've seen other comments that suggest that enharmonic notes have different harmonic functions, this confuses me. How can two notes or scales that are made up of exactly the same pitches have different harmonic functions?


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## tdc

I believe the reason why the above quote makes sense is because it deals with _microtonal _enharmonic relations where the pitches likely are not exactly the same.

post #47 here
Share and Learn About Techniques of Modern Music

Aside from this type of situation are there any instances where enharmonic notes or scales with exactly the same group of pitches have different harmonic roles?


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## EdwardBast

Consider the two short chord successions below. The first chords in each are enharmonic equivalents, F-A-C-Eb and F-A-C-D# respectively, but in music of the common practice era, one would expect them to occur in different keys and to resolve differently. The former chord would likely be the dominant 7th chord in Bb major or Bb minor, the latter is a chord of the augmented 6th in the key of A minor. In the first chord the active, dissonant intervals are a minor 7th (F-Eb) and a diminished 5th (A-Eb). These intervals one expects to contract; The Eb in both cases descends, the A would ascend because it functions as the leading tone. By contrast, in the second chord the active intervals are an augmented 6th (F-D#) and an augmented 4th (A-D#). These intervals one expects to expand; The D# in both cases resolves upward to E, the A resolves down to G#.

So, short answer: The way the enharmonically equivalent chords are notated suggests their resolution and function. Minor 7ths above the root want to come down, augmented 6ths want to ascend. If they are notated in contradiction of their function, they are less readily readable and comprehensible for the player.









Does that help?


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## tdc

EdwardBast said:


> Does that help?


Yes that makes sense. Thank you.


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## mikeh375

Understanding the same principles Edward has outlined above will help you also understand how spelling a diminished 7th in different ways can lead you into new and not necessarily closely related keys. Because of enharmonic practice, any note of a diminished 7th chord can be spelt and function as, a leading note, a 7th, flattened 9th, depending on the bass, which does not need to be part of the chord. I.e. a dim7th on f sharp has eflat or d sharp in its construction, and so if you play the chord with a d natural bass, the e flat becomes a flattened 9th and should resolve down theoretically, but if you play the same chord with a b natural bass note, the e flat should be spelt as a d sharp and then becomes the leading note into the keys of E.
Clear as mud eh?


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## BabyGiraffe

Edward and Mikeh are not right...

Renaissance, Baroque, Classical and early Romanticism = variety of Meantone tunings 
https://en.wikipedia.org/wiki/Meantone_temperament.
https://en.wikipedia.org/wiki/Quarter-comma_meantone - this one is the most famous, but wasn't as popular among string players, they prefer sharper fifths, so 1/5 meantone is better for strings (the difference is like 2 cents.)

I don't even know why we still use extended meantone notatation for 12 equal... It's not the optimal way to notate it unlike something like 31, but who cares...
I suggest retuning a digital piano to some historical 12note system and listen to how different keys and scales will sound. Not every key will be in tune and augmented sixths will have septimal intervals. You will understand why composers were using different accidentals (some of them had split keyboards harpsichords or organs, so even more than 12 meantone pitches were employed in some compositions).


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## mikeh375

BabyGiraffe said:


> Edward and Mikeh are not right...
> 
> Renaissance, Baroque, Classical and early Romanticism = variety of Meantone tunings
> https://en.wikipedia.org/wiki/Meantone_temperament.
> https://en.wikipedia.org/wiki/Quarter-comma_meantone - this one is the most famous, but wasn't as popular among string players, they prefer sharper fifths, so 1/5 meantone is better for strings (the difference is like 2 cents.)
> 
> I don't even know why we still use extended meantone notatation for 12 equal... It's not the optimal way to notate it unlike something like 31, but who cares...
> I suggest retuning a digital piano to some historical 12note system and listen to how different keys and scales will sound. Not every key will be in tune and augmented sixths will have septimal intervals. You will understand why composers were using different accidentals (some of them had split keyboards harpsichords or organs, so even more than 12 meantone pitches were employed in some compositions).


We are BG, on the not unreasonable assumption of equal temperament and textbook theory.


