Envelope (mathematics)

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

Envelope of a family of curves

Let each curve C_t in the family be given as the solution of an equation f_t(x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=f_t(x, y) and assume F is differentiable.

The envelope of the family C_t is then defined as the set of points for which

for some value of t, where is the partial derivative of F with respect to t.

Note that if t and u, t≠u are two values of the parameter then the intersection of the curves C_t and C_u is given by

or equivalently

Letting u→t gives the definition above.

An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0.

Synthesizer

A sound synthesizer (usually abbreviated as "synthesizer" or "synth", also spelled "synthesiser") is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones. Synthesizers may either imitate the still existing sounds (instruments, vocal, natural sound, etc.), or generate new electronic timbres not existing before. They are often played with a musical keyboard, but they can be controlled via a variety of other input devices, including music sequencers, instrument controllers, fingerboards, guitar synthesizers, wind controllers, and electronic drums. Synthesizers without built-in controllers are often called sound modules, and are controlled via MIDI or CV/Gate using a controller device.

Synthesizers use various methods to generate signal. Among the most popular waveform synthesis techniques are subtractive synthesis, additive synthesis, wavetable synthesis, frequency modulation synthesis, phase distortion synthesis, physical modeling synthesis and sample-based synthesis. Other less common synthesis types (see #Types of synthesis) include subharmonic synthesis, a form of additive synthesis via subharmonics (used by mixture trautonium), and granular synthesis, sample-based synthesis based on grains of sound, generally resulting in soundscapes or clouds.

Envelope detector

An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the envelope of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls. The diode in series rectifies the incoming signal, allowing current flow only when the positive input terminal is at a higher potential than the negative input terminal.

Most practical envelope detectors use either half-wave or full-wave rectification of the signal to convert the AC audio input into a pulsed DC signal. Filtering is then used to smooth the final result. This filtering is rarely perfect and some "ripple" is likely to remain on the envelope follower output, particularly for low frequency inputs such as notes from a bass guitar. More filtering gives a smoother result, but decreases the responsiveness; thus, real-world designs must be optimized for the application.

Definition of the envelope

Velocifero

Velocifero is the fourth studio album by English electronic band Ladytron and their first to be released by Nettwerk. The album was made available on the iTunes Store as of 19 May 2008, followed by a physical release on 2 June in the United Kingdom and on 3 June elsewhere. Velocifero peaked at number seventy-five on the UK Albums Chart, becoming the band's second highest-charting album to date in the UK (after Gravity the Seducer reached number seventy-two in 2011). It was also their first album to chart on the US Billboard 200, reaching number 131.

According to band member Reuben Wu, "velocifero" literally means "bringer of speed", and is also the name of a classic retro-styled scooter. "Black Cat" and "Kletva" are both sung entirely in Bulgarian. "Kletva" (which means "oath") is a cover of a song from a solo album by Kiril Marichkov of Bulgarian rock band Shturtzite. Daniel Hunt provided additional vocals on "Versus".

Velocifero produced three singles: "Ghosts", "Runaway" and "Tomorrow".

Tomorrow (song from Annie)

"Tomorrow" is a song from the musical Annie, with music by Charles Strouse and lyrics by Martin Charnin, published in 1977. The number was originally written as "The Way We Live Now" for the 1970 short film Replay, with both music and lyrics by Strouse.

Production history in the musical

It had appeared in prominence in productions of the musical throughout its history: it was heard in several versions in the original 1977 Broadway production, including ending the show. It was the entry and concluding credits score for the 1982 film adaptation.

Discussion

The song apparently pronounces an optimistic view of life through its main themes, 'hang on until tomorrow' and 'when I'm stuck with a day that's grey and lonely, I just stick out my chin and grin'. It appears to be in unison with another song in the score, "You're Never Fully Dressed Without a Smile", which again pronounces the importance of smiling, though in a much lighter and cheerful tone.

The lyrics were likely influenced by the song "There's a Great Big Beautiful Tomorrow", written by the Sherman Brothers for Disney's Carousel of Progress attraction for the 1964 New York World's Fair. The songs have very similar themes, stressing the importance of looking forward optimistically to a shining future, and in some places use almost identical lyrical phrases (compare "just a dream away" and "only a day away," for example).

Wake Up...It's Tomorrow

Wake Up...It's Tomorrow is the second album by the American psychedelic rock band Strawberry Alarm Clock, released on Uni Records, in 1968 (see 1968 in music). It came as the group was challenged with continuing the success of their debut single, the psychedelic classic and number one hit, "Incense and Peppermints" and the LP that followed. Their second effort would expand upon the arrangements of its predecessor's unique blend of vocal harmonies, psychedelia, and pop music. In addition, the band's atypical lineup, which featured two bassists, would be altered to address studio and performing issues.

Upon release, Wake Up...It's Tomorrow failed to chart, though it is considered Strawberry Alarm Clock's apex of musical creativity, as the band members were free to pen the majority of the album's material. The pop-oriented song "Tomorrow" preceded the release of the album, earning the group their second and final Top 40 hit. An additional follow-up single, "Sit with the Guru", also managed to chart on the Billboard Hot 100.