Discussing the 12 Tones

There are many different dialects when discussing music. Some are used in the context of College courses, some in the context of live performance practices, some in the context of constructing music on a computer. The terms used in the college music theory class often don't translate directly to methods used when creating music with computers. We'll discuss three different dialects of 'music speak'.

Dialect 1 - College Music Theory

• Uses 7 letters with chromatic modifiers of # and b, (sometimes x and bb) to name the 12 tones
• Context of the note determines it's name according to the rules of standard notation.
• different note/chord progression terminology:
• 1 2 3 4 5 6 7 (8)
• tonic, supertonic, mediant, subdominant, dominant, submediant, leading tone
• I,II,III,IV,V,VI,VII

Dialect 2 - standard notation with number analysis (often used in Jazz)

• Uses letters for key center, then uses 1 2 3 4 5 6 7 8 to define a major scale. chromatic modifier of # and b used as needed.
• i.e. D dorian is defined as
• Key: D
• Dorian scale formula: 1 2 b3 4 5 6 b7 8
• Even though there are no flats in D dorian, the formula is compared to D major and when lowering (b) the third (F#) and seventh (C#) the results are natural notes. So in these formulas the chromatic signs don't mean literal sharps or flats but should be interpreted as 'raised' or 'lowered' as compared to a major scale of this key.
• The names are resolved by using the key and the number combined. i.e. b3 in the key of D is the note F (because F# is the regular 3, so b3 means 'lower the F# to F')
• chord progression terminology: I,II,III,IV,V,VI,VII

Dialect 3 - MIDI pitch classes, 12 numbers mapped to 12 notes (per octave)

• This system is based on the MIDI specification and is the simplest of the dialects. Each note has a single number (no enharmonics).
• Note are categorized as a pitch class from the following equation:
• pitch_class = MIDI_note_number % 12 ('%' is the remainder operator)
• pitch_class will equal one of the following. [0 1 2 3 4 5 6 7 8 9 10 11] and they are mappped to [C C# D D# E F F# G G# A A# B] respectively.
• NOTE: The use of sharp names is arbitrary, there is no information regarding enharmonic names in this system. That is another layer of the language not fully defined in this system.

It seems like it should be very easy to talk about notes and chords in a system that uses 12 tones per octaves. However the standard language to discuss music first developed during a time when there were only 7 tones per octave used in the music of the day. (I'd love to hear a recording from that time to verify, time travel required.) Our current names of A B C D E F G were used without our current chromatic signs of sharps (#) and flats (b). The interval of a half step was between B-C and E-F only. The distance between all other adjacient notes was a whole step. Just like the white keys on a piano.

Wikipedia states that 'from the late 12th century to about 1600' a practice known as musica ficta was used where composers and/or performers where playing or singing notes that were different from what the score displayed. This was first done at final cadences of a composition. During the next centuries musica ficta became more common and it began to change the system into our 12 tones per octave system. The 12 tone system became the standard system during the time of the Baroque period. This very slow evolution meant that the old 7 letter names system needed to remain and a new method was needed to notate these new notes. At first it was a simple mark made above the note on the score. The performers knew what to do, at least the good ones. It took centuries but eventually the notation of sharps and flats became part of the system and #'s and b's were used to fill in the 5 extra notes used in between C-D, D-E, F-G, G-A, A-B.

Gosh, too bad they didn't name each of these chromatic notes with one single name. But I think they were influenced by the alphabetical pattern established by A-G and couldn't just call them [red orange yellow green blue] (or whatever) and be done with it. [Sigh]

The 12 tones per octave system in place today:

• uses a sharp (#) to raise the note a half step. The sharp symbol is placed before the note in standard notation but it follows the letter when written in text (C#).
• uses a flat (b) to lower the note a half step. The flat symbol is placed before the note in standard notation but it follows the letter when written in text (Db).
• Any letter can be sharped or flatted, which means that some white key can have a sharp or flat name in addition to it's 'natural' name.
• ALL five black keys have two names, C#/Db, D#/Eb, F#/Gb, G#/Ab, A#/Bb
• Four white keys have two names B/Cb, B#/C, Fb/E, E#/F

This results in a system of 21 names that can map to the 12 tones. Almost all of the notes will have 2 names. And as the system adheres to the alphabetical sequence when expressing harmonic and melodic minor, some double sharps (x) are required in the keys of D#, G# and A# minor. And in some rare cases double flats (bb) might even be used (i.e. Gb minor chord - spelled Gb Bbb Db). It has become more complicated than a simple 12 tones per octave issue. Just to cover the edge cases, many notation software programs allows as many as 35 different names to be mapped to the 12 tones. And choosing the correct names becomes part of the Music Theory knowledge base.

When using the MIDI number system, the structure of scales and chords are easily expressed without ambiguity. The multiple names issue becomes a separate layer. This page addresses some of this multi-layer aspect in the music language.

Comparing Names and Numbers

The following 3 groups of 4 rows show the various dialects 'translations'. Notice that in each group the only dialect that is always the same is dialect 3 (MIDI pitch class numbers). The standard notation numbers (row 2) and names (row 1) may be different for the same pitch. Notice also that the interval names (row 4) change depending on which letter name (row 1) is used for the note. It is easiest to process music structures for computers using the third dialect. The naming layer isn't always needed (sometimes the 'correct' names don't matter, only the correct sound.)

The first group use sharp names for the chromatic tones.

• Row 1 = dialect 1 - letter names
• Row 2 = dialect 2 - numbers 1-7 define major scale formula, chromatic sign with numbers define notes outside of major scale.
• Row 3 = dialect 3 - 0-11 half steps, also used as MIDI pitch classes
• Row 4 = dialect 1 & 2 - interval names from the first note to the other notes when using these letter names.