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## Schoenberg

Try writing out and playing the score of the C# major prelude and fugue in either book of the WTC in Db major.
You will find that it is much different than the original.


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## mikeh375

Schoenberg said:


> Try writing out and playing the score of the C# major prelude and fugue in either book of the WTC in Db major.
> You will find that it is much different than the original.


My dearest Arnold,

The difference is only in the spelling. The sound is exactly the same on a modern piano in equal temperament - it might be harder to read because of double sharps/flats perhaps (sorry not looked them up, but I have played them all at one time), but it is the same sound. 
The theory of enharmonic writing is a practice used by many composers (me included) as a means of opening up new fields of tonality to work in, in other words it is, apart from being theoretically sound, a compositional tool and has been used as such for a long time.
As the op has asked a question, re-spelling a note (especially in a dim7th chord with different bass notes) alters the notes function and therefore resolution as explained above by Edward and me. Just try it to hear the flexibility of the technique.

Issues regarding tuning are not relevant in the enharmonic practice I and Edward describe here, which is classic textbook harmony based on voice leading principles, although in actual orchestral performance, there will be slight deviations in the approach to tuning at times, especially in the strings.

Regardless, the theory and explanation is solid.


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## BabyGiraffe

mikeh375 said:


> Issues regarding tuning are not relevant in the enharmonic practice I and Edward describe here, which is classic textbook harmony based on voice leading principles, although in actual orchestral performance, there will be slight deviations in the approach to tuning at times, especially in the strings.


Classic textbook harmony is based on *historical *practices!!! People weren't playing in equal temperament. 
There are very big differences in what is playable and what sounds good in different systems and how you should notate something in them.
In any good book (Hanson, Straus, Kostka, Francoli, Forte, Perle etc) based on 12 equal you will encounter integer notation and spelling (but not in the sheet music scores, because noone managed to popularize any such system in the mainstream teachings), because it makes more sense than pythagorean/meantone accidentals. 
Mikeh mentions chord's function. "Functional" theory is popular in very Eastern and Northern Europe (you won't find a single modern textbook on it in UK or USA where they teach using Austrian, not German theories ), but it's very limited in scope (not like the traditional or Schenkerian theories are much better). I suggest reading the original Riemann/Oettingen's books. They are based on just intonation and diatonic structures - basically useless as a flexible, global theory.


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## mikeh375

BabyGiraffe said:


> Classic textbook harmony is based on *historical *practices!!! People weren't playing in equal temperament.
> There are very big differences in what is playable and what sounds good in different systems and how you should notate something in them.
> In any good book (Hanson, Straus, Kostka, Francoli, Forte, Perle etc) based on 12 equal you will encounter integer notation and spelling (but not in the sheet music scores, because noone managed to popularize any such system in the mainstream teachings), because it makes more sense than pythagorean/meantone accidentals.
> Mikeh mentions chord's function. "Functional" theory is popular in very Eastern and Northern Europe (you won't find a single modern textbook on it in UK or USA where they teach using Austrian, not German theories ), but it's very limited in scope (not like the traditional or Schenkerian theories are much better). I suggest reading the original Riemann/Oettingen's books. They are based on just intonation and diatonic structures - basically useless as a flexible, global theory.


I agree with a lot of that BG, no need to shout. However, equalT is mostly relevant to the op's query about enharmonic practice. Hansons theories can be adapted to enharmonic practice. Take one of Hansons involuted chords in perhaps a synthetic mode, re-spell a note or two and the voice leading implications become apparent and useful for a composer looking to develop material.
Whether the chord is spelt as integers, letters or in a traditional way, enharmonic technique expands the vocabulary. As a technique (theory) it is not limited in scope, rather the users imagination is limited if they cannot see the potential for new material in the practice. Are you a composer? If so try some moves with differently spelt dim7th chords to see how flexible they are. Also I mention functionality of notes, not chords.
Generally speaking, equalT is implied in most classic harmony books and that is where I am coming from to keep it straight forward and uncomplicated..I mean it is complicated enough as it is right?

