Aux marches du palais
(cf. La Flamande. Jehan Chardavoine, 1576).
Original:
Dans la cour du palais, il y a une Flamande. Elle a tant d'amoureux qu'elle ne sait lequel prendre...
All children raised in the modern Western tradition are repeatedly exposed at an early age
to the progression of the seven notes in the key of C-major
(used in all nursery-school songs).
Very few (about 0.01% or one in ten thousands) will ever associate
each note with its exact pitch in absolute terms (this rare ability is called
perfect pitch). Most, however, will eventually
learn the relative position of the notes in the scale.
Those 7 notes form an uneven progression within the regularly-spaced
12 notes of the chromatic scale,
which is the basis of all current music in the Western world.
Let's brush up on the basics of that system and explore how it was born from
two opposing forces: The desire for universal keyboards
and the need for harmony
(two tones are harmonious only when their frequencies are
nearly in the same ratio as two small integers).
Music Theory Distilled:
(10:33,
11:32,
4:08) by Casey Connor (2018).
(2020-07-27) The many Greek tonoi.
A codification inherited from the late Bronze Age.
(2018-02-23) Occidental origins of diatonicity, in the Middle Ages.
Musical notation helped crystallize the evolution of Western art music.
From AD 600 to 1500 or so (when neumes
came into wide use) Gregorian chants were mostly transmitted orally.
The Church recognized eight official modes (the first and the last one being identical)
each nominally spanning eight notes (hence the name octave
for the doubling interval so spanned).
The common note starting and ending an octave was called the final
(now dubbed tonic or tonal center)
because of the rule that a chant must end on that note
to provide a sense of resolution. That expectation
is created by giving prominence to the tonic, not necessarily by starting with it
(e.g., Dies Irae is in D-Dorian but starts with the ominous four notes F-E-F-D).
The eight Gregorian modes span a combined range of two vocal octaves.
In this ancient scheme, the lowest note involved is an "A",
which may ultimately explain how the notes first got their alphabetical names,
already found in
De Musica
(c. AD 510) by
Boëthius (c.477-524).
G4 and A4 are shared by all modes. The latter became the
standard for tuning. The former is what the
treble clef points to
(G-clef, clef de sol ).
Traditional Symmetrical 8-way Definition of the Church Modes :
Each even-numbered plagal mode
is named by adding the prefix hypo- to the authentic
odd-numbered mode which precedes it.
In De Harmonica (c. AD 880)
Hucbald specified the plagal mode as running from
one fourth below the authentic mode's final to one
fifth above it, as tabulated above.
The last mode (Hypomixolydian) is identical to the first one (Dorian).
The arbitrary distinction between authentic and plagal modes is now all but lost and three
new words have been coined for the former hypo- modes,
namely the two most popular modern modes
(ionian and aeolian; better known as Major and Minor) and the least popular one
(locrian).
As all of the above Gregorian modes are diatonic, they can
be matched with the modern modes of the major scale.
Actually, the Medieval modes denoted not only a mode (in the modern sense)
but a specific key
(at a fixed pitch) according to the following equivalences:
Modern equivalents of the 7 unique Medieval Gregorian modes
The rotation placing Dorian = Hypomixolydian in the middle, rather than at both
ends, reveals a deeper symmetry (mirror inversion of the
interval structure) among the seven distinct modes.
This ordering also puts them in strict order of brightness,
with B-Locrian last and least (all but unused).
The Beginning of Western Musical Notation :
The musical staff
was devised by Guido of Arezzo early in the 11-th century.
He used a 4-line staff and recommended the use of yellow and red ink.
(2018-02-16) Duration of notes and rests
Binary progression of standard durations (dotting prolongs by 50%).
A note is an elementary music element with nearly constant pitch and prescribed duration.
A quarter-note is represented by a black oval with either an upward stem to the right
or a downward stem to the left (the French just call it "a black"; une noire).
A half-note has twice the duration and is represented by a void oval with a stem
(French: une blanche).
A whole note is a void oval without stem; it's worth two half-notes or four quarter-notes.
Moving in the other direction, a flag on the stem of a quarter-note reduces its duration
by a factor of two and makes it an eighth-note (a quaver
to the British, une croche to the French).
Two flags indicate a sixteenth (French: double-croche).
Occasionally, three flags are used to denote a thirty-second (French: triple-croche).
Four or five flags are rarely used.
The shortest value ever used in the classical repertoire is denoted with six flags.
It's the two hundred fifty sixth note, which the French call a sextuple croche.
The British name is demisemihemidemisemiquaver.
Note Durations (and the corresponding silence periods)
A dot after a symbol extends its duration by 50%. A double-dot by 75%.
Two or more consecutive flag-bearing notes are beamed together with as many beams
touching each note's stem as the number of flags it ought to have.
Rest symbols are allowed between beamed notes.
How to Read Music (14:39)
by Sarah Jeffery (Team Recorder, 2019-02-14).
(2018-02-16) Tempo
How fast to play.
In Western culture, a musical piece is a sequence of monophonic or polyphonic tones,
timed by regular beats.
The tempo is either indicated by a traditional Italian locution or given precisely
in beats per minute (bpm).
Metronomes
are traditionally marked at the following values, in bpm:
40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60,
63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100,
104, 108, 112, 116, 120, 126, 132, 138, 144, 152,
160, 168, 176, 184, 192, 200, 208.
Traditional
Beats per Minute
Larghissimo
24 bpm and below
Grave
24 bpm - 40 bpm
Largo
40 bpm - 60 bpm
Larghetto
60 bpm - 66 bpm
Adagio
66 bpm - 76 bpm
Andante
76 bpm - 100 bpm
Moderato
100 bpm - 120 bpm
Allegro
120 bpm - 168 bpm
Presto
168 bpm - 200 bpm
Prestissimo
200 bpm and over
There's little need to perform below 30 bpm
(one beat every other second)
which is roughly the slowest tempo at which the human brain still links
the elements of a sequence as parts of a whole. Slower changes in tonality are
perceived as separate discrete events and the melody is just lost in time.
At the other extreme, too fast a tempo will not give the brain enough time
to grasp subdivisions in individual beats.
Ultimately, when something changes more than 20 times per second
(20 Hz or 1200 rpm) it's simply heard as a buzz.
That's when rapid clicks morph into a continuous pitch.
Historically, the earliest scientific unit of time
chosen by Galileo
was roughly the shortest time interval at which he couldn't perceive two percussions as distinct
(about 11 ms in modern terms).
At face value, that would imply that we perceive as separate
two cycles of a 9 Hz sound, if the conditions are right
(although 540 bpm is musically meaningless).
(2018-02-16) Beats (counts) bars (measures) and phrases.
4/4 common time :
4 beats to a bar and 4 or 8 bars to most phrases.
Beats are regularly-spaced time intervals.
In dance-music, they follow the kick drum.
Otherwise, a metronome can be used,
which delivers regular clicks and visual cues.
A whole number of beats make up a measure
(also called bar because the limits of all measures are indicated by vertical bars on
sheet music). That number depends on the time signature,
discussed in the next section.
A rhythm where some notes are stressed on the upbeat (between main downbeats)
is called syncopated.
In the rare cases where a note straddles two measures, it is said to be offbeat.
On sheet music, this is indicated at the beginning of the first
staff by two superposed numbers which summarize the rhythm.
