**Arithmetic rational** **divisions of octave** or **ARDO** (which is simplified as **ADO)**** **is an intervallic system considered as

arithmetic sequence with divisions of system as terms of sequence.

** **If the first division is __ R1__ (wich is ratio of C/C) and the last ,

__(wich is ratio of 2C/C), with common difference of__

**Rn**

**d**(which is **1/C**), we have :

** ****R2 = R1+d **

** R3= R1+2d **

** R4 = R1+3d **

** ……… **

** Rn = R1+(n-1)d**** **

Each consequent divisions like **R4** and **R3** have a difference of **d** with each other.The concept of division here is a bit different from **EDO** and other systems (which is the difference of cents of two consequent degree). In **ADO**, a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12-ADO system.

For any **C-ADO** system with **cardinality** of **C**, we have ratios related to different degrees of **m** as :

(C+m/C)

For example , in **12-ADO** the ratio related to the first degree is 13/12 .

**12-ADO **can be shown as series like: **12:13****:14:15:16:17:18:19:20:21:22:23:24** or **12 13 14 15 16 17 18 19 20 21 22 23 24 **

**.**

For an **ADO** intervallic system with **n** divisions we have unequal divisions of length by dividing string length to** n** unequal divisions based on each degree ratios.If the first division has ratio of **R1** and length of **L1** and the last, **Rn** and **Ln** , we have: **Ln = 1/Rn** and if **Rn >........> R3 > R2 > R1** so :

**L1 > L2 > L3 > …… > Ln**

**This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in 12-ADO system . On the contrary , we have equal divisios of length in ****EDL system**:

** **

__Relation between harmonics and ADO system__

**ADO** (like **EDL)** is based on **Superparticular ratios** and **harmonic series**. Have a look at 12-ADO in this picture:

The above picture shows that

**ADO**system is classified as :

** **- System with unequal **epimorios**** (****Superparticular****)** divisions.

- System based on ascending series of superparticular ratios with descending sizes.

- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).

**- An spreadsheet showing relation between harmonics , superparticular ratios and ADO system**

__Relation between Otonality and ADO system__

** **

We can consider **ADO** system as **Otonal system** .**Otonality** is a term introduced by **Harry Partch** to describe chords whose notes are the overtones (multiples) of a given fixed tone.

Considering ADO , an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the **harmonic series**. nominator here is called "**Numerary nexus**".

An Otonality corresponds to an **arithmetic series** of frequencies or a **harmonic series** of wavelengths or distances on a **string instrument**.

**- Fret position calculator (excel sheet ) based on EDL system and string length**

__- How to approximate EDand ADO systems with each other?Download this file__

__Related to ADO__

**Magic of Tone and the Art of Music by the late Dane Rhudyar**