ARDO (which is simplified as ADO) refers to Arithmetic Rational Divisions of the Octave. it is an intervallic system considered as an arithmetic sequence with divisions of the system as terms of a sequence.

If the first division is [math]R_1[/math] (which is ratio of C/C) and the last , [math]R_n[/math] (which is ratio of 2C/C), with common difference of d

(which is 1/C), we have :

[math] R_2 = R_1 + d \\ R_3= R_1 + 2d \\ R_4 = R_1 + 3d \\ \vdots \\ R_n = R_1 + (n-1)d [/math]

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 ton 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
• The Overtone Series

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