In simple time (as opposed to compound time, discussed next)
the top number gives the number of beats per bar.
The bottom number says which type of note
counts for one beat
(2 for a half-note per beat, 4 for a quarter-note per beat, 8 for an eighth-note per beat).
For example, 3/4 is often read "3 beats per bar and every quarter-note gets a beat".
March time is simply 1/2.
The 4/4 signature is called common time.
The 3/4 time-signature is waltz time.
Less common time signatures include 5/4, of which a prime example
(with offbeat notes) is
Take Five by the late
Dave Brubeck (1920-2012).
When the time signature's top number
is a proper multiple of three
(6, 9, 12, 15, 18, 24, etc.) each beat is understood to be divided
into three equal divisions. The number itself indicates how many such divisions there are
in a bar (not the number of beats per bar,
as is the case with simple time).
Thus, a bar in 6/8 has two beats divided into
three equal parts, worth an eighth-note each.
A famous example of 6/8 is the
folk songRising Sun Blues, popularized as
House of the Rising Sun,
by The Animals (1964).
(2018-02-16) Triplets and offbeat Triplets.
Trained musicians have trouble playing offbeat Triplets at a slow tempo.
A Triplet is usually just a group of three notes of equal durations
meant to be played with the same total duration as two
notes of the indicated kind.
A Triplet is indicated by a bracket with the numeral 3.
Thus, the duration of a Whole-note is divided equally into
three Triplet-halves.
Likewise, a Half is split into three equal Triplet-quarters.
More generally, a triplet bracket (i.e., a bracket with the numeral 3)
reduces all note durations within it by a factor of 2/3.
The bracket itself is optional if the notes are already beamed together.
Likewise, a bracket (or a beam)
bearing the numeral n reduces the duration of the notes it spans by a fixed fraction of
denominator n. That construct is known as a
tuplet (or an n-tuplet, when n is specified).
Numerical Greek prefixes (and/or some Latin alteration thereof)
can also be used:
Triplets. Factor of 2/3 (always).
Pentuplets, Quintuplets or Quintolets.
Factor of 4/5 (3/5 in compound time).
(2018-02-12) Perfect Pitch (Absolute Pitch)
The codified language of Western music has its native speakers.
About one in 10000 people have developed native familiarity with the language
of music by being exposed to its complexity at a very young age.
The most striking ability they develop is called perfect pitch
(or absolute pitch) which is the ability to instantly
name a note or a combination of notes with perfect accuracy without
the benefit of prior tuning.
This ability cannot be acquired later in life.
A take-home message of this diagram is that D, G and A have no other names.
The syllabic names below (first column) are used in
Romance and Slavic languages. In English,
Sol is pronounced So
and Si is called Ti
(thus avoiding a possible confusion with the letter "C", for
Ut or Do).
Equally-Tempered Frequencies of Western Notes, in Hz
(A440 pitch)
C4 is called
middle C and standard
concert A
(A4 , 440 Hz) is dubbed A above middle C.
Each octave starts at a C and ends with the B above it.
Treble C (C5) is better known to Opera afficionados as
Tenor High C
since it's the highest note in the classical male repertoire.
On an 88-key piano, the lowest note is A0
( 27.5 Hz ). The highest is
C8 ( 4186.009 Hz )
which is the lowest note of Octave 8, in ISO numbering.
In the abovescientific pitch notation
(SPN, used everywhere with growing popularity)
the ISO number of each octave is used after the name of a pitch
(following the # or b accidentals, if any)
to denote a particular tone unambiguously.
Subscripting is optional: A4 and A4 both mean 440 Hz.
Other competing systems are still in use, which are mutually incompatible to some degree.
In all cases, tones in the same octave (always from C to B, mercifully) are denoted alike
and we give only the notation corresponding to "C" (Do, Ut) in the following
table. Musicians routinely speak of a particular tone by identifying the "C"
just below it (e.g., 440 Hz is "A above middle-C" or concert A, less often "middle A").
Competing ways of naming an octave (C to B) and/or the C tone it starts with :
The "low" octave,
between contra and bass, is also called "great bass".
The usual female classical opera repertoire extends to
Soprano Bb (Bb5; one whole tone below C6; Soprano C, High C or Top C).
Legendary prima donnas have routinely gone beyond this, up to and including G#6.
The highest note ever sung in a regular performance at New-York's Met Opera
was A above high-C (A6 , 1760 Hz) by soprano
Audrey Luna as the very first note of Leticia in
The Exterminating Angel by Thomas Adès (2017).
She
can sustain C above high-C (C7 , 2093 Hz) at full voice.
In 2003, Maria Carey hit a G7 (3136 Hz) during a public rendition of
The Star Spangled Banner.
For the Guinness
book of world records, singer Adam Lopez
smashed his own previous record for a male vocalist
(D7 , 2349 Hz) by almost a full octave.
when he hit a C#8 (4435 Hz)
in front of a live studio audience (2008).
That's one semitone beyond the piano range.
An urban legend says that the record for a female vocalist is a G10
by Brazilian singer Georgia Brown.
This is silly; a G10 would be squarely ultrasonic
and inaudible (25 kHz). In the video, she does hit a very respectable
A#7 (3729.31 Hz). Three semitones above Maria Carey's
G7 ,
not three octaves above it. (Did someone confuse semitones and octaves?)
(2020-06-18) Legacy Pitch.
Alternate choices of frequencies for "A above middle C" (A4 ).
The tuning fork was invented in 1711 by
John Shore (c.1662-1752).
Before that time, the only records of bygone tunings are extant organ pipes.
There's a slight complication with organ pipes: Their
frequencies are proportional to the square root of the absolute temperature
of ambiant air. For Church organs, the "standard" temperature was then assumed
to be 59°F = 15°C = 288.15 K
(much cooler than what scientists use as "room temperature"). Any actual measurement
should be adjusted to that temperature, according to the square-root law.
The frequency of steel tuning-forks varies with temperature in the opposite direction
(they sound flatter when hot) but about 20 times less so than wind instruments
(about 86 ppm/°C, which translates musically to 0.15 cent/°C).
That's negligible under ordinary circumstances.
It might be possible to devise an alloy
to suppress the temperature-dependence of tuning-forks, which would be intermediate between
invar (64% Fe, 36% Ni) and
elinvar (52% Fe, 36% Ni, 12% Cr).
To the best of my knwledge, this has never been perfected
(as there are cheaper ways to generate a very accurate sound frequency, in this electronic age).
Likewise, 432 Hz gives a whole value in Hz to any A above A0 = 27 Hz
(compared to A0 = 27.5 Hz under the standard A440 tuning system).
The 415 and 466 Hz tuning are recent conventions corresponding to just
one semitone lower or higher than the A440 modern standard (1936).
This appeals to musicians who want a better historical authenticity
for periods when standard tuning was notoriously higher or lower than today.
That solution is also compatible with electronic instruments which routinely offer the ability
to transpose up or down by a whole number of semitones.
Part of this is a fad.
Most of us, mere mortals not blessed or cursed with perfect pitch,
can't tell the difference when everything is transposed by a semitone.
However, it's true that a violin tuned one semitone higher will have
a slightly richer harmonic content which can be perceptible (at least
by the same kind of people, with or without perfect pitch, who can tell
a Stradivarius from a lesser instrument).