The second group uses flat name for the chromatic tones and 'weird' names (Fb, E#, Cb) for the third, fourth and seventh scale degrees. Although this is not the correct spelling for a major scale, standard notation does allow this type of notation. When unusual name choices are used (such as Cb instead of B) it can create the strange (but legal) interval of C-Cb (either an A1 or d8 depending on if the notes are a half step apart or 11 half steps apart).

• Row 1 = dialect 1 - letter names
• Row 2 = dialect 2 - numbers 1-7 define major scale formula, chromatic sign with numbers define notes outside of major scale or 'weird' spelling of major scale tones (i.e. b4, #3).
• Row 3 = dialect 3 - 0-11 half steps, also used as MIDI pitch classes
• Row 4 = dialect 1 & 2 - interval names from the first note to the other notes when using these letter names.

The third group uses additional 'weird' names. The names of B# and E# are incorrect for C major but they are legal names in the standard notation language.

• Row 1 = dialect 1 - letter names
• Row 2 = dialect 2 - numbers 1-7 define major scale formula, chromatic sign with numbers define notes outside of major scale or 'weird' spelling of major scale tones (i.e. #7, #3).
• Row 3 = dialect 3 - 0-11 half steps, also used as MIDI pitch classes
• Row 4 = dialect 1 & 2 - interval names from the first note to the other notes when using these letter names.

Notice that depending on the letter names used, an interval of size 10 half steps might be called an Augmented 6th or a Minor 7th. This is true of any interval size, depending on the names for those note, the interval name will change accordingly.

Scale formulas

The major scale is often expressed as an interval formula. In the typical College course it is common to talk about 'half steps' and 'whole steps' (or 'semitones' and 'wholetones'). Sometimes numbers are used: 1 for half step, 2 for whole step. From a starting note, one should apply the interval formula to find the other notes of the scale. The following shows the formula for an ascending major scale in two different systems. (Descending scales will use the pattern in reverse.)

• W W H W W W H - distance between adjacient tones (W = whole step, H = half step)
• 2,2,1,2,2,2,1 - distance between adjacient tones (2 = 2 half steps, 1 = 1 half step)

When using a formula with the computer, the second method (using numbers) works well but often a third method is used. Starting with 0 (representing the first note) each number represents the distance in half steps from the starting note.

• 0,2,4,5,7,9,11,12 (half steps distances from a root)

Natural Minor translated to three different dialects:

• W H W W H W W - distance between adjacient tones (W = whole step, H = half step)
• 2,1,2,2,1,2,2 - distance between adjacient tones (2 = 2 half steps, 1 = 1 half step)
• 0,2,3,5,7,8,10,12 (half steps distances from a root)

The additional layer of naming the notes is discussed on the Play Major Scales and Play Minor Scales pages.

Chord formulas

All of the following methods lead to the same result: which is a major triad.

Chord formulas are often espressed as selected scale degrees from a scale.

• The first, third and fifth notes of a major scale define a major triad structure.

Another method is using intervals from the root of the chord.

• A major triad is defined as, a starting note 'root' and a 'third' whose interval from the root is a Major third (4 half steps) and a 'fifth' whose interval from the root is a Perfect 5th (7 half steps).

Another method is a stack of intervals.

• A major triad is defined as, a starting note 'root' and a 'third' whose interval from the root is a Major third (4 half steps) and a 'fifth' whose interval from the third is a Minor 3rd (3 half steps).

A method useful for computer generated music is an array of half step values. This is the computer friendly version of the second method above.

• A major triad is defined as a MIDI root number processed with offsets of [0,4,7].
• This is NOT the same as MIDI pitch class numbers (unless you are in the key of C).
• This array of [0,4,7] represents the interval structure of the major triad. It is tranposed to any starting root by adding the MIDI number for that root to each element in the triad formula.
• The note C is from the MIDI pitch class of 0 which means that all MIDI numbers that are evenly divisible by 12 (with a remainder of 0) are letter C in the various octaves. The MIDI numbers of 0, 12, 24, 36, 48, 60, 72, etc. they're all C notes. Middle C is MIDI number 60. Using a root of MIDI number 60 the triad formula yields the MIDI numbers 60, 64, 67 which are C, E and G respectively: C major triad.
• What if the root of the chord is MIDI number 61? It would yield the MIDI numbers 61, 65, 68. What are the names of those notes? Well it depends, is 61 named C# or Db? The MIDI numbers will create the correct sounds for both C# major triad and Db major triad, but MIDI numbers don't help determine the names. That is a separate layer of logic that needs it's own computer code to resolve the naming issues according to the rules of standard notation.
• The naming layer is discussed on the Play Chords page.

Another method useful for computer generated music is an array of scale degree values. This is just like the first method implemented in a computer friendly manner.

• A major triad is defined as the 1st, 3rd and 5th [0,2,4] elements (zero indexed) of a major scale formula: [0,2,4,5,7,9,11,12]
• This process will yield the 1st, 3rd and 5th elements of the scale formula which is the interval structure of [0,4,7]. This is then used with a MIDI root number to create a specific major triad as described above.
• The naming layer is discussed on the Play Chords page.

Tone.js

Tone.js doesn't work directly with MIDI numbers but it has methods to translate MIDI numbers to frequencies or pitch/octave notation suitable for use with Tone.js. In the other pages in this series discussing Tone.js you'll see some methods that use MIDI numbers to calculate the structures, then using either the methods from Tone.js or some homebrewed code to manage the naming layer. It's useful to understand all of these dialects and be able to make any translation from one to another if needed.

Back to the Tone.js Setup page.