In music theory, an interval is a difference in pitch between two sounds.
An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.In Western music, intervals are most commonly differences between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C♯ and D♭. Intervals can be arbitrarily small, and even imperceptible to the human ear.
In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.
In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). Examples include the minor third or perfect fifth. These names identify not only the difference in semitones between the upper and lower notes but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭.
Please walk me step by step on how to do it (we don't have imaginary numbers so don't bring that up)
Also how to put signs on the numbers line when I get minus in the root? (non solveable equation)
Sorry for my English.
Exercise:
My solutions:
For events to be simultaneous, the invariant interval must be bigger than zero (spacelike). I got $$I = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 = -(0-1)^2 + (0-2)^2 + (0-0)^2 + (0-0)^2 = -1 + 4 = 3 >0$$. Which is indeed greater than zero, to find the...
Hello, PF
This is Theorem 8 of Chapter 4 of the ninth edition of Calculus, by Robert A. Adams: "Existence of extreme values on open intervals". I have an alternative and easier proof, based on epsilon-delta arguments, but it's not mine, and I don't understand it completely.
The fact is that...
Hello,
I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR.
All us know the physical meaning of timelike paths: basically they are paths followed through...
My initial guess was to calculate the upper and lower value, and then average those two values, but I don't know whether this is correct to make the uncertainty interval symmetric.
After I calculated the average value, I subtracted it form the upper and lower value, and obtained the symmetric...
Now I don't really know much about the subject, I'm primarily just peaking into my textbook to see how to solve this or that exercise. I believe I can figure out how to solve the third question. However I couldn't find how to solve the first two. I know how to find a 95% confidence interval for...
In Abner Shimony's paper "The Reality of the Quantum World", the choice between particle detector and wave interference detector is said to be made "after the photon had interacted with the beam splitter".
A: Isn't it true that, at light speed, time is not passing for the photon? And so, with...
Hey! :o
I want to determine the following sets:
$\displaystyle{\bigcap_{1\leq n\in \mathbb{N}}\left (-\frac{1}{n},\frac{1}{n}\right )}$
$\displaystyle{\bigcup_{n\in \mathbb{N}}\left (-n,n\right )}$
$\displaystyle{\bigcap_{n\in \mathbb{N}}\left (n, 10n^2+50\right )}$
I have done the...
Hello! Can someone help me understand how are confidence intervals for some parameters of a fit different from the errors on the parameters obtained, for example, from the error matrix. I read Bevington and the whole book he mentions that we can use the error from the error matrix to define the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need further help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:
In...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:
In the...
Closed and Bounded Intervals are Compact ... Sohrab, Proposition 4.1.9 ... ...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Proposition...
Hello,
I am attempting to correctly interpret what a confidence interval means.
This is what I know: a confidence interval is a a continuous interval of values with a lower bound and an upper bound centered around a sample mean.For example given a certain population, we are interested in the...
Homework Statement
Let ##\mathscr{F}## be a finite family of open or closed intervals in the line ##\mathbb{R}^1##. Show by an elementary proof that if any ##2## of them intersect, then all of them intersect.
Homework EquationsThe Attempt at a Solution
Here is my attempt, where for now I'm...
Homework Statement
Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated.
If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible...
2000
(a) Find the solution of the given initial value problem in explicit form.
$$xdx+ye^{-x}dy=0, \quad y(0)=1$$
\begin{align*}\displaystyle
xdx&=-ye^{-x}dy \\
\frac{x}{e^{-x}}\, dx&=-y\, dy\\
xe^x\, dx&=-y\, dy
\end{align*}
(b) Plot the graph of the solution
$\quad...
Hello.
I am bewildered by so many different notions of probability distribution percentages, i.e. the proportion of values that lie within certain standard deviations from the mean.
(1) There is a Chebyshev's inequality:
- for any distribution with finite variance, the proportion of the...
I think I have clarified one of my questions about Friedman’s metric. When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?
I thought it was the former. But in cosmology...
\begin{equation} {x}^{6}\ln\left({x}\right)\end{equation}
Find the interval where the function is concave up and down.
The answers I got are:
Concave up: (ln(2), infinity)
Concave down: (0, ln(2))
and for the life of me I cannot figure out why they aren't correct.
any help? (heart)
Homework Statement
In the photo[/B]
https://imgur.com/a/tf4jI
Homework Equations
vf^2 = v0^2 + 2ad
d = v0t + 1/2 a t^2
3. The Attempt at a Solution [/B]
Soo, I am really having difficult on where to start, however, I believe I need to utilize the fact that at the top of the path, the v0 is...
Hi All,
I am having trouble finding a good ref. for finding confidence intervals for population proportions
for small sample sizes; n<30. I have seen suggestions to use simulations, t-distributions, etc. .
Any ref. , please?
Thanks.
Homework Statement
Show that ##(n,n+1) \cap (k,k+1)## is empty, provided that ##n \neq k##.
Homework EquationsThe Attempt at a Solution
[/B]
WLOG, take ##k < n##. Then ##k -n \ge 1## is some natural number. If ##x \in (n,n+1) \cap (k,k+1)##, then ##-(n+1) < -x < -n## and ##k < x < k+1##...
I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 5: Continuous Functions ...
I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show each interval on the number line.