This once drove
pitch inflation.
(2018-02-23) Musical staves and clefs. Grand staff.
A staff consisting of 5 lines (4 spaces) can be extended with ledger lines.
Between the bass and treble staff, there would normally be room for just a single ledger line
corresponding to middle C. However, the two staves are normally interpreted
on the piano by the two hands (bass staff for the left hand and treble staff for the right hand)
and they are printed with enough room between them to allow for several
ledger lines. Middle C and the adjoining notes are printed either with
the bass staff or with the treble staff,
depending on which hand is meant to play them.
(2018-02-18) 25, 32, 37, 44, 49, 54, 61,
64, 73, 76 or 88 keys.
A grand-piano keyboard has 88 keys (some rare pianos have 92 or 97).
The Imperial
Bösendorfer Model 290 (290 cm) was introduced in 1909 spanning eight octaves
(97 keys from C0 to C8)
as suggested by composer
Ferruccio Busoni (1866-1924) to
accomodate some organ works of
Johann Sebastian Bach (1685-1750):
Some 32-foot registers of large organs do go down to C0 at 16.35 Hz.
As shown at left, the extension of nine extra bass keys is signaled visually by five dark-brown tops
on keys which could be expected to be white.
Organ manuals almost always go from C to C. Full-sized ones span
five octaves (61 keys) more rarely six (73 keys) or
seven (85 keys)
as found only in a few very large organs meant to play C0
(16 Hz or so) the lowest note in the classical repertoire,
which is felt more than it is heard :
Historically, most organ keyboards spanned only four
octaves (49 keys).
Small 37-key manuals are also found.
The pedalboards of traditional organs have between 12 and 32 keys.
Twelve sizes of electronic keyboards are widely available
in the sixteen different layouts illustrated below.
As these keyboards can often be shifted at will by a whole number of octaves,
the highlighted positions may not always play as
middle C or
concert A.
Both ends of the 64-key keyboard match the layout of a grand piano.
This pattern helped make allWurlitzer electric pianos
popular, from 1954 to 1984 (besides a rare 44-key simplified classroom model).
The design was revived in the recently-discontinued
Roland RD-64 (introduced in 2013) which was
unique in its class,
with 64 weighted hammer-action keys. With controls to the left, the RD-64 is about as long as a 73-key keyboard.
(2017-04-09) Relative Pitch & Tone Intervals
Ratios of sound frequencies.
Harmony is perceived when two tones are heard whose frequencies are
in a ratio close to the ratio of two small integers.
The smaller the integers, the greater the impression of harmony.
Thus, the octave is the most harmonious interval (2:1 ratio)
besides unison (1:1 ratio).
The fifth (3/2 ratio) is not far behind.
Pure Natural Consonant Intervals :
Two consonant tones are characterized by frequencies in a
simple ratio (i.e., the ratio of two small integers).
A musical interval is a frequency ratio.
Natural consonant intervals are thus ratio of small integers.
The most important ones have traditional musical names:
Pure Intervals and Equal-Tempered Approximations Thereof
Note the perfect vertical symmetry of the above table with respect to the horizontal
line which separates the two kind of tritones. The product of the ratios (from the first column) in symmetrical pairs
is equal to 2. Equivalently, their values in semitones (last columns)
add up to 12. Musicians call that symmetry an inversion
(French: renversement).
For example, the septimal major sixth is the inversion of the
septimal minor third (both of those are quite rare).
One percent of a semitone is called a cent.
The fourth and the fifth. have excellent chromatic
approximations, which are less than 2 cents off
(that's impossible to detect by ear). Likewise for the whole tone (9:8)
and minor seventh (16:9) which are less than 4 cents off.
The other important musical intervals appearing in unshaded entries have good
enough chromatic counterparts, which are respectively off by only
11.7, 15.6 and 13.7 cents,
for the half-tone, minor third,
major third and their respective inverses.
This is the happy coincidence (a minor mathematical miracle)
which made equal-tempered Western music possible.
A chromatic whole step is also close enough to the 10:9 natural harmony (lesser tone).
The septimal intervals (involving the seventh harmonic of fundamental sounds)
are rarely found outside of Blues nowadays,
starting with the lesser septimal tritone (7:5)
also known as the septimal of Huygens
and its inverse, the greater septimal tritone (10:7) also called
Euler's tritone.
For intervals greater than an octave (but less than two octaves) musicians often
talk about compound intervals and will use a locution like
compound major third (compound M3) instead of
major tenth (M10).
Lesser musical names have been given to some intervals greater than an octave, including:
11:5 Neutral Ninth. (A neutral second is 12:11. Very close to 1.5 semitones.)
Besides unison and octaves, there are 11 possible intervals in
equal-tempered Western music. Some are good approximations to
the consonant pure intervals.
Others are dissonant. Here's the complete list:
m2: Minor second. Half step = 1 semitone. Twelfth of an octave.
M2: Major second. Whole step = 2 semitones. (10/9 or 9/8 ratio).
m3: Minor third. Augmented second. Quarter of an octave. (» 6/5)
(2020-08-20) Euler's Gradus Function (Euler, 1730)
Suavitatis Gradus = Degree of harmony (suavity).
Arguably, music builds on the ancient notion of
commensurability:
Two numbers are commensurable when they are proportional
to two integers. They are consonant when those two integers are small.
Consonance is thus not a quality that commensurable quantities possess or not;
it's something they have to a greater or lesser degree (Latin: gradus).
Euler's musical treatise
Tentamen Theorae Novae Musicae was completed in 1730 (when he was 23)
but only published in 1939. In it,
Euler tried to quantify how pleasant
two integers are when they measure musical frequencies heard together.
This work didn't attract much interest at the time.
It was said to be too mathematical for musicians and too musical for mathematicians.
For the elusive thing he was trying to quantify,
Euler used the Latin term suavitas,
which generally indicates how something appeals to the senses or the mind.
Possible translations include:
Charm, appeal, beauty, attractiveness, pleasantness, agreeableness or sweetness.
In a musical context, that could also be called tunefulness, melodiousness, harmoniousness, consonance,
smoothness or softness (as opposed to harshness or spiciness).
Take your pick. I'll use the direct descendant "suavity" in the rest of this article.
It's clear at the outset that any two-variable function f grading the
consonance of two numbers must be symmetrical and invariant by scaling
(as the unit of frequency is entirely arbitrary):
f (p,q) = f (q,p) = f (kp,kq) = f (p/q, 1)
using the single-variable grading function g(x) = f (x, 1) symmetry implies:
g (x) = g(1/x)
Presented as a consequence of interchanging two tones whose consonance is being compared,
this symmetry is obvious. However, this implies the nontrivial fact that
inverting all tones in a musical piece gives a totally different piece which is equally
harmonious. A fact which was well-known to
Johann Sebastian Bach (1685-1750),
who made extensive use of that
symmetry in some of his compositions.
contemporary authors have dubbed related topics negative harmony.
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
gE (n)
1
2
3
3
5
4
7
4
5
6
11
5
13
8
7
5
17
6
19
7
9
12
23
6
9
g0 (n)
0
1
2
2
4
3
6
3
4
5
10
4
12
7
6
4
16
5
18
6
8
11
22
5
8
As gE(n) is always equal to g0(n)+1,
either function is equally suited for comparisons.