I should add that I am referring to within tonal/expanded tonal practice too. Atonality and serial are different and it makes no real sense for enharmonic practice to be employed, rather just convenient spelling for comprehension perhaps, although that too is often better for being influenced by voice leading when it comes to practicality.


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## drmdjones

For what it's worth, I have to agree with EdwardBast and Mikeh375 on this one.


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## millionrainbows

"Are there any instances where enharmonic notes or scales with exactly the same group of pitches have different harmonic roles?"

There are twelve possibilities.

Using white notes in the key of C:

C-E-G can be seen as a C major triad (root-maj3-5);

C-E-G can be seen as an A minor (min3, 5, b7) with no root;

C-E-G can be seen as Fmaj 9 (5, Maj7, 9)

C-E-G can be seen as Bb13 b5 (9, b5, 13)

C-E-G can be seen as G sus 13 (sus4, 13, root)

C-E-G can be seen as Bsus+ b9 (b9, sus4, aug5)

C-E-G can be seen as Gsus 13 (sus4, 13, root)

C-E-G can be seen as F#7 b5 b9 (b5, b7, b9)

C-E-G can be seen as E+ #9 (aug5, root, #9)

C-E-G can be seen as Eb13 b9 (13, b9, Maj3)

C-E-G can be seen as D9 sus (b7, 9, sus4)

C-E-G can be seen as C#maj7 b5 #9 (maj 7, #9, b5)


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## tdc

millionrainbows said:


> "Are there any instances where enharmonic notes or scales with exactly the same group of pitches have different harmonic roles?"
> 
> There are twelve possibilities.
> 
> Using white notes in the key of C:
> 
> C-E-G can be seen as a C major triad (root-maj3-5);
> 
> C-E-G can be seen as an A minor (min3, 5, b7) with no root;
> 
> C-E-G can be seen as Fmaj 9 (5, Maj7, 9)
> 
> C-E-G can be seen as Bb13 b5 (9, b5, 13)
> 
> C-E-G can be seen as G sus 13 (sus4, 13, root)
> 
> C-E-G can be seen as Bsus+ b9 (b9, sus4, aug5)
> 
> C-E-G can be seen as Gsus 13 (sus4, 13, root)
> 
> C-E-G can be seen as F#7 b5 b9 (b5, b7, b9)
> 
> C-E-G can be seen as E+ #9 (aug5, root, #9)
> 
> C-E-G can be seen as Eb13 b9 (13, b9, Maj3)
> 
> C-E-G can be seen as D9 sus (b7, 9, sus4)
> 
> C-E-G can be seen as C#maj7 b5 #9 (maj 7, #9, b5)


These notes aren't enharmonic though. Those are different harmonic roles for notes that are spelled the same way. That said, I've never mapped out the different harmonic possibilities for a given triad like this and it is rather interesting to think about.


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## millionrainbows

tdc said:


> These notes aren't enharmonic though. Those are different harmonic roles for notes that are spelled the same way. That said, I've never mapped out the different harmonic possibilities for a given triad like this and it is rather interesting to think about.


Well, if you spelled them all correctly, some of them would be enharmonic spellings.

C-E-G can be seen as Bsus+ b9 (b9, sus4, aug5) would be (C, E, f double-sharp)

C-E-G can be seen as Gsus 13 (sus4, 13, root) would be (B#, E, G)

C-E-G can be seen as F#7 b5 b9 (b5, b7, b9) would be (C, E, Fb)

C-E-G can be seen as E+ #9 (aug5, root, #9) would be (B#, E, G)

C-E-G can be seen as Eb13 b9 (13, b9, Maj3) would be (C, Fb, G)

C-E-G can be seen as C#maj7 b5 #9 (maj 7, #9, b5) would be (B#, D double-sharp, G)


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## nobilmente

String players are known for making context-sensitve adjustments to the notes they play: for instance, a G# as a leading tone to A may be raised in pitch to accentuate the resolution, whereas Ab would perhaps go the other way. String players typically see only a single line, so the vertical chordal context may not always be self-evident in the way it is in a keyboard score. They have their ears of course, and usually they use them...


