Understanding terminology relating to the formula for the relationship between length of a string and its pitch

I am teaching mathematics, and I don't have any background in music. There is a cool example in the textbook that is related to the violin. There are a couple of technical words in that problem that I don't understand. I believe these question might be silly, but I would appreciate if someone could explain a bit. Here are my questions:

Q1: What does it mean by the phrase " the unstopped length" of a violin?

Q2: How many notes are there in between A and C#?

Q3: What is special about the multiplier of 0.944? Is there any explanation behind this choice?

If someone can help me with these three questions, I can solve my math problem. Thanks so much.

NB: If this question is inappropriate for this site, I will delete this post.

Here is the real problem statement:

A violin string is stopped so that the resulting string length makes a desired musical note. In order to make the next higher note,† the string must be shortened using a factor of \$2^{−1/12}\$. That is, the current length is multiplied by 0.944. The length of an unstopped string is 32 centimeters.

(a) Find a formula for an exponential function that gives the length L, in centimeters, of a string that is stopped to make a tone n notes higher than the unstopped string.

(b) One of the unstopped strings makes an A note. To what length (in centimeters) must the string be stopped in order to make C♯, which is 4 notes higher? (Round your answer to two decimal places.) cm

• Q1 - do you mean the unstopped length of the string? Q2 - do you mean between A and the C# immediately above? (It might help to post the whole question, if you can, to provide context). Commented Sep 29, 2021 at 22:04
• Reinforcing @topomorto's comment, the answer to Q2 is context dependent. I've given the most likely number in my answer, but other numbers are possible. Please post the text of the question you're trying to answer. Commented Sep 29, 2021 at 22:20
• Thanks so much. I have edited my post. I included the original problem statement. Commented Sep 29, 2021 at 22:25
• Really sorry to be pedantic, but as it stands this question title and question are not useful for future readers. So I’m going to vote to close. Both the questions asked here, about stopped strings and the inclusive counting of pitches are useful, but need to be asked separately and with titles that relate to the questions. Like I say, I like the content, but it needs to be useful for other readers. Commented Sep 29, 2021 at 22:40
• @BobBroadley I certainly agree that a quick edit to the title is in order, but I see it as one question that happens to have a rather complex answer ("how many notes are there") and one that just happens to be an incidental clarification in support of that question. But BocaPeer, sure, no harm in splitting into two questions. Commented Sep 29, 2021 at 22:48

• The "unstopped" length means the longest vibrating length of the string, known as the "open" string. Other pitches are produced by pressing ("stopping") the string at various points along its length.

• The number of notes between A and C# (inclusive) is 5: A A# B C C#.

• 2^(-1/12) approx. equals 0.944. The real question is: why is the twelfth root of 2 important. Briefly, strings whose lengths are related by a 2:1 ratio are consider to produce the "same" pitch. This is known as the "octave". Most of the music we commonly encounter is based on dividing that octave into 12 "intervals" of equal ratio. That requires the twelfth root of 2. (For anyone who'd like to get into the mathematical weeds on this, see Mathematical difference between white and black notes in a piano. Why are there twelve notes in an octave? also provides very useful history — and a link to the aforementioned post.)

• @BokaPeer Without knowing the question, I suggest opening a new post. Worst case, it'll be closed with a request to add it on to this current one. Commented Sep 29, 2021 at 22:39
• @BokaPeer If the question is unrelated it's best as a separate post; if it's related to this one you can edit your post to include it. I wanted to chime in just to say that, although the math passage doesn't use it, a common phrase to describe this length is "sounding length, to avoid confusion (since there are portions of a string that don't vibrate, like the amount wound around a peg). Commented Sep 29, 2021 at 22:39
• It seems likely there's a Q&A on the site discussing in more detail how 12-TET works and the math behind it. Would appreciate help in tracking it down. Commented Sep 29, 2021 at 23:08
• @bokapeer It's ok if you "feel out" your question out here in comments. Then someone will suggest if it's best asked as another question, added to this one, suitable for another SE site, unsuitable etc
– Rusi
Commented Sep 30, 2021 at 10:38
• @BokaPeer here's a possible list to start with (or look on wikipedia pages for references too) math.stackexchange.com/questions/36683/… ; and the videos and essays here: ams.org/publicoutreach/math-and-music Commented Oct 4, 2021 at 15:02

I try a more didactic approach to answer the questions:

• An "open string" vibrates over its full length. The string is shortened by stopping it, meaning to press it down with a finger at a given place. Then only the part between the finger and the bow will vibrate.

• Between A and C# are 4 semitones; this is already stated in your quote. Musicians include the starting note in their count of an interval however, but only tones of the respective scale are counted, so it is a major third. (This is not asked by your question, but my best guess to interpret Aarons respective answering part.)

• The number 0.944 and its computation on the factor 2 (culture-independent octave frequency multiplier) was already stated. If you shorten a string to 0.944 of previous length, it will sound a semitone higher. If dividing the original frequency by it, one gets the frequency of the semitone higher.