However g0 possesses the rare property of being a
totally additive function which is to say that it ressembles
a logarithm:
g0 ( x y ) = g0 (x) + g0 (y)
By contrast gE seems flawed:
gE ( x y ) = gE (x) + gE (y) + 1
g0 is the difference of two standard (totally) additives functions, best
defined by their effects on the power of a prime (pn ):
The perfect fifth (3/2) isn't close enough to a tritone and
the integers in the ratio 17/12 (1.416666...) are not small enough
to qualify as harmony.
So, the closest harmonious approximation to a tritone is 7/5 = 1.4.
That unusual interval is technically called the lesser septimal tritone.
It's used in Blues.
Its musical inverse (10/7) is the greater septimal tritone.
The ratio of those two tritones is 49/50 = 0.98.
The lesser one is thus exactly 2% below the greater one.
That makes them about 0.35 semitones apart.
(2018-02-19) Pentatonic, diatonic (heptatonic) and chromatic scales.
Current keyboards were made for the key of C major (or A minor).
In the modern equal-temperament formally introduced by J.S. Bach,
the chromatic scale consists of the twelve pitches whose frequencies
form a geometric progression of constant ratio 1.059463... (the twelfth root of 2)
modulo factors of any power of two (that's the learned way to say that frequencies
separated by any whole number of octaves represent the same pitch and have the same name
in Western art music).
The qualifier chromatic also apply to any subset thereof which
is not strictly contained in a diatonic scale, as described next.
Diatonic Scale :
The Western diatonic scale is the heptatonic scale formed
by the white keys of the piano, or any transposition thereof.
It includes only seven notes per octave. The first one is called tonic,
the fifth one dominant.
There's more to a scale than the notes it consists of.
The tonic plays a central role (the melody normally starts and ends with the tonic).
The dominant is a close second in importance.
The specific seven notes which appear in the octave of a diatonic scale are called its
key signature.
Most famously, C-major and A-minor
share the same key signature (you play them using only the white keys on a piano).
There are 12 major scales and 12 (natural) minor scales.
Six of those (3 major, 3 minor) have two
enharmonic names
(played alike but denoted differently). So, the 24 common diatonic scales have 30 different names:
A 7-note diatonic scale is named after its first note (the tonic). E.g., C-major. Modes which share the same tonic are said to be parallel.
The above is best called the natural minor scale
(or descending minor scale). It's a true diatonic scale
(in the key of A, it uses only the white keys of the piano) but it suffers
from a flaw in ascending order in the sense that the last note, called
the subtonic, is no longer just a semitone
away from the tonic (as is the leading tone in the major scale)
but a whole tone away...
To remedy that, the harmonic minor scale is introduced
which raises the subtonic one semitone, back to leading tone status. However,
this fix introduces an unusual large gap of three semitones between
the previous note (the submediant) and the subtonic/leading tone.
Another flavor of minor scale, called the melodic minor scale
is introduced which closes the gap by raising that previous note too.
All told, the melodic minor scale is obtained from the major
scale simply by lowering just the third note (so that the last four notes
are separated by a whole tone).
This is sometimes called the ascending melodic minor scale
and it's so often used in Jazz that it's most commonly known as the
Jazz minor scale.
The semi-obsolete locution descending melodic minor scale
simply refers to the classical natural minor scale we've already discussed
(which is merely the sixth mode of the major scale,
also known as Aeolian).
Neither the harmonic nor the melodic minor scales obey the diatonic pattern :
Harmonic Minor Scale
Melodic Minor Scale
1
2
3
4
5
6
7
1
2
3
4
5
6
7
F
G
Ab
Bb
C
Db
E
F
G
Ab
Bb
C
D
E
C
D
Eb
F
G
Ab
B
C
D
Eb
F
G
A
B
G
A
Bb
C
D
Eb
F#
G
A
Bb
C
D
E
F#
D
E
F
G
A
Bb
C#
D
E
F
G
A
B
C#
A
B
C
D
E
F
G#
A
B
C
D
E
F#
G#
E
F#
G
A
B
C
D#
E
F#
G
A
B
C#
D#
B
C#
D
E
F#
G
A#
B
C#
D
E
F#
G#
A#
F#
G#
A
B
C#
D
E#
F#
G#
A
B
C#
D#
E#
C#
D#
E
F#
G#
A
B#
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
Cb
Db
Eb
Fb
G
Ab
Bb
Cb
Db
Eb
F
G
Eb
F
Gb
Ab
Bb
Cb
D
Eb
F
Gb
Ab
Bb
C
D
Bb
C
Db
Eb
F
Gb
A
Bb
C
Db
Eb
F
G
A
It's usually considered undesirable to have different accidentals on the key signature
of modern sheet music, as would happen in three of the above cases
(G harmonic or melodic, and D harmonic). The problem can be solved in those cases
by using only flats in the key and raising the last note with an individual sharp whenever it occurs.
(2018-02-26) The Seven
Diatonic
Modes (modes of the major scale)
A key is given by a
tonic and a mode
(e.g., C-major or F-Lydian).
The modern diatonic modes described below are named after ancient Greek
harmoniai
(or tonoi) and/or the medieval church modes. However, those concepts are only loosely related.
By definition, a diatonic scale consists of seven steps; two half-tones (H) separated by alternating groups
of two or three whole-tones (W). There are seven ways to choose a starting point in such
a progression. Each such way is called a mode.
The most common ones are the aforementioned major and minor modes, also called
Ionian and Aeolian. Here's the complete list:
105 names for the 84 diatonic scales:
7 modes in 12 keys (15 allowed names). (For
each mode, the tonic can be one of the 12 choices in column "1".)
1
IV: Lydian
7
18
1
I: Ionian (= Major)
7
17
1
V: Mixolydian
7
16
1
II: Dorian
7
15
1
VI: Aeolian (= Minor)
7
14
1
III: Phrygian
7
13
1
VII: Locrian
7
12
F
G
A
Bb
C
D
E
F
G
A
Bb
C
D
E
C
D
E
F
G
A
B
C
D
E
F
G
A
B
G
A
B
C
D
E
F#
G
A
B
C
D
E
F#
D
E
F#
G
A
B
C#
D
E
F#
G
A
B
C#
A
B
C#
D
E
F#
G#
A
B
C#
D
E
F#
G#
E
F#
G#
A
B
C#
D#
E
F#
G#
A
B
C#
D#
B Cb
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E# F
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B# C
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
C
Db
Eb
F
G
Ab
Bb
C
Db
Eb
F
G
Eb
F
G
Ab
Bb
C
D
Eb
F
G
Ab
Bb
C
D
Bb
C
D
Eb
F
G
A
Bb
C
D
Eb
F
G
A
M
m
m
M
M
m
mo
M
m
m
M
M
m
mo
1st
2nd
3rd
4th
5th
6th
7th
<<< mode of the major scale.
In Numericana tables like this one,
the next-to-last line gives the
quality (major, minor, diminished, augmented) of
the in-scale triad rooted in the corresponding column.
The last line repeats the rank of the mode named & numbered as a 1-7 span at the top of the table.
Two scales which share the same diatonic mode are said to be modes of each other.
For example, F-Lydian is the 4-th mode of C-major (i.e., C-Ionian) because
its tonic (F) is the 4-th note in the C-major scale. Likewise,
G-Mixolydian is the fifth mode of C-major or the second mode of F-Lydian.
