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A short history  of tunes:

(2020-07-27)  
A codification inherited from the late Bronze Age.

Lydia, Phrygia, Locris, Ionia, Aeolis, Doris, 

(2018-07-27)  
An overlooked ingredient in the evolution of music.

(2018-02-23)  
Musical notation helped crystallize the evolution of Western art music.

From AD 600 to 1500 or so  (whenneumes 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 noteto provide a sense of resolution.  That expectationis 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.

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 ).

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 fromonefourth below the authentic mode's final  to onefifth 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 threenew 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 canbe 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:

The rotation placing  Dorian = Hypomixolydian  in the middle,  rather than at bothends,  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)  
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 rightor 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 durationby 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.

Note Durations   (and the corresponding silence periods)

Duration		American	British	French
	8	Larga	Large	Maxime (silence de maxime)
	4	Quadruple note (quadruple rest)	Longa (longa rest)	Longue (silence de longue)
	2	Double note (double rest)	Breve (breve rest)	Carrée (bâton de pause)
	1	Whole note (whole rest)	Semibreve (semibreve rest)	Ronde (pause)
	1/2	Half note (half rest)	Minim (minim rest)	Blanche (demi-pause)
	1/4	Quarter note (quarter rest)	Crotchet (crotchet rest)	Noire (soupir)
	1/8	Eighth note (eighth rest)	Quaver (quaver rest)	Croche (demi-soupir)
	1/16	Sixteenth note (sixteenth rest)	Semiquaver (semiquaver rest)	Double croche (quart de soupir)
	1/32	32nd note (32nd rest)	Demisemiquaver	Triple croche
	1/64	64th note (64th rest)

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.

(2018-02-16)  
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 preciselyin 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 linksthe elements of a sequence as parts of a whole.  Slower changes in tonality areperceived 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 timeto 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)  
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.

Time signature  (French:chiffrage)

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 tocompound 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).

Pink Floyd's Money  (1973) alternates  3/4  and  4/4,  withoffbeat notes.

Compound Time :

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 dividedinto three equal divisions.  The number itself indicates how many such divisions there arein 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 intothree equal parts,  worth an eighth-note each. A famous example of  6/8  is the folk song Rising Sun Blues,  popularized as House of the Rising Sun, by The Animals  (1964).

(2018-02-16)  
Trained musicians have trouble playing offbeat Triplets at a slowtempo.

A Triplet  is usually just a group of three notes of equal durationsmeant 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 intothree 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 ofdenominator  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  incompound time).
• Heptuplets,  Septuplets, Septolets  or 

(2018-02-13)  
Eight possible performance marks : ppp   pp   p   mp   mf   f   ff   fff

(2018-02-13)  
Describing sound in terms of a superposition of frequencies.

(2018-02-12)  
The codified language of Western music has its native speakers.

About one in 10000 people have developed native familiarity with the languageof 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 instantlyname a note or a combination of notes with perfect accuracy withoutthe 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 inRomance 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).

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 above scientific pitch notation (SPN, used everywhere with growing popularity) the  ISO  number of each octave is used after the name of a pitch (following the # orb 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 alikeand we give only the notation corresponding to "C"  (Do, Ut)  in the followingtable.  Musicians routinely speak of a particular tone by identifying the "C"just below it  (e.g., 440 Hz  is "A above middle-C" orconcert A,  less often "middle A").

Competing ways of naming an octave  (C to B)  and/or the  C  tone it starts with :

Octave #	0	1	2	3	4	5	6	7	8
Scientific	C0	C1	C2	C3	C4	C5	C6	C7	C8
Yamaha	C-1	C0	C1	C2	C3	C4	C5	C6	C7
Helmholtz (1863)	C,, ,,C CCC C2 2C	C, ,C CC C1 1C	C	c	c' c1	c'' c2	c''' c3	c'''' c4	c''''' c5
LilyPond	c,,,	c,,	c,	c	c'	c''	c'''	c''''	c'''''
Alternate

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). Shecan  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 Guinnessbook 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)  
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.