1. | x - 1 | < or = 1/2
Solution:
-1/2 < or = x - 1 < or = 1/2
(-1/2) + 1 < or = x < or = (1/2) + 1
1/2 < or = x < or = 3/2
----[1/2-------3/2]----...
The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show each interval on the number line.
1. | x | < 2
Solution:
-2 < x < 2
This can be written as (-2, 2).
----(-2-----0----2)----
Correct?
2. | x | > 0
Solution:
The point x lies to the...
The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the intervals on a number line.
(A) |x - 4| < 4
(B) |x + 5| >= 2
For (A), I did the following:
-4 < x - 4 < 4
I now add 4 to each term.
0 < x < 8
On the number line, I would need to...
The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the intervals on a number line.
(A) |x| < 2
(B) |x| > 0
For (A), I must plot -2 < x < 2 on the number line. In interval notation it is written (-2, 2). Is this right?
For (B), we have...
Homework Statement
A 49.0 kg sprinter, starting from rest, runs 58.0 m in 9.90 s at constant acceleration.
What is the sprinter's power output at 1.90 s , 3.00 s , and 5.70 s ?
Homework Equations
##d=v_it + \frac{1}{2}at^2##
##F=ma##
##W=Fd##
##P=\frac{W}{t}##
The Attempt at a Solution
I got...
Homework Statement
Verify that if an interval is bounded it must be one of the four following types: ##(a, b)##, ##[a, b]##, ##(a, b]##, or ##[a, b)##.
Homework EquationsThe Attempt at a Solution
I don't quite get what I should actually prove here.
Do I have to see if, for ##A \subseteq S##...
Hi.. I am stuck up with a double integration where one of the integration limit is infinity. I know quadpack (qagi) can handle integration over infinite intervals. But how to make it work for the double integration. Or if there is any other routine that can handle both double integration and...
This is a kind of silly-sounding question I never realized puzzled me until moments ago, when I looked up the algorithm for spherical coordinates in n dimensions.
In two dimensions, we have polar coordinates, consisting of r from 0 to ∞, and θ from 0 to 2π. In spherical coordinates, we have a...
Homework Statement
Question:
A student used a digital stopwatch three time 10 oscillations of a pendulum. The timings were 13.52s, 13.64s and 13.58s
a) Calculate i the average time for 10 oscillations, ii the time period of the oscillations.
b) Estimate the accuracy of the timings, giving a...
Homework Statement
Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n##
Homework Equations
What does that last statement mean? Union for n greater than or equal to one times the interval?
The Attempt at a Solution
I can't understand the question.
The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?
$\tiny\text{LCC 206 8.8.11 Infinite Intervals of Integration}$
$$\displaystyle
I=\int_{1}^{\infty} {x}^{-2} \,dx = 1$$
$$I=\left[\frac{1}{x}\right]_1^\infty=\left| 0-1 \right|=1$$
$\text{the only way apparently to get 1 is to use absolute value ?}$
$\tiny\text{from Surf the Nations math study...
Okay, I am studying Baby Rudin and I am in a lot of trouble.
I want to show that a closed interval [a,b] is compact in R. The book gives a proof for R^n but I am trying a different proof like thing.
Since a is in some open set of an infinite open cover, the interval [a,a+r_1) is in that open set...
Suppose that
f(x) = (x^2 + 10)(4 - x^2).
(A) Find all critical values of f.Critical value(s) =
(B) Use interval notation to indicate where f(x) is increasing. Increasing: =
(C) Use interval notation to indicate where f(x) is decreasing. Decreasing: =
D) Find the x-coordinates of all local...
The question says find apex, low point and the monotonic properties of the functions. a) b) c)...
To find intervals, I use the abc-formula. Example:
f(x) = 3x^3 - 3x
d/dx * f(x) = 3 * 3x^2 - 3, here a=3*3, b= -3 and c=0 (because there is none)
x1 = ( -b + sqrt(b^2 + 4*ac) ) / 2a
x2 = ( -b -...
Homework Statement
Find the regions of increase and decrease for ##f(x) = -3 \cot^{-1}(x)##
2. The attempt at a solution
My instructor and I disagree. The answer he gave was ##(-\infty, \infty)##, while my answer was ##(-\infty, 0) \cup (0, \infty)##.
My reasoning is this: ##-3\cot...
Hello, I'm doing:
Find absolute min/max of f(x)=x^3-12x^2-27x+8
First I found derivative and when I solved, I got x=9 and x=-1.
So I have to find max/min for 3 different intervals:
a) [-2,0]
And I thought absolute max=-1 and absolute min = none?
b) [1,10]
max= none
min= none
c) [-2,10]...
Let $I \subseteq \mathbb{R}$ be an interval. Prove that
1. If $x, y \in I$ and $ x \le y$ then $[x,y] \subseteq I$.
2. If $I$ is an open interval, and if $x \in I$, then there is some $\delta > 0 $ such that $[x-\delta, x+\delta] \subseteq I$.
Hi All,
The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this:
"There is no uniqueness in decimal expansions of real numbers, specially if one wishes to compare numbers (and their decimal expansions) extremely close of one another."
Is this...
Hi, I’m having a problem with a particular case of binomial proportion.
I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...
The HadCRUT4 global surface temperature anomalies in tabular form include confidence intervals.
The GISS surface temperature data is available in tabular text form here http://data.giss.nasa.gov/gistemp/tabledata_v3/GLB.Ts+dSST.txt
- but it doesn't include any confidence intervals.
Can...