In modern practice, this type of reference is most commonly used with respect to the
relevant major scale only, as indicated in the last line of the above table.
Dorian is the only palindromic mode (2122212).
In all other cases, a mirror mode is obtained by inversion:
Mixolydian (2212212) and Aeolian (2122122).
Ionian (2212221) and Phrygian (1222122).
Lydian (2221221) and Locrian (1221222). Brightest and darkest.
For each of those mirror modal pairs, the ascending version of one
is the same as the descending version of the other.
Béla Bartók (1881-1945)
remarked that all 12 chromatic tones are obtained by mixing
some pairs of diatonic modes on the same tonic (e.g., C-Lydian with C-Phrygian).
More generally, the above table cab be used to count how many tones are
obtained when mixing two diatonic modes on the same tonic.
Number of tones obtained by mixing two modes
IV
I
V
II
VI
III
VII
F
Lydian
IV
7
8
9
10
11
12
12
C
Ionian
I
8
7
8
9
10
11
12
G
Mixolydian
V
9
8
7
8
9
10
11
D
Dorian
II
10
9
8
7
8
9
10
A
Aeolian
VI
11
10
9
8
7
8
9
E
Phrygian
III
12
11
10
9
8
7
8
B
Locrian
VII
12
12
11
10
9
8
7
Among many other things, this provides a name for a half-dozen special octatonic scales,
which are actually the first 6 modes of the Bebop (dominant) scale:
Lydian-Ionian adds Bb to F-Lydian or F# to C-Ionian. Mode IV of the Bebop scale
Ionian-Mixolydian adds Bb to C-Ionian or F# to G-Mixolydian. Mode I of the Bebop scale.
Mixolydian-Dorian adds Bb to G-Mixolydian or F# to D-dorian. Mode V of the Bebop scale
Dorian-Aeolian adds Bb to D-dorian or F# to A-Aeolian. Mode II of the Bebop scale.
Aeolian-Phrygian adds Bb to A-Aeolian or F# to E-Phrygian. Mode VI of the Bebop scale.
Phrygian-Locrian adds Bb to E-Phrygian or F# to B-Locrian. Mode III of the Bebop scale.
(2020-07-20) Naming Non-Diatonic Scales and Modes
A systematic nomenclature to define precisely lesser-known modes.
The diatonic scale (especially its Major and Minor modes)
is the backdrop for almost all classical music and a good chunk of modern tunes,
although 6 of the common diatonic modes are now in common use
(locrian is left out).
Only four heptatonic scales in the Western chromatic system can be expressed
canonically in all 12 keys, by naming all seven notes (letters)
once and only once, with at most one sharp or one flat each. Namely:
This property, which is taken for granted by most casual students of Western music,
is indeed a rare one. It fails whenever the scale includes two consecutive
half-steps (HINT: For every mode, there's a key where
A and G are included with the tone between them, which can be called neither
A# nor Gb without repeating a letter).
There are only two other heptatonic scales for which this
doesn't happen (Hungarian major and
Romanian major) but they both fail in two keys,
for less obvious reasons.
This fact is lost on most composers and almost all practicing musicians, with
little or no consequences: Transposing a piece in writing for all possible keys
is rarely required, if ever.
So, there's little or no obstacle to experimentation with a
huge number of exotic scales.
The most popular ones eventually get a colorful name.
For others, the standard practice is to use a known name
(preferably one of the diatonic modes
and indicate what modification(s), sharp of flat, is to be applied to what
degree(s).
Some are queasy about using degree 1 in this scheme.
I beg to differ but accomodate those concerns by putting such names
inside square brackets.
The systematic exploration of non-diatonic scales and their modes started in 1907
with the investigations of synthetic scales
by Ferruccio Busoni (1866-1924) who first considered all
scales which could be derived from a diatonic mode by lowering or raising a single degree by one semitone.
Each of the 7 diatonic modes features 5 skipped notes which
can be included in one of two ways. Therefore, 70 names can be attributed
to new scales this way (for a total of 77 names) However,
some of those scales may have more than one name (as is the case for any mode of the
melodic minor scale).
For example, the Assyrian scale (mode II
of the melodic minor scale) possesses two distinct systematic names:
either Dorian b2 or Phrygian #6. In the key of G, that's:
G Ab Bb C D E F
As G-Dorian is G A Bb C D E F and G-Phrygian is G Ab Bb C D Eb F.
Conversely, some of the 66 different heptatonic scales can't be named at all with just one modification
from a diatonic mode.
The scope was later expanded to include all possible scales
and their respective modes.
Dominant means the 3 is major and the 7 is flat.
The whole problem is of greater theoretical interest than of practical worth. J. Murray Barbour (1929)
(2020-06-26) Defining Brightness and Darkness
A nice quantification of an elusive concept.
The white numbers which appear in the rightmost column of our modal charts
for heptatonic chromatic scales (including the seven diatonic modes
from the previous section) are a numerical evaluation of
brightness computed as follows:
For each of the seven notes allowed in a given mode,
we count the number of disallowed chromatic tones
between the root and itself. Adding all those counts give
a sum of seven integers (always starting with 0 for the root itself)
totaling between 0 and 30 (those extremes are reached for two
modes of the pathological scale where consecutives notes are separated
by either one semitone or seven).
For example, the Lydian mode of the major scale has a brightness of
18 (namely 0+1+2+3+3+4+5) whereas the brigthness of the major scale
itself (or rather Ionian, its first mode) is only 17 = 0+1+2+2+3+4+5.
This evaluation can be proved to be correct for diatonic scales
through the miracle of the circle of fifths
which shows that the above count actually measures precisely
set inclusion...
In other words, changing diatonic modes with a fixed root
will never raise some notes and lower others.
It does one or the other, for all individual notes in all pieces of music
ever written without individual accidentals.
That wonderful fact isn't true for the modes of non-diatonic
scales analysed below.
Yet, we may keep the above measure as a standard indication
of the otherwise elusive concept of brightness, under the debatable
simplification that brightness only depends on the difference between
the number of notes raised and the number of notes lowered.
We may boldly use that number to compare the
brightness of different modes of the same scale. Any scale.
The brightness of an n-tonic scale goes from
0 to (n-1)(12-n). That is:
0 for the monotonic scale.
0 to 10 for a ditonic scale,
0 to 18 for a tritonic scale.
0 to 24 for a tetratonic scale.
0 to 28 for a pentatonic scale.
0 to 30 for an hexatonic scale.
0 to 30 for an heptatonic scale.
0 to 28 for an octatonic scale.
0 to 24 for an enneatonic scale.
0 to 18 for a decatonic scale.
0 to 10 for an hendecatonic scale.
0 for the dodecatonic scale.
The brightnesses of two inverse n-tonic modes add up to (n-1)(12-n).
Linear Brightness is Tonal. Quadratic Clarity is Atonal.
The perceived brightness of a rooted scale is
probably a clever balance between the two.
For the classification of modes within a scale,
I am only retaining the linear score described above
which is easy to compute and meets expectations
in the only case everybody agrees on (diatonic modes).
Barely more complicated would be the introduction of
the variance (or its square root, the
deviation).
Any mode could be represented as a point in the plane with linear brightness
as the x-coordinate and quadratic brightness as the y-coordinate.