415	Hz		Baroque low-pitch  (modern convention).
422	Hz		Classical pitch  (Mozart, Beethoven, ...).
423.2	Hz		Dresden Opera House  (1815).
430.54	Hz		Philosophical pitch. Sauveur  pitch (1713).
432	Hz		Promoted as "Verdi tuning" by LaRouche followers.
435	Hz		French standard  (1859-02-16)  until 1936.
440	Hz		Normal pitch (since 1936).  Stuttgart pitch.  A440.  ISO 16.
442	Hz		Symphonic pitch.
452.5	Hz		Philarmonic pitch.  British Army.  Brass-bands.
457	Hz		US military high-pitch.
466	Hz		Renaissance high-pitch  (modern convention).
506	Hz		Great organ at Halberstadt  (Nicholas Faber, 1361).

The only  (musically irrelevant)  virtue of the philosophical pitch  is to make any  "C"  awhole number of Hz, down to  C_-4 = 1 Hz,  using:

A₄   =   2 ^8.75 Hz  =   430.5389646099018460319362438314... Hz

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 justone semitone lower or higher than the A440 modern standard  (1936). This appeals to musicians who want a better historical authenticityfor periods when standard tuning was notoriously higher or lower than today. That solution is also compatible with electronic instruments which routinely offer the abilityto 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 withperfect 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 havea slightly richer harmonic content which can be perceptible  (at leastby the same kind of people, with or without perfect pitch, who can tellaStradivarius from a lesser instrument). This once drove pitch inflation.

(2018-02-23)  
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 interpretedon 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 withthe bass staff or with the treble staff, depending on which hand is meant to play them.

(2018-02-18)  
A grand-piano keyboard has 88 keys  (some rare pianos have 92 or 97).

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  C (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 availablein 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 .

8 white keys	One-dollar piano	C-C
25 keys	Smallaccordion	C-C
32 keys	Casio Rapman (1991)	F-C
32 keys	M-Audio Keystation Mini 32	C-G
37 keys		F-F
37 keys		C-C
44 keys		F-C
49 keys		C-C
54 keys		C-F
61 keys		C-C
64 keys		A-C
		73
		73
		73
		76

Both ends of the 64-key  keyboard match the layout of a grand piano. This pattern helped make all Wurlitzer 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)  
Ratios of sound frequencies.

Harmony is perceived when two tones are heard whose frequencies arein 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 asimple 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

Ratio	English	French	Approximation	Semitones
1 : 1	Unison	Unisson	1	0
16 : 15	Half-tone	Demi-ton	1.05946309436	1  (+0.11731)
10 : 9	Lesser tone		1.12246204831	2  (0.17596)
9 : 8	Whole tone			2  (+0.03910)
8 : 7	Septimal tone			2  (+0.31174)
7 : 6	Septimal Minor Third		1.18920711500	3  (0.33129)
6 : 5	Minor third	Tierce mineure		3  (0.15641)
5 : 4	Major third	Tierce majeure	1.25992104989	4  (+0.13686)
4 : 3	Fourth	Quarte	1.33483985417	5  (0.01955)
7 : 5	Lesser SeptimalTritone		1.41421356237	6  (0.17488)
10 : 7	Greater SeptimalTritone			6  (+0.17488)
3 : 2	Fifth	Quinte	1.49830707688	7  (+0.01955)
8 : 5	Minor Sixth	Sixte mineure	1.58740105197	8  (0.13686)
5 : 3	Major Sixth	Sixte majeure	1.68179283051	9  (+0.15641)
12 : 7	Septimal Major Sixth			9  (+0.33129)
7 : 4	(Septimal)Harmonic Seventh		1.78179743628	10  (0.31174)
16 : 9	Minor Seventh			10  (0.03910)
9 : 5	GreaterMinor Seventh			10  (+0.17596)
15 : 8	Major Seventh		1.88774862536	11  (0.11731)
2 : 1	Octave	Octave	2	12

Note the perfect vertical symmetry of the above table with respect to the horizontalline which separates the two kind of tritones.  The product of the ratios (from the first column) in symmetrical pairsis 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 chromaticapproximations, 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 goodenough 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 minormathematical 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 ofHuygens 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 oftentalk 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.)
• 9:4   Major Ninth.
• 7:3   Minimal Tenth.
• 3:1   (Bohlen-Pierce) Tritave.