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## Guest

EdwardBast said:


> Consider the two short chord successions below. The first chords in each are enharmonic equivalents, F-A-C-Eb and F-A-C-D# respectively, but in music of the common practice era, one would expect them to occur in different keys and to resolve differently. The former chord would likely be the dominant 7th chord in Bb major or Bb minor, the latter is a chord of the augmented 6th in the key of A minor. In the first chord the active, dissonant intervals are a minor 7th (F-Eb) and *a diminished 5th (A-Eb)*. These intervals one expects to contract; The Eb in both cases descends, the A would ascend because it functions as the leading tone. By contrast, in the second chord the active intervals are an augmented 6th (F-D#) and an *augmented 4th (A-D#)*. These intervals one expects to expand; The D# in both cases resolves upward to E, the A resolves down to G#.
> 
> So, short answer: The way the enharmonically equivalent chords are notated suggests their resolution and function. Minor 7ths above the root want to come down, augmented 6ths want to ascend. If they are notated in contradiction of their function, they are less readily readable and comprehensible for the player.
> 
> View attachment 112508
> 
> 
> Does that help?


Very clear.
Just a couple of additional points, if they can help: the augmented 6th in the second chord is more precisely a *German* augmented 6th (compare with their *Italian & French variants*); the vertical interval E-flat to A in the first chord is an *augmented 4th* (not a diminished 5th); the vertical interval D# to A in the second chord is a *diminished 5th* (not an augmented 4th).


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## Guest

EdwardBast said:


> Consider the two short chord successions below. The first chords in each are enharmonic equivalents, F-A-C-Eb and F-A-C-D# respectively, but in music of the common practice era, one would expect them to occur in different keys and to resolve differently. The former chord would likely be the dominant 7th chord in Bb major or Bb minor, the latter is a chord of the augmented 6th in the key of A minor. In the first chord the active, dissonant intervals are a minor 7th (F-Eb) and a diminished 5th (A-Eb). These intervals one expects to contract; The Eb in both cases descends, *the A would ascend because it functions as the leading tone*. By contrast, in the second chord the active intervals are an augmented 6th (F-D#) and an augmented 4th (A-D#). These intervals one expects to expand; The D# in both cases resolves upward to E, the A resolves down to G#.
> 
> So, short answer: The way the enharmonically equivalent chords are notated suggests their resolution and function. Minor 7ths above the root want to come down, augmented 6ths want to ascend. If they are notated in contradiction of their function, they are less readily readable and comprehensible for the player.
> 
> View attachment 112508
> 
> 
> Does that help?


And that is perfectly true, but...
In Bach's chorales (the "371"), he occasionally treats the first note of the cadence as a leading tone and makes an unprecedented move to the subdominant, with the leading tone proceeding to the fifth of the next chord, and not rising to the expected tonic (RM 322).
In the inner voices (alto & tenor) he usually always treats the leading tone in opposition to what the harmony treatises tell us, where he resolves the leading tone not onto the tonic but onto the fifth or third of the resolution chord. Naughty boy, Bach was.


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## EdwardBast

TalkingHead said:


> And that is perfectly true, but...
> In Bach's chorales (the "371"), he occasionally treats the first note of the cadence as a leading tone and makes an unprecedented move to the subdominant, with the leading tone proceeding to the fifth of the next chord, and not rising to the expected tonic (RM 322).
> In the inner voices (alto & tenor) he usually always treats the leading tone in opposition to what the harmony treatises tell us, where he resolves the leading tone not onto the tonic but onto the fifth or third of the resolution chord. Naughty boy, Bach was.


Yes, the only reasonable way to get a full triad for the final chord, but not really relevant to the discussion.



TalkingHead said:


> Very clear.
> Just a couple of additional points, if they can help: the augmented 6th in the second chord is more precisely a *German* augmented 6th (compare with their *Italian & French variants*); the vertical interval E-flat to A in the first chord is an *augmented 4th* (not a diminished 5th); the vertical interval D# to A in the second chord is a *diminished 5th* (not an augmented 4th).


I revoiced the chords after writing the post so the intervals ended up inverted.


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