Experts should then be asked to draw an arrow from
one point the another whenever the former is clearly brighter than the latter.
To my knowledge, this has never been done.
A sensible formula might emerge this way.
A priori, I'd guess that
subjective brightness is an increasing function
of both the (linear) mean and (quadratic) deviation.
We may use a linear appoximation to the ideal function so described:
f (x,y) = a x + b y
where
a ≥ 0 and b ≥ 0
The Garklein Theorem :
Two modes of an heptatonic scale have different linear brightness scores.
(2020-06-22) The two most common non-diatonic minor scales.
Modes of the melodic minor scale. Modes of the harmonic minor scale.
The natural minor scale is diatonic.
As such, it's just another mode of the major scale
(namely, the 6th mode of the major scale, also called Aeolian).
Therefore, all modes of the natural minor scale would also be
modes of the major scale and they are not considered separately.
However, neither the harmonic minor scale,
nor the melodic minor scale are diatonic and they are
not modes of each other either. Therefore,
both give rise to a full set of 7 distinct modes in 12 possible keys.
We may tabulate them as we did the more common diatonic modes
(i.e, the modes of the major scale).
2 1 2 2 2 2 1 2 1 2 2 2 2 1 :
Melodic Minor Scale All 7 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
1
III: Lydian augmented
7
19
1
IV: Lydian dominant
7
17
1
I: Jazz minor scale
7
16
1
V: Hindu scale
7
15
1
II: Phrygidorian, Assyrian
14
1
VI: Half diminished
7
13
1
VII: Altered dominant
7
11
F
G
Ab
Bb
C
D
E
F
G
Ab
Bb
C
D
E
C
D
Eb
F
G
A
B
C
D
Eb
F
G
A
B
G
A
Bb
C
D
E
F#
G
A
Bb
C
D
E
F#
D
E
F
G
A
B
C#
D
E
F
G
A
B
C#
A
B
C
D
E
F#
G#
A
B
C
D
E
F#
G#
E
F#
G
A
B
C#
D#
E
F#
G
A
B
C#
D#
B
C#
D
E
F#
G#
A#
B
C#
D
E
F#
G#
A#
F#
G#
A
B
C#
D#
E#
F#
G#
A
B
C#
D#
E#
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B# C
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
Cb
Db
Eb
F
G
Ab
Bb
Cb
Db
Eb
F
G
Eb
F
Gb
Ab
Bb
C
D
Eb
F
Gb
Ab
Bb
C
D
Bb
C
Db
Eb
F
G
A
Bb
C
Db
Eb
F
G
A
m
m
M+
M
M
mo
mo
m
m
M+
M
M
mo
mo
1st
2nd
3rd
4th
5th
6th
7th
<<< mode of melodic minor.
The 7 modes of the melodic minor scale are known under various names:
Dorian #7. Ionian b3. (Ascending) melodic minor scale.
Jazz minor.
The Hindu scale is palindromic. The other six modes
come in three mirror pairs consisting of a bright mode and a dark mode, increasingly different:
Jazz minor scale and phrygidorian.
Acoustic scale and aeolocrian.
Lydian augmented (brightest) and altered scale (darkest mode).
With an extra chromatic passing tone between the 5th and 6th degrees ,
mode I becomes an octatonic scale known as Bebop melodic minor.
The other proper Bebop scales are all modes of two other scales,
unrelated to the melodic minor scale.
Nevertheless, we may consider Bebop modes obtained from the other heptatonic modes listed above
by allowing a passing tone at the
blue-shaded position .
Only mode I is widely accepted:
In the description of scales, the term harmonic
indicates the presence of at least one step consisting of an augmented second
(3 semitones). Such a step is best abbreviated "A" in the alphabetical version of
interval structures (where H is a half-step of 1 semitone
and W is a whole-step of 2 semitones.
The scientific name for that is sesquitone,
but the abbreviation "S" is unused.
Rick Beato
insists on calling Ultralocrian "Super Locrian bb7" because he always
presents modes in the key of C (although B# is called for here)
which leads to Bbb. That's a missed opportunity
to stress that double alterations are never needed for modes of the four
normative scales (diatonic, melodic minor, harmonic minor and harmonic major).
The harmonic minor scale has no axis of symmetry.
The mirror inverses of its modes are modes of the
harmonic major scale, described next...
(2020-06-27) Harmonic Major Scale
The mirror inverse of the harmonic minor scale.
This is arguably either the most exotic of the normative heptatonic Western scales
or the least exotic of the exceptional ones. It was introduced in 1853 by
Moritz Hauptmann (1792-1868).
The current name of the scale was popularized by
Nikolai Rimsky-Korsakov (1844-1908)
in his Practical Manual of Harmony (1885).
2 2 1 2 1 3 1 2 2 1 2 1 3 1 :
Harmonic Major Scale All 7 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
1
VI : Lydian Augmented #2
20
1
IV : Lydian diminished
7
17
1
I: Harmonic major
7
16
1
V : Mixolydian b2
7
15
1
II : Blues heptatonic
7
14
1
III : Phrygian b4
7
12
1
VII : Locrian b7
7
11
F
G
A
Bb
C
Db
E
F
G
A
Bb
C
Db
E
C
D
E
F
G
Ab
B
C
D
E
F
G
Ab
B
G
A
B
C
D
Eb
F#
G
A
B
C
D
Eb
F#
D
E
F#
G
A
Bb
C#
D
E
F#
G
A
Bb
C#
A
B
C#
D
E
F
G#
A
B
C#
D
E
F
G#
E
F#
G#
A
B
C
D#
E
F#
G#
A
B
C
D#
B
C#
D#
E
F#
G
A#
B
C#
D#
E
F#
G
A#
F#
G#
A#
B
C#
D
E#
F#
G#
A#
B
C#
D
E#
C#
D#
E#
F#
G#
A
B#
C#
D#
E#
F#
G#
A
B#
Ab
Bb
C
Db
Eb
Fb
G
Ab
Bb
C
Db
Eb
Fb
G
Eb
F
G
Ab
Bb
Cb
D
Eb
F
G
Ab
Bb
Cb
D
Bb
Cb
D
Eb
F
Gb
A
Bb
Cb
D
Eb
F
Gb
A
1st
2nd
3rd
4th
5th
6th
7th
...mode of the harmonic major scale.
The 7 modes of the harmonic major scale are called:
(2020-08-13) Other heptatonic scales
They can't be given signatures in all 12 keys without double alterations.
The four heptatonic scales given so far
(diatonic, melodic minor, harmonic minor and harmonic major)
are the only ones for which a simple key signature
can be given in all 12 keys (without using double alterations or worse).
Seasoned musicians may be unfazed by this fact
(which we prove elsewhere)
but this is a good pretext to transfer the
discussion of other heptatonic scales
to a dedicated page, where you'll also find
an exhaustive discussion of Bebop scales,
which have heptatonic and octatonic aspects
(the eighth note is just a passing tone
but they are usefully classified as modes of just three
octatonic scales).
The whole-tone scale
is the only anhemitonic
hexatonic scale.
The keys of E# =F, G, A, B=Cb, C# =Db and D# =Eb are modes of F.
Likewise, F# =Gb, A# =Bb, B# =C, D, E=Fb and D# =Eb are modes of C.
Both sets are equimodal, so the whole-tone scale has only one mode.