Besides unison and octaves,  there are  11  possible intervals inequal-tempered Western music.  Some are good approximations tothe 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)
• M3:  Major third. Third of an octave.  (Approx. 5/4 ratio.)
• P4:  Fourth.  Five semitones.  (Approx. 4/3 ratio.)
• d5, A4:  Diminished fifth or augmented fourth.  Tritone.  6 semitones.
• P5:  Fifth.  Seven semitones.  (Approx. 3/2 ratio.)
• m6:  Minor sixth.  Eight semitones.  (Approx. 8/5 ratio.)
• M6:  Major sixth.  Nine semitones.  (Approx. 5/3 or 12/7 ratio.)
• m7:  Minor seventh.  Ten semitones.  (Approx. 9/5 or 16/9 ratio.)
• M7:  Major seventh.  Eleven semitones.  (Approx. 15/8 ratio.)
• P8:  Octave.  Twelve semitones.  (Exactly 2/1 ratio.)

(2020-08-20)  
Suavitatis Gradus   =   Degree of harmony  (suavity).

Arguably,  music builds on the ancient notion of commensurability: Two numbers are commensurable  when they are proportionalto 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.

It's clear at the outset that any two-variable function f  grading theconsonance 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 thatinverting all tones in a musical piece gives a totally different piece which is equallyharmonious.  A fact which was well-known to Johann Sebastian Bach (1685-1750),who made extensive use of thatsymmetry in some of his compositions. contemporary authors have dubbed related topics negative harmony.

As  g_E(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 ressemblesa logarithm:

g₀ ( x y )   =   g₀ (x)  +  g₀ (y)

By contrast g_E  seems flawed:  g_E ( x y )   =   g_E (x)  +  g_E (y)  +  1

g₀  is the difference of two standard (totally) additives functions,  bestdefined by their effects on the power of a prime  (pⁿ):

• A001414:  "Sum of prime factors"  sopfr (pⁿ)  =  n p
• A001222:  "Number of prime factors"   (pⁿ)  =  n

(2018-02-15)  
Two conflicting musical intervals: Perfect fifth  (3:2)  & octave  (2:1).

A perfect  fifth is a musical interval corresponding toa frequency ratio of exactly 3:2. Twelve of those intervals is slightly over  seven octaves:

1.5 ¹²   =   129.746337890625           vs.           2 ⁷   =   128

Therein lies the secret of Western music; this simple statement is the reason why we have  12  notesbut only use  7  of them in a given key.

(2018-02-17)  
Diabolus in Musica.  The sound of a police siren.  Half an octave.

As an octave is a factor of  2, half  an octave is the square root of two.

1.4142135623730950488016887242...

The convergents of the square root of  2 are easily obtained from its continued fraction expansion [1;2,2,2,2,2,2,2...].  They are:

1,  3/2,  7/5,  17/12,  41/29,  99/70, 239/169,  577/408,  1393/985,  ...

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 enoughto 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)  
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 frequenciesform 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 frequenciesseparated by any whole number of octaves represent the same pitch and have the same namein Western art music).

The qualifier chromatic  also apply to any subset thereof whichis not strictly contained in a diatonic scale,  as described next.

Diatonic Scale :

The Western diatonic scale is the heptatonic  scale formedby 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 are  12  major scales and  12  (natural) minor scales. Six of those  (3 major, 3 minor)  have twoenharmonic names (played alike but denoted differently).  So,  the 24 common diatonic  scales have 30 different names:

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 suffersfrom a flaw in ascending order in the sense that the last note,  calledthe 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 introducedwhich raises the subtonic one semitone, back to leading tone status.  However, this fix introduces an unusual large gap of three  semitones betweenthe 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 notesare 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.

It's usually considered undesirable to have different accidentals on the key signatureof 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 casesby using only flats in the key and raising the last note with an individual sharp whenever it occurs.

Pentatonic Scales :

One particular example of a pentatonic scale is  C-D-E-G-A. Another one is formed by the black keys of the piano,  starting with  F#.

Many examples of similar pentatonic scales exist outside of Western music.

(2018-02-26)  Modes  (modes of the major scale)
A key  is given by a tonic  and a mode (e.g.,  C-major  or  F-Lydian).

By definition, a diatonic scale consists of seven steps; two half-tones (H) separated by alternating groupsof two or three whole-tones (W).  There are seven ways to choose a starting point in sucha progression.  Each such way is called a mode. The most common ones are the aforementioned major and minor modes,  also calledIonian and Aeolian.  Here's the complete list:

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)  becauseits 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 therelevant 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 oneis the same as the descending version of the other.

