In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.

The months of the year are modulo 12; 4 months after November is March, 13 months after January is February, and so on (if you want to use the modulo function, you'd need to number the months from 0 to 11, which is a little unconventional).

A keyboard with keys labelled A to G, with the D keys highlighted.

The naming of musical notes is modulo 7; if you start at the note "D" on a piano, and count up 7 white notes, you'll end up back on "D", which is the same note an octave higher. This raises the interesting fact that there are 7 different notes in an octave; it gets its name because if you count the notes at the start and end ("D" in this case, there are 8 notes).

Actually, there are really 12 different notes in an octave if you count the black keys (if you go 12 semitones up from the note "D", you'll end up at D again), so in this sense the 12 different notes on a piano are a lot like the 12 months or the 12 ticks on a 12-hour clock; but it would be really confusing if we referred to the notes as "January, February,...".

The cyclic number relationship comes up when playing notes in computer programs, which often use a digital representation of piano notes called MIDI (for example, these numbers are used in Scratch). In this system, middle C is note number 60, C# is 61, and D is 62. So an octave above 62 is 62+12 = 74. Every 12th number corresponds to the same note name.

In music, an octave or perfect octave is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The word "octave" comes from a Latin root meaning "eight". It seems an odd name for a frequency that is two times, not eight times, higher.

Actually, there are really 12 different notes in an octave if you count the black keys (if you go 12 semitones up from the note "D", you'll end up at D again), so in this sense the 12 different notes on a piano are a lot like the 12 months or the 12 ticks on a 12-hour clock; but it would be really confusing if we referred to the notes as "January, February,...".

The cyclic number relationship comes up when playing notes in computer programs, which often use a digital representation of piano notes called MIDI (for example, these numbers are used in Scratch). In this system, middle C is note number 60, C# is 61, and D is 62. So an octave above 62 is 62+12 = 74. Every 12th number corresponds to the same note name.

A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale.

Here is the visual representation of different Keys on the Keyboard.

No matter what instrument you play, or even if you write all of your music digitally, it pays to understand the foundations of music theory. When you’re writing music, if you don’t understand what notes create harmonies, and why, you’ll be fighting an uphill battle. Read more...

Recommendations

Phonascus is an open source (free) tool designed to help you learn how music works. Included are a number of skills and exercises for music students, including both theory lessons as well as those designed to help you tune your ears to what's happening in a composed piece of music. Learn to identify scales, intervals, notes, chords, and cadences, and explore included interactive books on music theory, all from one program.

LenMus aims at providing a free suite of integrated products to satisfy the needs of music students and teachers. It aims at becoming fully cross-platform and to offer, also, a good tool to write interactive texts on music (eBooks), with an embedded integrated music score editor.

Phonascus is developed on Launchpad, and is made available as open source under version 3 of the GPL. Compiled versions for both Linux and Windows are available for download.

Phonascus allows you to practice specific skills and exercises. The different activities can be customized to meet your needs. Phonascus allows you to work at your own pace, providing interactive feedback until mastery of each concept is achieved. Phonascus includes exercises on:

• Identifying and building key signatures.

• Building and spelling intervals.

• Identifying scales.

• Identifying chords.

• Identifying cadences.

• Music reading: scores composed by the program. Never repeated.

• Clefs reading

These exercises are also fully customizable, to allow you to focus on those points in which you are having difficulties or to trim the exercise to your study level.

• In 'learning' mode the program analyses your answers and schedule questions to systematically covering all the subject and to focus on those questions that are troubling you. This mode is the most systematic one and asked questions are adapted your learning needs, to minimize your study time and optimise your learning rate. Your performance data is saved and the next time you return again to the exercise, the program takes care of asking questions to ensure an optimal learning path. The result is, ideally, a reduction in the amount of time needed to study a subject and the assurance that the subject has been systematically reviewed. This mode is based on the 'spaced repetition' or 'Leitner method'.

• In 'practising' mode the program uses your performance data, saved in learning mode, to choose questions. It selects questions at random but giving more probability to those that are troubling you. Your performance data in this mode is not saved. This mode is useful when you have finished your daily assignment and you would like to practise more.

• In 'exam' mode, neither your saved performance data not your answers to previous questions are taken into account to formulate the next question. At any moment, all possible questions have the same probability of being asked. This mode is useful for testing your knowledge before taking an examination, but is less useful for learning.

• The 'quiz' mode is similar to the 'exam' mode but two answer counters are displayed and questions are accounted in both counters: one in first counter and the next one in the second counter. This mode is useful to work in pairs or in teams at classroom.