The 10 key signatures of Whole-tone :
5 key signatures for the class of F
E# F F F F
F#
G G G G G
A A A A A
Bb
B B B B Cb
C
C# C# C# Db Db
D
D# D# Eb Eb Eb
E
E# F F F F
F#
5 key signatures for the class of C
F
F# F# F# F# Gb
G
G# G# G# Ab Ab
A
A# A# Bb Bb Bb
B
B# C C C C
C#
D D D D D
E E E E E
F
F# F# F# F# Gb
Adding a seventh note to the whole-tone scale always yields one of the modes of Neapolitan Major.
For either mode (I or II, corresponding to the column of tonics indicated on the bottom line)
you always have a choice of two differents lines
omitting a different letter (shown on a grey background)
whose line on the staff will remain empty.
The two choices are transcribed with totally different key signatures
but the six tones actually played are the same in the end.
Here are the two enharmonic
results for the key of C:
C D# E G Ab B C or C Eb Fb G Ab B for mode I.
C Db E F G# A only for mode II. (Unless you want to use B#.)
Both choices of key signatures yield the same tones and brightness.
(2018-04-19) Diminished Scales. Double-diminished chords.
Two scales: Whole-half diminished scale and Half-whole diminished.
Octatonic symmetrical scale, where the 8 notes are obtained by repeating four times the same
two-note progression spanning three semitones (either
semitone-tone or tone-semitone). four times.
That gives two modes:
HWHWHWHW Half-whole diminished. Also called dominant.
WHWHWHWH Whole-half diminished.
There are only three distinct keys:
C = Eb = Gb = A
B = D = F = Ab
Bb = Db = E = G
No avoid notes. All the chords are interchangeable...
(2017-04-09) Chords and tertian harmony (stacking thirds).
The way simultaneous sets of notes are organized in Western music.
The tightest chords are triads consisting of three distinct notes,
characterized be by the two intervals which separate the root
from the other two notes. The first of those is a third;
either a major third (M3; worth 3 semitones) or a minor third (m3; worth 2 semitones).
The second one can be a perfect fifth (P5; worth 7 semitones), a diminished fifth (d5;
worth 6 semitones) or an augmented fifth (A5; worth 8 semitones).
That yields four different qualities of triads:
The four different qualities of triads (in closed position)
A triad can be denoted by the roman numeral corresponding to its root;
UPPER-case for a major triad.
Lower-case for a minor triad.
UPPER-case with a "+" superscript for an augmented triad.
Lower-case with a "o" superscript for a diminished triad.
The seventh chords are four-note chords (tetrads) derived from the above triads.
Spread Triads, Chord Inversions and Figured Bass :
The above describes the most basic form of a triad, called close-position,
root-position. Other versions of the same chord are obtained
by a so-called inversion which consists in displacing
one of the three notes by a full octave.
Modulation is the essential part of the art. Without it there is little music,
for a piece derives its true beauty not from the large number of fixed modes
which it embraces but rather from the subtle fabric of its modulation. Charles-Henri de Blainville
(1767)
According to Andy Chamberlain, that's what
Jacob Collier has been promoting using a silly compound prefix.
What we'll call here telescoping Lydian is what Collier dubs
super-ultra-hyper-mega-meta Lydian.
The idea hinted at by Collier and articulated by Chamberlain is to
consider the first 7 notes of some heptatonic mode (e.g., Lydian) and prolong
that with the notes corresponding to the same mode rooted at the n-th degree,
provided the initial seven notes are not affected by the switch.
This works with either n=4 or n=5 (not both) for all diatonic modes.
Telescoping F-Lydian
F
G
A
B
C
D
E
C
D
E
F
G
A
B
G
A
B
C
D
E
F#
D
E
F#
G
A
B
C#
A
B
C#
D
E
F#
G#
E
F#
G#
A
B
C#
D#
B Cb
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E# F
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
C
Db
Eb
F
G
Eb
F
G
Ab
Bb
C
D
Bb
C
D
Eb
F
G
A
F
G
A
B
C
D
E
The highlighted last line is identical to the first, which corresponds to
the repetition of the 48-note pattern (not 49) with a period of 7 octaves.
By starting the full pattern at the beginning of the relevant line,
you obtain telescoping Lydian in any of the twelve possible keys.
Now, if we impose the additional requirement of a perfect naming of the notes
(repeating the letters A,B,C,D,E,F,G in sequence) the 7-octave pattern can't be repeated
indefinitely (that would require 49 notes instead of 48). The maximal perfect pattern
contains just 67 tones, over 9½ octaves:
Telescoping Diatonics
E
F#
G#
A
B
C#
D#
Cb
Db
Eb
Fb
Gb
Ab
Bb
Gb
Ab
Bb
Cb
Db
Eb
F
Db
Eb
F
Gb
Ab
Bb
C
Ab
Bb
C
Db
Eb
F
G
Eb
F
G
Ab
Bb
C
D
Bb
C
D
Eb
F
G
A
F
G
A
B
C
D
E
C
D
E
F
G
A
B
G
A
B
C
D
E
F#
D
E
F#
G
A
B
C#
A
B
C#
D
E
F#
G#
E
F#
G#
A
B
C#
D#
B
C#
D#
E
F#
G#
A#
F#
G#
A#
B
C#
D#
E#
C#
D#
E#
F#
G#
A#
B#
G#
A#
B#
C#
D#
E#
G
The musical explorations of Jacob Collier (b. 1994)
Jacob Collier is from a musical family and has perfect pitch.
He is an alumnus from the
Purcell School for Young Musicians.
He dropped out of the Jazz piano class at the
Royal Academy of Music (where his mother is a professor).
Describing him as an autodidact is a
misleading term,
possibly damaging for young people who are still unsure about the benefits of a formal education.
When successful, education may seem unnecessary a posteriori, but this ain't so.
Collier has been sharing his creation on YouTube since 2012 (at age 18).
He has received accolades from the music industry (including several
Grammys).
(2018-02-15) Musical Instrument Digital Interface (MIDI, 1980)
Protocol for transmitting and recording keyboard performances.
A time-stamped MIDI event correspond to depressing a certain key at a certain velocity
and for a certain duration.
A 7-bit MIDI note number n (between 0 and 127) corresponds to the following frequency:
f = 2 (n-69) / 12 × 440 Hz
This is to say that note 69 is concert-A (440 Hz; A above middle-C)
by definition. Middle-C is 60. The lowest note on the piano is number 21
(A0 , 27.5 Hz). The highest is 108 (C8 , 4186.01 Hz).
The MIDI numbers span almost 11 octaves,
from C-1 (8.176 Hz)
to G9 (12543.854 Hz):
The 128 MIDI note numbers :
Organ Register
32'
16'
8'
4'
2'
1'
6''
3''
ISO Octave
-1
0
1
2
3
4
5
6
7
8
9
Do
C
0
12
24
36
48
60
72
84
96
108
120
C#
1
13
25
37
49
61
73
85
97
109
121
Ré
D
2
14
26
38
50
62
74
86
98
110
122
D#
3
15
27
39
51
63
75
87
99
111
123
Mi
E
4
16
28
40
52
64
76
88
100
112
124
Fa
F
5
17
29
41
53
65
77
89
101
113
125
F#
6
18
30
42
54
66
78
90
102
114
126
Sol
G
7
19
31
43
55
67
79
91
103
115
127
G#
8
20
32
44
56
68
80
92
104
116
La
A
9
21
33
45
57
69
81
93
105
117
A#
10
22
34
46
58
70
82
94
106
118
Si
B
11
23
35
47
59
71
83
95
107
119
Typically, very low or very high codes are silenced for different ranges in voices which are intended
to represent actual musical instruments.