Mixed Modes andPolymodal chromaticism :

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 areobtained 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)  
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:

• Thediatonic scale  (including Major and Natural Minor).
• Themelodic minor scale.
• Theharmonic minor scale.
• Theharmonic major scale  (the mirror-inverse of harmonic minor).

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 consecutivehalf-steps  (:  For every mode, there's a key whereA and G are included with the tone between them, which can be called neitherA# nor Gb without repeating a letter). There are only two other heptatonic scales for which thisdoesn't happen  (Hungarian major andRomanian major)  but they both fail in two keys, for less obvious reasons.

This fact is lost on most composers and almost all practicing musicians, withlittle or no consequences:  Transposing a piece in writing for all possible keysis 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 whatdegree(s).

Some are queasy about using degree 1 in this scheme. I beg to differ but accomodate those concerns by putting such namesinside square brackets.

• [ Mixolydian #1 ]   =   Ultralocrian.
• [ Mixolydian b1 ]   =   Mode V of Major b5.
• [ Locrian b4 ]   =   Super-Locrian.
• [ Aeolian b1 ]   =   Lydian Augmented #2.

William Zeitler (1954-)has taken the radical creative step of assigning separate greek-sounding names to all possible modes.The words he coined remain unused,  except as welcome identifiers for some obscure entries of the comprehensive electronic catalog  put together by Ian Ring.

The systematic exploration of non-diatonic scales and their modes started in 1907with the investigations of synthetic scales by Ferruccio Busoni (1866-1924)  who first considered allscales 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 whichcan be included in one of two ways.  Therefore,  70  names can be attributedto 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 themelodic minor scale). For example,  the Assyrian scale  (mode IIof 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 modificationfrom 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)  
A nice quantification of an elusive concept.

The white numbers which appear in the rightmost column of our modal chartsfor heptatonic chromatic scales (including the seven diatonic modesfrom 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 tonesbetween the root and itself.  Adding all those counts givea sum of seven integers  (always starting with 0 for the root itself) totaling between 0 and 30  (those extremes are reached for twomodes of the pathological scale where consecutives notes are separatedby either one semitone or seven).

For example, the Lydian mode of the major scale has a brightness of18 (namely 0+1+2+3+3+4+5) whereas the brigthness of the major scaleitself (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 scalesthrough 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 rootwill never raise some notes and lower others. It does one or the other, for all individual notes in all pieces of musicever written without individual accidentals.

That wonderful fact isn't true for the modes of non-diatonicscales analysed below. Yet, we may keep the above measure as a standard indicationof the otherwise elusive concept of brightness,  under the debatablesimplification that brightness only depends on the difference betweenthe number of notes raised and the number of notes lowered.

We may boldly use that number to compare thebrightness 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 isprobably a clever balance between the two. For the classification of modes within a scale, I am only retaining the linear score described abovewhich is easy to compute and meets expectationsin the only case everybody agrees on  (diatonic modes).

Barely more complicated would be the introduction ofthe variance (or its square root, thedeviation). Any mode could be represented as a point in the plane with linear brightnessas the x-coordinate and quadratic brightness as the y-coordinate. Experts should then be asked to draw an arrow fromone 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 thatsubjective brightness is an increasing functionof 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)  
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 bemodes 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 arenot 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).

The 7 modes of the melodic minor scale are known under various names:

1. Dorian #7.  Ionian b3.  (Ascending) melodic minor scale. Jazz minor. 
2. Phrygian #6.  Dorian b2.  Dorian b9.  Phrygidorian. Assyrian.
3. Lydian #5.  [ Phrygian b1.] Lydian augmented.
4. Mixolydian #4.  Lydian b7.  Lydomixian.  Lydian dominant.  Super Lydian. Acoustic scale.  Overtone scale. Bartók scale. Simpsons Scale. Prometheus heptatonic. Vachaspati  (64thmela).
5. Aeolian #3.  Mixolydian b6. Aeolian dominant. Hindu scale.  Mixaeolian.   Melodic Major. Major-Minor.
6. Locrian #2.  Aeolian b5.  Minor b5. Semilocrian.  Minor Locrian. Half diminished. Aeolocrian.  Altered diminished.  Overtone Inverse.
7. [ Ionian #1.]  Locrian b4. Altered dominant scale.  Super Locrian.

