Chromatic vs. Diatonic Half Steps

June 5, 2007 at 1:38 AM

There seems to be some confusion on the difference between a chromatic vs. a diatonic half step. In her blog entry Starling-DeLay Day Five: Kurt Sassmannshaus' violin master class Laurie quotes Professor S. who quotes Dorothy Delay "All 1/2 steps are not created equal," Prof. S said. Dorothy DeLay simplified it somewhat, she said that "if the notes have the same name, the half step is wide."

The first part is true but the second is not. A chromatic half step (e.g. g to g# or aflat to a etc.) is smaller than a diatonic half step. (e.g. G to A flat or G# to A etc.).

I haven't the skill or background to demonstrate this myself so I will appeal to authority. Earlier I reviewed How Equal Temperament Ruined Harmony (and Why You Should Care) by Ross W. Duffin in a blog entry.

One of the major points of this book is that a chromatic semitone is smaller than a diatonic semitone. Chapter 3 of the book deals directly with this.

Duffin adds some interesting quotes.

From Leopold Mozart ...according to their proper ratios, notes with flat signs are a comma higher than those in the same position with a sharp sign. For example D flat is higher than C#, A flat higher than G#, G flat than F# and so on.

From Johann Joachim Quantz The large semitone has five commas, the small one only four. Therefor E flat must be a comma higher than D#....

I would be interested to know if anyone can cite other authorities that would support the Delay-Sassmanshaus assertion of the opposite.

From Stephen Brivati
Posted on June 5, 2007 at 1:51 AM

Greetings,
that`s interesting. You`ve cited good references and know the technical side better than me by far. All I can say, which is not what you asked for, is that the intonation I use is as DeLay describes. Thus in the key of D major c sharp as the leading note is very close to d. D flat on the otehr hand as a minor third of b flat minor would be considerably flatter.
Can that be reconciled with what you are saying?
Cheers,
Buri

From janet griffiths
Posted on June 5, 2007 at 6:28 AM

Perhaps rather than debating whether a c sharp is higher or lower than a d flat it would be simpler to say that intonation should always be in the context of the key that is being played in and its role or fuction melodically and harmonically thus even a b when it is the leading note in c major has a slightly different role than when it is the dominant in e major

From Corwin Slack
Posted on June 5, 2007 at 6:37 PM

Since the principles of tuning are fairly well founded and actually quite objective it may be better to find the principle on which that is based.

The diatonic major scale has two sizes of whole steps. The whole step between the second and third degrees of the scale has a ration of 10/9 or 182 cents. The whole step between the sixth and seventh degree of the scale has (and all the other whole steps) have ratios of 9/8) or 204 cents. (There are 200 cents in an equal tempered whole step.)

Now I am jumping completely out of my league (i.e. I may not be stating the correct principle) but perhaps we want to play a sharper leading tone because of that relationship. There may be other acoustic issues at work.

As an aside, my teacher has a rule of thumb for adjusting to just intonation. He says in sharp keys play sharps flat and in flat keys play flats sharp. This rule takes into account our desire to nmaximize the sonority of the open strings. It adjusts for the two sizes of whole step that particularly affect the 3rd degree of the scale.

From Peter Ouyang
Posted on June 6, 2007 at 12:42 AM

The system of tuning described by Sassmanshaus is what almost all classically trained violinists use when playing scales. People try to come up with formalized systems to describe things like this (so-called "Pythagorean tuning", for example), but it boils down to one simple idea -- in scale passages, most people like to hear narrow half-steps because it tends to make leading tones sound more like leading tones. For example, we like to hear a high C-sharp in D major so the C# sounds like it "wants to resolve" to D.

When we hear arpeggios and chords, sometimes people like to hear slightly different tuning, again depending on the context. Take C# in an A major triad. If you tune the C# as a pure major third relative to the A, it's actually quite flat compared to equal temperament -- about 14 cents. Where you place the C# affects the quality of the chord. When it's closer to the pure major third, the chord will probably sound more "blended" , but if the C-sharp is tuned higher, the chord will sound a bit "brighter" or perhaps a little "jangly." It looks like the Leopold Mozart quote is describing a system of "just intonation", where for example in A major the C# gets lowered so you get a pure A major triad.

From Pauline Lerner
Posted on June 6, 2007 at 1:19 AM

I like Janet's explanation. The key in which the piece is played determines the relationships of the notes.

From Corwin Slack
Posted on June 6, 2007 at 2:59 AM

Sassmmanshaus is correct if we tuned in Pyathagorean scale. But the Pythagorean scale is not useful in modern harmonic style due to the wolf 5th.

Duffin's book describes and acknowledges expressive tuning i.e. sharp leading tones(and its basis in the Pythagorean scale) but he says that while the ear may prefer it in melodic passages it destroys harmony altogether.

I have frequently seen people comment (here on v.com) that the old violinists played out of tune. Duffin analyzed a recording of Joachim and found that he played very well in tune using just intonation.

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Chromatic vs. Diatonic Half Steps

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