For example, with the built-in Grand Piano of
Ableton Live, no sound corresponds to codes
below 21 (A0, the lowest key on the piano) or above 108 (C8, the
highest key on the piano). That means silencing most of the keys in both of
the extreme octave shiftts provided by some small keyboards (like the M-Audio
Keystation Mini 32).
The only Stradivarius violin still in as new condition is the
Messiah (1716)
which has almost never been played. It was the favorite of the
Mirecourt-born Parisian luthier
Jean-Baptiste Vuillaume (1798-1875)
who bought it in 1854 from the heirs of the violin dealer
Luigi Tarisio (1796-1854).
Vuillaume once owned 24 other Stradivari. He made an ornate tailpiece and carved pegs
to upgrade the Messiah, which is now kept at the
Ashmolean_Museum
in Oxford.
Vuillaume made about 3000 instruments, including direct copies of his revered Messiah.
Some makers have achieved a great reputation for specific violin parts:
(2020-06-18) Recorder (French: flûte à bec )
Sopranino, soprano, alto, tenor, basset, great bass, contrabass, etc.
Sometimes just called flutes (of which they are now the most common kind)
recorders have been known by that name since
1388, or earlier.
Many languages which don't use a straight adaptation of the word recorder
employ a locution which translates as block flute, refering to the
internal construction of the mouthpiece, featuring a solid wooden
fipple plug.
That's often called a cedar block, because it's almost exclusively
made from pencil cedar, the moth-repellent wood famously used for
cedar chests.
That wood is actually best called junniper
(French: genévrier) because it's not technically a cedar at all.
Juniperus virginiana,
commonly goes by many other names:
eastern redcedar, Virginian juniper, eastern juniper, red juniper, pencil-cedar, aromatic-cedar, etc.
It's used for recorder blocks because of its excellent resistance to rot in humid enviroments.
Modern recorders come in the sizes listed below from smallest to largest,
usually with baroque-english fingering (as opposed to German fingering).
The quoted tuning refers to the lowest note the instrument can produce
(always using scientific notation).
One of those is dubbed Big babe
on account of its size (about 2.8 m tall) and as a
mnemonic for its low note (Bb).
It can play more than one octave lower than the cello but
several people who play
it keep misrepresenting that as just "one semitone lower
than the lowest tone of a cello" (which is C2, not C1 !). This goes to show that
they know more about recorders than string instruments...
Big Babe belongs to the Dutch
Royal Wind Music,
a 13-player ensemble
featuring that sub-contrabass and two
sextets which are playing one octave apart.
The above is mostly the streamlined modern lineup,
which allows recorder players to play any instrument by mastering only two fingering systems
(C and F). Historically however,
there are traces of a traditional nomenclature where successive members of the recorder family
were uniformly separated by a perfect fifth (incidentally, this makes the nominal
key sufficient to identify the octave, with room to spare for an unplayable
instrument in Eb = D# at either end of the spectrum).
This scheme, which was probably never really enforced,
applies to the three deepest recorders in use. It drifts from there
but names remain almost recognizable for smaller instruments...
Unused. Would be way too small for a real chromatic instrument.
Sopranissimo in Ab = G#6 (four tones above C6 Garklein).
Sopranino in Db = C#6 (four tones above F5 ).
Soprano in Gb = F#5 (three tones above C5 ).
Alto in B4 (three tones above F4 ).
Tenor in E4 (two tones above C4 ).
One tone above D4 voice flute.
In a pinch, professional recorder players can play one semitone lower than the nominal
pitch of their instrument by closing all finger holes
(including "hole 0" for the left thumb on the underside) and partially blocking the end of the pipe "hole 8";
that's what the first line of the table below is intended to convey.
The other lines indicate how each note of the chromatic scale can be played
(there are often other solutions) to cover 2¾ octaves.
Garklein Soprano Tenor Great Bass
Baroque Fingering
Sopranino Alto Basset Contrabass
0
1
2
3
4
5
6
7
8
B
0
1
2
3
4
5
6
7
8-
E
C
0
1
2
3
4
5
6
7
F
C# / Db
0
1
2
3
4
5
6
7-
F# / Gb
D
0
1
2
3
4
5
6
G
D# / Eb
0
1
2
3
4
5
6-
G# / Ab
E
0
1
2
3
4
5
A
F
0
1
2
3
4
6
7
A# / Bb
F# / Gb
0
1
2
3
5
6
B
G
0
1
2
3
C
G# / Ab
0
1
2
4
5
6-
C# / Db
A
0
1
2
D
A# / Bb
0
1
3
5
6
D# / Eb
B
0
1
E
C
0
2
F
C# / Db
1
2
F# / Gb
D
2
G
D# / Eb
2
3
4
5
6
G# / Ab
E
0-
1
2
3
4
5
A
F
0-
1
2
3
4
6
A# / Bb
F# / Gb
0-
1
2
3
5
B
G
0-
1
2
3
C
G# / Ab
0-
1
2
4
C# / Db
A
0-
1
2
D
A# / Bb
0-
1
2
5
6 (7-)
D# / Eb
B
0-
1
2
4
5
E
C
0-
1
4
5
F
C# / Db
0-
1
3
4
6
7-
8
F# / Gb
D
0-
1
3
4
6
7-
G
D# / Eb
0-
2
3
( 5 6 7-)
G# / Ab
E
0-
2
3
4
5
6
A
F
0-
1
2
4
5
8
A# / Bb
F# / Gb
0-
1
2
4
5
B
G
0-
1
4
C
The open fingerings are defined to be the patterns where the first holes
are closed and the last ones are open.
They tend to be easier, more reliable and more stable than the other patterns,
which are called forked fingerings (where open and closed holes alternate).
In 1926, the German instrument maker
Peter Harlan (1898-1966)
introduced a modified version of the recorder,
where the fourth fingerhole is larger than the fifth, in an effort to make
the instrument faster to learn.
That modification yields an open fingering
for the first F in a C-descant (so that all simple 1-octave tunes
in C-Major can be played with open fingerings only).
That so-called German fingering is still used
in the educational market. However, only the simplest songs
are easier to play on a German instrument and the early habits so acquired
are a hindrance to the use of better recorders, which all use Baroque fingering.
Playing ALL the notes (12:02)
by Sarah Jeffery (Team Recorder, 2017-09-28).
Sounding good on a soprano (21:45)
by Sarah Jeffery (Team Recorder, 2016-04-28).
Hitting the high notes (12:36)
by Sarah Jeffery (Team Recorder, 2016-06-29).
Hand position (15:30)
by Sarah Jeffery (Team Recorder, 2016-08-09).
Trills Tutorial (15:05)
by Sarah Jeffery (Team Recorder, 2016-09-30).
Alternative Fingerings (16:50)
by Sarah Jeffery (Team Recorder, 2016-10-20).
Tuning (16:50)
by Sarah Jeffery (Team Recorder, 2017-10-05).