The Hindu scale  is palindromic.  The other six modescome 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 aboveby allowing a passing tone at the blue-shaded position . Only mode  I  is widely accepted:

1. Bebop Melodic Minor.
6. Half-diminished Bebop.

Harmonic Minor Scale :

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 ofinterval structures  (where H is a half-step of 1 semitoneand W is a whole-step of 2 semitones. The scientific name  for that is sesquitone, but the abbreviation "S" is unused.

The double-harmonic major scale  has two of those. The modes of the harmonic minor scale  only have one.

Dominant  means the 3 is major and the 7 is flat.

The 7 modes of the harmonic minor scale are known under these names:

1. Aeolian #7.  Minor #7.  Harmonic Minor.
2. Locrian #6. 
3. Ionian #5.  Ionian augmented.
4. Dorian #4 (or #11). Ukrainian Dorian.  Romanian Minor.  Souzinak.
5. Phrygian #3.  Phrygian Major. Phrygian dominant. Hijaz.  Jewish.  Spanish.  Dorico Flamenco.
6. Lydian #2 (or #9).
7. [ Mixolydian #1.]  Ultralocrian.

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)  
The mirror inverse of the harmonic minor scale.

This is arguably either the most exotic of the normative heptatonic Western scalesor 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).

The 7 modes of the harmonic major scale are called:

1. Ionian b6.  Major b6. Harmonic Major.
2. Dorian b5.  Blues heptatonic.  Kartzihiar. [1645]
3. Phrygian b4.
4. Lydian b3.  Lydian diminished.
5. Mixolydian b2.  Harmonic Minor Inversed.
6. [ Aeolian b1.]  Lydian Augmented #2.
7. Locrian b7.

(2020-08-13)  
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 signaturecan 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 findan 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 threeoctatonic scales).

The following articles have been transferred.

• Double-harmonic major scale.  Each of its  7  modes has one cursed  key.
• Hungarian major scale  and Romanian major scale.
• Neapolitan Major is palindromic.  Neapolitan minor is Major b2  inverted.
• The Blues leading-tone scale  is the inverse of the Persian scale.
• The Major flat-5 scale  and its interesting modes.
• Half a dozen Bebop modes  in three distinct scales.
• Enigmatic scales:  Minor, ascending, descending and  (octonic)  mixed.

(2018-04-13)  
Six notes per octave.

The single mode of the whole-tone scale (brightness 15):

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.

Adding a seventh note to the whole-tone scale always yields one of the modes ofNeapolitan Major.

The two modes of the augmented scale :

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 linesomitting 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#.)

(2018-04-19)  
Two scales:  Whole-half diminished scale and Half-whole diminished.

Octatonic symmetrical scale,  where the 8 notes are obtained by repeating four times the sametwo-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)  
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 rootfrom 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;worth6 semitones) or an augmented fifth (A5; worth 8 semitones). That yields four different qualities  of triads:

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 obtainedby a so-called  inversion  which consists in displacingone of the three notes by a full octave.

Secondary Chords :

(2020-08-17)  

(2018-02-25)  
Musical punctuation.  Dominant and tonic.

(2021-04-25)  

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)

(2018-03-20)  
A few basic principles composers have been using.

(2020-08-05)  
Composing a musical piece without human intervention.

(2020-07-23)  
Changing the sound of a tune without affecting its structure.

(2018-02-16)  
Transforming a melody by moving one or more note to a different octave.

(2018-02-12)  
The Lydian Chromatic Concept of Tonal Organization  (LCCTO).

George Russell published the first edition of LCCTO in 1953 (age 30)and spent nearly fifty years refining it. He published the final edition in 2001:

The Lydian Chromatic Concept of Tonal Organization:
The Art and Science of Tonal Gravity  (George Russell,2001)

(2020-08-08)  

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 toconsider the first 7 notes of some heptatonic mode  (e.g., Lydian)  and prolongthat 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 alldiatonic 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 tothe 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 repeatedindefinitely  (that would require 49 notes instead of 48).  The maximal perfect patterncontains 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 hasperfect 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)  
Protocol for transmitting and recording keyboard performances.

A time-stamped MIDI event correspond to depressing a certain key at a certain velocityand 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):

Typically,  very low or very high codes are silenced for different ranges in voices which are intendedto represent actual musical instruments. For example,  with the built-in Grand Piano  of Ableton Live,  no sound corresponds to codesbelow  21  (A0, the lowest key on the piano)  or above  108  (C8,  thehighest key on the piano).  That means silencing most  of the keys in both ofthe extreme octave shiftts provided by some small keyboards  (like the M-AudioKeystation Mini 32).

(2018-02-26)  
Beyond twelve tones per octave and back to the origins of music.

(2020-09-04)  

(2018-02-26)  
Guitars and others.

The standard tuning  for the open tones of a six-string guitar is:

E₂  A₂  D₃  G₃  B₃  E₄  

(2018-02-26)  
Violin, viola, violoncello (or cello)  and double-bass (or bass).

The smallest related instruments (violino piccolo &pochette) are all but forgotten.  The standard tuning of the 4 strings in modern instruments is:

• Violin  (all fifths tuning):  G3,  D4,  A4,  E5.
• Viola  (tuned in fifths):  C3,  G3,  D4,  A4.
• Violoncello  (tuned in fifths):  C2,  G2,  D3,  A3.
• Double Bass  (all fourths tuning) : E1,  A1,  D2,  G2.
  • 5-String Double Bass: B0,  E1,  A1,  D2,  G2    (most often).
  • 6-String Double Bass:  B0,  E1,  A1,  D2,  G2,   C3.

Famous violin makers of the past include:

• Nicola Amati (1596-1684).  Cremona.
• Francesco Rugieri (1628-1698).  Cremona.
• Antonio Stradivari (1644-1737).  Cremona.
• Carlo Bergonzi (1683-1747).  Milan.  Cremona.
• Guarneri del Gesù (1698-1744).  Cremona.
• Carlo Giuseppe Testore (c.1665-1738).  Milan.
• Domenico Montagnana (1686-1750).  Venice.
• Gaetano and Pietro Sgarabotto (1878-1959 & 1903-1990).

Some makers have achieved a great reputation for specific violin parts:

• Bridges : Aubert Lutherie, in Mirecourt,  France.

(2020-06-18)  
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 theinternal construction of the mouthpiece,  featuring a solid wooden fipple plug.

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).

• Garklein recorder in C₆. Sopranissimo  or piccolo recorder.
• Sopranino recorder in F₅.  Exilent.
• Soprano recorder in C₅. Descant.  Formerly "fifth flute".  Most popular instrument in basic music education. 
• Alto recorder in F₄. Treble flute.  Consort flute.  (Common) flute.
• (Voice flute in D₄.  Largest keyless  recorder.)
• Tenor recorder in C₄.  Large enough to require keys.
• Bass recorder in F₃.  Basset, ordinary small bass.
• Great bass recorder in C₃. Also called quart-bass recorder.
• Contrabass in F₂.
• Sub-greatbass recorder  in C₂. See Juho Myllylä  (2020-01-18).
• Sub-contrabass recorder in F₁. The largest model sold by Paetzold.
• Proper Sub-contrabass recorder  in Bb₀. Only three exist today. 

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 familywere uniformly separated by a perfect fifth  (incidentally,  this makes the nominalkey sufficient to identify the octave, with room to spare for an unplayableinstrument  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 therebut names remain almost recognizable for smaller instruments...

0. Unused.  Would be way  too small for a real chromatic instrument.
1. Sopranissimo in Ab = G#₆  (four tones above  C₆ Garklein).
2. Sopranino in Db = C#₆  (four tones above F₅).
3. Soprano in Gb = F#₅  (three tones above  C₅).
4. Alto in B₄  (three tones above  F₄).
5. Tenor in E₄  (two tones above  C₄). One tone above  D₄ voice flute.
6. Basset in A₃  (two tones above  F₃).
7. Greatbass in D₃  (one tone above  C₃).
8. Contrabass in G₂  (one tone above  F₂).
9. Sub-greatbass in C₂.
10. Contrabass in F₁.
11. Sub-contrabass in Bb₀.  (Modern creation by Adriana Breukink.)
12. Unused.  Would require extreme lung capacity.

In a pinch,  professional recorder players can play one semitone lower than the nominalpitch 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.

The open fingerings  are defined to be the patterns where the first holesare 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 makethe instrument faster to learn. That modification yields an open fingering for the first F in a C-descant  (so that all simple 1-octave tunesin C-Major can be played with open fingerings only). That so-called German fingering  is still usedin the educational market.  However, only the simplest songsare easier to play on a German instrument and the early habits so acquiredare a hindrance to the use of better recorders,  which all  use Baroque fingering.